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Chapter 3
Experimental Errors and
Error Analysis
This chapter is largely a tutorial on handling experimental errors of measurement. Much of the material has been extensively tested with science undergraduates at a variety of levels at the University of Toronto.
Whole books can and have been written on this topic but here we distill the topic down to the essentials. Nonetheless, our experience is that for beginners an iterative approach to this material works best. This means that the users first scan the material in this chapter; then try to use the material on their own experiment; then go over the material again; then ...
provides functions to ease the calculations required by propagation of errors, and those functions are introduced in Section 3.3. These error propagation functions are summarized in Section 3.5.
3.1 Introduction
3.1.1 The Purpose of Error Analysis
For students who only attend lectures and read textbooks in the sciences, it is easy to get the incorrect impression that the physical sciences are concerned with manipulating precise and perfect numbers. Lectures and textbooks often contain phrases like:
For an experimental scientist this specification is incomplete. Does it mean that the acceleration is closer to 9.8 than to 9.9 or 9.7? Does it mean that the acceleration is closer to 9.80000 than to 9.80001 or 9.79999? Often the answer depends on the context. If a carpenter says a length is "just 8 inches" that probably means the length is closer to 8 0/16 in. than to 8 1/16 in. or 7 15/16 in. If a machinist says a length is "just 200 millimeters" that probably means it is closer to 200.00 mm than to 200.05 mm or 199.95 mm.
We all know that the acceleration due to gravity varies from place to place on the earth's surface. It also varies with the height above the surface, and gravity meters capable of measuring the variation from the floor to a tabletop are readily available. Further, any physical measure such as can only be determined by means of an experiment, and since a perfect experimental apparatus does not exist, it is impossible even in principle to ever know perfectly. Thus, the specification of given above is useful only as a possible exercise for a student. In order to give it some meaning it must be changed to something like:
Two questions arise about the measurement. First, is it "accurate," in other words, did the experiment work properly and were all the necessary factors taken into account? The answer to this depends on the skill of the experimenter in identifying and eliminating all systematic errors. These are discussed in Section 3.4.
The second question regards the "precision" of the experiment. In this case the precision of the result is given: the experimenter claims the precision of the result is within 0.03 m/s
1. The person who did the measurement probably had some "gut feeling" for the precision and "hung" an error on the result primarily to communicate this feeling to other people. Common sense should always take precedence over mathematical manipulations.
2. In complicated experiments, error analysis can identify dominant errors and hence provide a guide as to where more effort is needed to improve an experiment.
3. There is virtually no case in the experimental physical sciences where the correct error analysis is to compare the result with a number in some book. A correct experiment is one that is performed correctly, not one that gives a result in agreement with other measurements.
4. The best precision possible for a given experiment is always limited by the apparatus. Polarization measurements in high-energy physics require tens of thousands of person-hours and cost hundreds of thousand of dollars to perform, and a good measurement is within a factor of two. Electrodynamics experiments are considerably cheaper, and often give results to 8 or more significant figures. In both cases, the experimenter must struggle with the equipment to get the most precise and accurate measurement possible.
3.1.2 Different Types of Errors
As mentioned above, there are two types of errors associated with an experimental result: the "precision" and the "accuracy". One well-known text explains the difference this way:
" " E.M. Pugh and G.H. Winslow, p. 6.
The object of a good experiment is to minimize both the errors of precision and the errors of accuracy.
Usually, a given experiment has one or the other type of error dominant, and the experimenter devotes the most effort toward reducing that one. For example, in measuring the height of a sample of geraniums to determine an average value, the random variations within the sample of plants are probably going to be much larger than any possible inaccuracy in the ruler being used. Similarly for many experiments in the biological and life sciences, the experimenter worries most about increasing the precision of his/her measurements. Of course, some experiments in the biological and life sciences are dominated by errors of accuracy.
On the other hand, in titrating a sample of HCl acid with NaOH base using a phenolphthalein indicator, the major error in the determination of the original concentration of the acid is likely to be one of the following: (1) the accuracy of the markings on the side of the burette; (2) the transition range of the phenolphthalein indicator; or (3) the skill of the experimenter in splitting the last drop of NaOH. Thus, the accuracy of the determination is likely to be much worse than the precision. This is often the case for experiments in chemistry, but certainly not all.
Question: Most experiments use theoretical formulas, and usually those formulas are approximations. Is the error of approximation one of precision or of accuracy?
3.1.3 References
There is extensive literature on the topics in this chapter. The following lists some well-known introductions.
D.C. Baird, (Prentice-Hall, 1962)
E.M. Pugh and G.H. Winslow, (Addison-Wesley, 1966)
J.R. Taylor, (University Science Books, 1982)
In addition, there is a web document written by the author of that is used to teach this topic to first year Physics undergraduates at the University of Toronto. The following Hyperlink points to that document.
3.2 Determining the Precision
3.2.1 The Standard Deviation
In the nineteenth century, Gauss' assistants were doing astronomical measurements. However, they were never able to exactly repeat their results. Finally, Gauss got angry and stormed into the lab, claiming he would show these people how to do the measurements once and for all. The only problem was that Gauss wasn't able to repeat his measurements exactly either!
After he recovered his composure, Gauss made a histogram of the results of a particular measurement and discovered the famous Gaussian or bell-shaped curve.
Many people's first introduction to this shape is the grade distribution for a course. Here is a sample of such a distribution, using the function .
We use a standard package to generate a Probability Distribution Function ( ) of such a "Gaussian" or "normal" distribution. The mean is chosen to be 78 and the standard deviation is chosen to be 10; both the mean and standard deviation are defined below.
We then normalize the distribution so the maximum value is close to the maximum number in the histogram and plot the result.
In this graph,
Finally, we look at the histogram and plot together.
We can see the functional form of the Gaussian distribution by giving symbolic values.
In this formula, the quantity , and . The is sometimes called the . The definition of is as follows.
Here is the total number of measurements and is the result of measurement number .
The standard deviation is a measure of the width of the peak, meaning that a larger value gives a wider peak.
If we look at the area under the curve from graph, we find that this area is 68 percent of the total area. Thus, any result chosen at random has a 68% change of being within one standard deviation of the mean. We can show this by evaluating the integral. For convenience, we choose the mean to be zero.
Now, we numericalize this and multiply by 100 to find the percent.
The only problem with the above is that the measurement must be repeated an infinite number of times before the standard deviation can be determined. If is less than infinity, one can only estimate measurements, this is the best estimate.
The major difference between this estimate and the definition is the . This is reasonable since if = 1 we know we can't determine
Here is an example. Suppose we are to determine the diameter of a small cylinder using a micrometer. We repeat the measurement 10 times along various points on the cylinder and get the following results, in centimeters.
The number of measurements is the length of the list.
The average or mean is now calculated.
Then the standard deviation is to be 0.00185173.
We repeat the calculation in a functional style.
Note that the package, which is standard with , includes functions to calculate all of these quantities and a great deal more.
We close with two points:
1. The standard deviation has been associated with the error in each individual measurement. Section 3.3.2 discusses how to find the error in the estimate of the average.
2. This calculation of the standard deviation is only an estimate. In fact, we can find the expected error in the estimate,
As discussed in more detail in Section 3.3, this means that the true standard deviation probably lies in the range of values.
Viewed in this way, it is clear that the last few digits in the numbers above for function adjusts these significant figures based on the error.
is discussed further in Section 3.3.1.
3.2.2 The Reading Error
There is another type of error associated with a directly measured quantity, called the "reading error". Referring again to the example of Section 3.2.1, the measurements of the diameter were performed with a micrometer. The particular micrometer used had scale divisions every 0.001 cm. However, it was possible to estimate the reading of the micrometer between the divisions, and this was done in this example. But, there is a reading error associated with this estimation. For example, the first data point is 1.6515 cm. Could it have been 1.6516 cm instead? How about 1.6519 cm? There is no fixed rule to answer the question: the person doing the measurement must guess how well he or she can read the instrument. A reasonable guess of the reading error of this micrometer might be 0.0002 cm on a good day. If the experimenter were up late the night before, the reading error might be 0.0005 cm.
An important and sometimes difficult question is whether the reading error of an instrument is "distributed randomly". Random reading errors are caused by the finite precision of the experiment. If an experimenter consistently reads the micrometer 1 cm lower than the actual value, then the reading error is not random.
For a digital instrument, the reading error is ± one-half of the last digit. Note that this assumes that the instrument has been properly engineered to round a reading correctly on the display.
3.2.3 "THE" Error
So far, we have found two different errors associated with a directly measured quantity: the standard deviation and the reading error. So, which one is the actual real error of precision in the quantity? The answer is both! However, fortunately it almost always turns out that one will be larger than the other, so the smaller of the two can be ignored.
In the diameter example being used in this section, the estimate of the standard deviation was found to be 0.00185 cm, while the reading error was only 0.0002 cm. Thus, we can use the standard deviation estimate to characterize the error in each measurement. Another way of saying the same thing is that the observed spread of values in this example is not accounted for by the reading error. If the observed spread were more or less accounted for by the reading error, it would not be necessary to estimate the standard deviation, since the reading error would be the error in each measurement.
Of course, everything in this section is related to the precision of the experiment. Discussion of the accuracy of the experiment is in Section 3.4.
3.2.4 Rejection of Measurements
Often when repeating measurements one value appears to be spurious and we would like to throw it out. Also, when taking a series of measurements, sometimes one value appears "out of line". Here we discuss some guidelines on rejection of measurements; further information appears in Chapter 7.
It is important to emphasize that the whole topic of rejection of measurements is awkward. Some scientists feel that the rejection of data is justified unless there is evidence that the data in question is incorrect. Other scientists attempt to deal with this topic by using quasi-objective rules such as 's . Still others, often incorrectly, throw out any data that appear to be incorrect. In this section, some principles and guidelines are presented; further information may be found in many references.
First, we note that it is incorrect to expect each and every measurement to overlap within errors. For example, if the error in a particular quantity is characterized by the standard deviation, we only expect 68% of the measurements from a normally distributed population to be within one standard deviation of the mean. Ninety-five percent of the measurements will be within two standard deviations, 99% within three standard deviations, etc., but we never expect 100% of the measurements to overlap within any finite-sized error for a truly Gaussian distribution.
Of course, for most experiments the assumption of a Gaussian distribution is only an approximation.
If the error in each measurement is taken to be the reading error, again we only expect most, not all, of the measurements to overlap within errors. In this case the meaning of "most", however, is vague and depends on the optimism/conservatism of the experimenter who assigned the error.
Thus, it is always dangerous to throw out a measurement. Maybe we are unlucky enough to make a valid measurement that lies ten standard deviations from the population mean. A valid measurement from the tails of the underlying distribution should not be thrown out. It is even more dangerous to throw out a suspect point indicative of an underlying physical process. Very little science would be known today if the experimenter always threw out measurements that didn't match preconceived expectations!
In general, there are two different types of experimental data taken in a laboratory and the question of rejecting measurements is handled in slightly different ways for each. The two types of data are the following:
1. A series of measurements taken with one or more variables changed for each data point. An example is the calibration of a thermocouple, in which the output voltage is measured when the thermocouple is at a number of different temperatures.
2. Repeated measurements of the same physical quantity, with all variables held as constant as experimentally possible. An example is the measurement of the height of a sample of geraniums grown under identical conditions from the same batch of seed stock.
For a series of measurements (case 1), when one of the data points is out of line the natural tendency is to throw it out. But, as already mentioned, this means you are assuming the result you are attempting to measure. As a rule of thumb, unless there is a physical explanation of why the suspect value is spurious and it is no more than three standard deviations away from the expected value, it should probably be kept. Chapter 7 deals further with this case.
For repeated measurements (case 2), the situation is a little different. Say you are measuring the time for a pendulum to undergo 20 oscillations and you repeat the measurement five times. Assume that four of these trials are within 0.1 seconds of each other, but the fifth trial differs from these by 1.4 seconds ( , more than three standard deviations away from the mean of the "good" values). There is no known reason why that one measurement differs from all the others. Nonetheless, you may be justified in throwing it out. Say that, unknown to you, just as that measurement was being taken, a gravity wave swept through your region of spacetime. However, if you are trying to measure the period of the pendulum when there are no gravity waves affecting the measurement, then throwing out that one result is reasonable. (Although trying to repeat the measurement to find the existence of gravity waves will certainly be more fun!) So whatever the reason for a suspect value, the rule of thumb is that it may be thrown out provided that fact is well documented and that the measurement is repeated a number of times more to convince the experimenter that he/she is not throwing out an important piece of data indicating a new physical process.
3.3 Propagation of Errors of Precision
3.3.1 Discussion and Examples
Usually, errors of precision are probabilistic. This means that the experimenter is saying that the actual value of some parameter is within a specified range. For example, if the half-width of the range equals one standard deviation, then the probability is about 68% that over repeated experimentation the true mean will fall within the range; if the half-width of the range is twice the standard deviation, the probability is 95%, etc.
If we have two variables, say and , and want to combine them to form a new variable, we want the error in the combination to preserve this probability.
The correct procedure to do this is to combine errors in quadrature, which is the square root of the sum of the squares. supplies a function.
For simple combinations of data with random errors, the correct procedure can be summarized in three rules. will stand for the errors of precision in , , and , respectively. We assume that and are independent of each other.
Note that all three rules assume that the error, say , is small compared to the value of .
If
z = x * y
or
then
In words, the fractional error in is the quadrature of the fractional errors in and .
If
z = x + y
or
z = x - y
then
In words, the error in is the quadrature of the errors in and .
If
then
or equivalently
includes functions to combine data using the above rules. They are named , , , , and .
Imagine we have pressure data, measured in centimeters of Hg, and volume data measured in arbitrary units. Each data point consists of { , } pairs.
We calculate the pressure times the volume.
In the above, the values of and have been multiplied and the errors have ben combined using Rule 1.
There is an equivalent form for this calculation.
Consider the first of the volume data: {11.28156820762763, 0.031}. The error means that the true value is claimed by the experimenter to probably lie between 11.25 and 11.31. Thus, all the significant figures presented to the right of 11.28 for that data point really aren't significant. The function will adjust the volume data.
Notice that by default, uses the two most significant digits in the error for adjusting the values. This can be controlled with the option.
For most cases, the default of two digits is reasonable. As discussed in Section 3.2.1, if we assume a normal distribution for the data, then the fractional error in the determination of the standard deviation , and can be written as follows.
Thus, using this as a general rule of thumb for all errors of precision, the estimate of the error is only good to 10%, ( one significant figure, unless is greater than 51) . Nonetheless, keeping two significant figures handles cases such as 0.035 vs. 0.030, where some significance may be attached to the final digit.
You should be aware that when a datum is massaged by , the extra digits are dropped.
By default, and the other functions use the function. The use of is controlled using the option.
The number of digits can be adjusted.
To form a power, say,
we might be tempted to just do
function.
Finally, imagine that for some reason we wish to form a combination.
We might be tempted to solve this with the following.
then the error is
Here is an example solving . We shall use and below to avoid overwriting the symbols and . First we calculate the total derivative.
Next we form the error.
Now we can evaluate using the pressure and volume data to get a list of errors.
Next we form the list of pairs.
The function combines these steps with default significant figure adjustment.
The function can be used in place of the other functions discussed above.
In this example, the function will be somewhat faster.
There is a caveat in using . The expression must contain only symbols, numerical constants, and arithmetic operations. Otherwise, the function will be unable to take the derivatives of the expression necessary to calculate the form of the error. The other functions have no such limitation.
3.3.1.1 Another Approach to Error Propagation: The and Datum
The reason why the output of the previous two commands has been formatted as is that typesets the pairs using ± for output.
A similar construct can be used with individual data points.
Datum[{70, 0.04}]Datum[{70, 0.04}]
Just as for , the typesetting of uses
The and constructs provide "automatic" error propagation for multiplication, division, addition, subtraction, and raising to a power. Another advantage of these constructs is that the rules built into know how to combine data with constants.
The rules also know how to propagate errors for many transcendental functions.
This rule assumes that the error is small relative to the value, so we can approximate.
or arguments, are given by .
We have seen that typesets the and constructs using ±. The function can be used directly, and provided its arguments are numeric, errors will be propagated.
One may typeset the ± into the input expression, and errors will again be propagated.
The ± input mechanism can combine terms by addition, subtraction, multiplication, division, raising to a power, addition and multiplication by a constant number, and use of the . The rules used by for ± are only for numeric arguments.
This makes different than
3.3.1.2 Why Quadrature?
Here we justify combining errors in quadrature. Although they are not proofs in the usual pristine mathematical sense, they are correct and can be made rigorous if desired.
First, you may already know about the "Random Walk" problem in which a player starts at the point = 0 and at each move steps either forward (toward + ) or backward (toward - ). The choice of direction is made randomly for each move by, say, flipping a coin. If each step covers a distance , then after steps the expected most probable distance of the player from the origin can be shown to be
Thus, the distance goes up as the square root of the number of steps.
Now consider a situation where measurements of a quantity are performed, each with an identical random error . We find the sum of the measurements.
, it is equally likely to be + as - , and which is essentially random. Thus, the expected most probable error in the sum goes up as the square root of the number of measurements.
This is exactly the result obtained by combining the errors in quadrature.
Another similar way of thinking about the errors is that in an abstract linear error space, the errors span the space. If the errors are probabilistic and uncorrelated, the errors in fact are linearly independent (orthogonal) and thus form a basis for the space. Thus, we would expect that to add these independent random errors, we would have to use Pythagoras' theorem, which is just combining them in quadrature.
3.3.2 Finding the Error in an Average
The rules for propagation of errors, discussed in Section 3.3.1, allow one to find the error in an average or mean of a number of repeated measurements. Recall that to compute the average, first the sum of all the measurements is found, and the rule for addition of quantities allows the computation of the error in the sum. Next, the sum is divided by the number of measurements, and the rule for division of quantities allows the calculation of the error in the result ( the error of the mean).
In the case that the error in each measurement has the same value, the result of applying these rules for propagation of errors can be summarized as a theorem.
Theorem: If the measurement of a random variable is repeated times, and the random variable has standard deviation , then the standard deviation in the mean is
Proof: One makes measurements, each with error .
{x1, errx}, {x2, errx}, ... , {xn, errx}
We calculate the sum.
sumx = x1 + x2 + ... + xn
We calculate the error in the sum.
This last line is the key: by repeating the measurements times, the error in the sum only goes up as [ ].
The mean
Applying the rule for division we get the following.
This completes the proof.
The quantity called
Here is an example. In Section 3.2.1, 10 measurements of the diameter of a small cylinder were discussed. The mean of the measurements was 1.6514 cm and the standard deviation was 0.00185 cm. Now we can calculate the mean and its error, adjusted for significant figures.
Note that presenting this result without significant figure adjustment makes no sense.
The above number implies that there is meaning in the one-hundred-millionth part of a centimeter.
Here is another example. Imagine you are weighing an object on a "dial balance" in which you turn a dial until the pointer balances, and then read the mass from the marking on the dial. You find = 26.10 ± 0.01 g. The 0.01 g is the reading error of the balance, and is about as good as you can read that particular piece of equipment. You remove the mass from the balance, put it back on, weigh it again, and get = 26.10 ± 0.01 g. You get a friend to try it and she gets the same result. You get another friend to weigh the mass and he also gets = 26.10 ± 0.01 g. So you have four measurements of the mass of the body, each with an identical result. Do you think the theorem applies in this case? If yes, you would quote = 26.100 ± 0.01/ [4] = 26.100 ± 0.005 g. How about if you went out on the street and started bringing strangers in to repeat the measurement, each and every one of whom got = 26.10 ± 0.01 g. So after a few weeks, you have 10,000 identical measurements. Would the error in the mass, as measured on that $50 balance, really be the following?
The point is that these rules of statistics are only a rough guide and in a situation like this example where they probably don't apply, don't be afraid to ignore them and use your "uncommon sense". In this example, presenting your result as = 26.10 ± 0.01 g is probably the reasonable thing to do.
3.4 Calibration, Accuracy, and Systematic Errors
In Section 3.1.2, we made the distinction between errors of precision and accuracy by imagining that we had performed a timing measurement with a very precise pendulum clock, but had set its length wrong, leading to an inaccurate result. Here we discuss these types of errors of accuracy. To get some insight into how such a wrong length can arise, you may wish to try comparing the scales of two rulers made by different companies — discrepancies of 3 mm across 30 cm are common!
If we have access to a ruler we trust ( a "calibration standard"), we can use it to calibrate another ruler. One reasonable way to use the calibration is that if our instrument measures and the standard records , then we can multiply all readings of our instrument by / . Since the correction is usually very small, it will practically never affect the error of precision, which is also small. Calibration standards are, almost by definition, too delicate and/or expensive to use for direct measurement.
Here is an example. We are measuring a voltage using an analog Philips multimeter, model PM2400/02. The result is 6.50 V, measured on the 10 V scale, and the reading error is decided on as 0.03 V, which is 0.5%. Repeating the measurement gives identical results. It is calculated by the experimenter that the effect of the voltmeter on the circuit being measured is less than 0.003% and hence negligible. However, the manufacturer of the instrument only claims an accuracy of 3% of full scale (10 V), which here corresponds to 0.3 V.
Now, what this claimed accuracy means is that the manufacturer of the instrument claims to control the tolerances of the components inside the box to the point where the value read on the meter will be within 3% times the scale of the actual value. Furthermore, this is not a random error; a given meter will supposedly always read too high or too low when measurements are repeated on the same scale. Thus, repeating measurements will not reduce this error.
A further problem with this accuracy is that while most good manufacturers (including Philips) tend to be quite conservative and give trustworthy specifications, there are some manufacturers who have the specifications written by the sales department instead of the engineering department. And even Philips cannot take into account that maybe the last person to use the meter dropped it.
Nonetheless, in this case it is probably reasonable to accept the manufacturer's claimed accuracy and take the measured voltage to be 6.5 ± 0.3 V. If you want or need to know the voltage better than that, there are two alternatives: use a better, more expensive voltmeter to take the measurement or calibrate the existing meter.
Using a better voltmeter, of course, gives a better result. Say you used a Fluke 8000A digital multimeter and measured the voltage to be 6.63 V. However, you're still in the same position of having to accept the manufacturer's claimed accuracy, in this case (0.1% of reading + 1 digit) = 0.02 V. To do better than this, you must use an even better voltmeter, which again requires accepting the accuracy of this even better instrument and so on, ad infinitum, until you run out of time, patience, or money.
Say we decide instead to calibrate the Philips meter using the Fluke meter as the calibration standard. Such a procedure is usually justified only if a large number of measurements were performed with the Philips meter. Why spend half an hour calibrating the Philips meter for just one measurement when you could use the Fluke meter directly?
We measure four voltages using both the Philips and the Fluke meter. For the Philips instrument we are not interested in its accuracy, which is why we are calibrating the instrument. So we will use the reading error of the Philips instrument as the error in its measurements and the accuracy of the Fluke instrument as the error in its measurements.
We form lists of the results of the measurements.
We can examine the differences between the readings either by dividing the Fluke results by the Philips or by subtracting the two values.
The second set of numbers is closer to the same value than the first set, so in this case adding a correction to the Philips measurement is perhaps more appropriate than multiplying by a correction.
We form a new data set of format { }.
We can guess, then, that for a Philips measurement of 6.50 V the appropriate correction factor is 0.11 ± 0.04 V, where the estimated error is a guess based partly on a fear that the meter's inaccuracy may not be as smooth as the four data points indicate. Thus, the corrected Philips reading can be calculated.
(You may wish to know that all the numbers in this example are real data and that when the Philips meter read 6.50 V, the Fluke meter measured the voltage to be 6.63 ± 0.02 V.)
Finally, a further subtlety: Ohm's law states that the resistance is related to the voltage and the current across the resistor according to the following equation.
V = IR
Imagine that we are trying to determine an unknown resistance using this law and are using the Philips meter to measure the voltage. Essentially the resistance is the slope of a graph of voltage versus current.
If the Philips meter is systematically measuring all voltages too big by, say, 2%, that systematic error of accuracy will have no effect on the slope and therefore will have no effect on the determination of the resistance . So in this case and for this measurement, we may be quite justified in ignoring the inaccuracy of the voltmeter entirely and using the reading error to determine the uncertainty in the determination of .
3.5 Summary of the Error Propagation Routines
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Chemometrics in analytical chemistry—part I: history, experimental design and data analysis tools
Feature Article
Published: 03 August 2017
Volume 409 , pages 5891–5899, ( 2017 )
Cite this article
Richard G. Brereton 1 ,
Jeroen Jansen 2 ,
João Lopes 3 ,
Federico Marini 4 ,
Alexey Pomerantsev 5 ,
Oxana Rodionova 5 ,
Jean Michel Roger 6 ,
Beata Walczak 7 &
Romà Tauler 8
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Chemometrics has achieved major recognition and progress in the analytical chemistry field. In the first part of this tutorial, major achievements and contributions of chemometrics to some of the more important stages of the analytical process, like experimental design, sampling, and data analysis (including data pretreatment and fusion), are summarised. The tutorial is intended to give a general updated overview of the chemometrics field to further contribute to its dissemination and promotion in analytical chemistry.
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Brereton, R.G., Jansen, J., Lopes, J. et al. Chemometrics in analytical chemistry—part I: history, experimental design and data analysis tools. Anal Bioanal Chem 409 , 5891–5899 (2017). https://doi.org/10.1007/s00216-017-0517-1
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DOI : https://doi.org/10.1007/s00216-017-0517-1
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Error is a measure of accuracy of the values in your experiment. It is important to be able to calculate experimental error, but there is more than one way to calculate and express it. Here are the most common ways to calculate experimental error:
Error Formula
In general, error is the difference between an accepted or theoretical value and an experimental value.
Error = Experimental Value - Known Value
Relative Error Formula
Relative Error = Error / Known Value
Percent Error Formula
% Error = Relative Error x 100%
Example Error Calculations
Let's say a researcher measures the mass of a sample to be 5.51 grams. The actual mass of the sample is known to be 5.80 grams. Calculate the error of the measurement.
Experimental Value = 5.51 grams Known Value = 5.80 grams
Error = Experimental Value - Known Value Error = 5.51 g - 5.80 grams Error = - 0.29 grams
Relative Error = Error / Known Value Relative Error = - 0.29 g / 5.80 grams Relative Error = - 0.050
% Error = Relative Error x 100% % Error = - 0.050 x 100% % Error = - 5.0%
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Errors in Chemical Analysis: Determinate and Indeterminate Errors
December 14, 2022
Chemical analysis
Table of Contents
Errors in chemical analysis are simply defined as the difference between a measured value and the true value. It denotes the estimated uncertainty in a measurement or experiment.
During chemical analysis, error in measurement occurs due to faulty calibration, standardization or random variation, or uncertainty in results. By frequent calibration standardization and analysis of the known sample, the error can be minimized but it is impossible to perform a chemical analysis totally free of errors. In every chemical reaction , we want to minimize errors.
Types of Errors in chemical analysis
Errors are mainly of three types in chemical analysis:
Random error (Indeterminate error)
Systematic error (Determinate error)
Gross error
1. Determinate errors
Determinate or systematic errors are those errors that have definite values and have some assignable cause. For every repeated measurement carried out in the same manner, these errors are consistently the same. Systematic errors usually introduce bias into the outcome of the measurement. The accuracy of the results is influenced by significant mistakes. These mistakes can also be identified and corrected because they are reproducible.
Bias measures the systematic error associated with analysis. It has a -ve sign if it causes results to be low and has positive sign if the results are high.
Types of determinate error (systematic error)
Personal error : During the measurement of the analytical experiments, there is often a need for personal judgment. For example: estimating the portion of the pointer between two scale divisions, the color of the solution at the endpoint, the level of the liquid of the mark in pipette or burette, etc. The main source of personal error is prejudice or bias. Human has a tendency to estimate scale reading to the precision in a set of results. We sometimes knowingly cause the result to fall closer to the true value. Number bias is also another important source of personal error. Colour blindness person can cause a personal error in volumetric analysis.
Operational error
Instrumental or reagent error : The measuring tools also have a certain amount of determinate error. Burette, pipette, and volumetric flask, for instance, always deliver slightly differently from what the scale indicates. These inconsistencies primarily result from using the glassware at a temperature that is different than the calibration since doing so damages the container wall when it is heated to dry because of contamination on the interior surface. Due to excessive use and the battery’s low voltage, electronic devices may also experience errors. The failure to accurately and often calibrate the instruments could possibly be the cause of the errors. Similar to how changing temperatures can affect numerous electronic components, errors can result from these fluctuations.
Methodic error : The non-ideal type of chemical and physical behavior of reagents and reactions on which an analysis is based can introduce methodic errors. The slowness of reaction, in the completeness of reaction, un-stability of chemical species, and possible occurrence of side reaction can cause methodic errors which may interfere with the measurement process. For example, in volumetric analysis, a small amount of excess reagent is necessary to change the color of an indicator to signify the completion of the reaction. The errors that are associated with the methods are often very difficult to detect and the most serious type of error out of all types of systematic error.
Constant or proportional error : Constant type of determinate errors are independent of the size of the sample analyzed. When there are constant mistakes, the relative error changes as the sample size is altered, but the absolute error remains constant. As the size of the quantity being measured shrinks, the effect of constant error becomes more pronounced. Meanwhile, Proportional errors decrease or increase in proportion to the size of the sample. The presence of interfering impurities in the sample is the main cause of the proportional error. Iodine is released, for instance, while estimating Cu ++ with potassium iodide. If the samples are contaminated with Fe +++ ions, I 2 is also produced from KI, and an error in the estimation of Cu ++ results. The amount of Fe +++ also doubles when the sample size is doubled, and error starts to become compulsive. Such errors are referred to as additive errors since the proportional error is sample size dependent.
2. Indeterminate errors (Random error)
Data are distributed more or less symmetrically around the mean value as a result of the indeterminate error. The precision of measurement reflects the random error. Hence, measurement precision is impacted by random or indeterminate errors.
Variable fluctuations that are unavoidable or unknown that may have an impact on the findings of experiments are what create indeterminate (or random) errors. Uncertainties in measurements can lead to random or indeterminate errors.
Errors of this kind, which are random or indeterminate, always exist in measurements. Such an error can never be completely ruled out. They are a significant factor in the determination of the analyte’s uncertainty. Most of the causes of random errors are impossible to pinpoint. Due to the low value of individual causes of such errors, they cannot be quantified even if their sources are known. However, the combined impact of all errors leads to large variability in the measurement at random.
3. Gross errors
Either too high or too low findings are the outcome of this kind of error. They are the outcome of human mistakes. Outliers, or results that appear to differ significantly from all other measured data in a set of repeated measurements, are frequently the result of gross error.
Minimization of errors
Indeterminate errors are beyond the control of the analyst, however, determinate errors can be minimized. Some of the common methods that can be employed for the minimization of errors are:
Calibration of apparatus : The instrument’s calibration is one method of reducing error. Instruments should always be regularly calibrated because they could alter as a result of water, corrosion, overuse, etc. The calibration can be done using (i) comparison with a standard, and (ii) external standard calibration.
Blank determination : A determination is performed under identical conditions by excluding the sample in order to reduce errors brought on by reagent impurities.
Independent method of analysis: This approach enables the parallel use of the technique that is being evaluated and a reliable analytical strategy. The independent approach ought to differ as little as possible from the investigative approach. This reduces the possibility that a common element in the sample will have the same effect on both approaches. By using the statistical test, we were able to determine whether the bias in the method being studied or the Undetermined errors in the two approaches was to blame for the difference in the results.
Control determination : To reduce errors, a standard material is employed in experiments under identical experimental conditions.
Dilution method : The dilution method is a vital technique for lowering interference errors. In this procedure, dilution is carried out in a way that interferent species have little to no impact below a specific concentration. This approach requires extreme caution because the sample is diluted, and the dilution may change how the sample’s analyte is measured.
Standard addition : The known amount of standard solution of analyte is added to one portion of the sample. The responses before and after the addition are measured and used to obtain the analyte concentration. The standard addition method assumes the linear relationship between response and analyte concentration.
Internal standard method : A known amount of reference species is added to all the samples, standard, and blank. The response signal is obtained as the ratio of the analyte signal to the reference species signal. It is utilized in chromatographic and spectroscopic analysis.
Parallel determination: To reduce the likelihood of unintentional errors, duplicate or triple determination is performed instead of a single determination.
Errors in chemical analysis Video
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Reviewed by Julie Vanegas, Assistant Professor, University of Texas Rio Grande Valley on 11/8/22
Throughout the book, every topic has a full explanation, which is imperative in order to provide guidance to both the student and the teacher at the same time. Furthermore, the exercises/examples are organized in such a manner that it is easy for... read more
Comprehensiveness rating: 5 see less
Throughout the book, every topic has a full explanation, which is imperative in order to provide guidance to both the student and the teacher at the same time. Furthermore, the exercises/examples are organized in such a manner that it is easy for the reader to understand and follow them.
Content Accuracy rating: 5
This is fantastic! During my Analytical Chemistry class, I am using this book to explain a topic in a variety of ways that are not conventional in explaining topics. As a result of the exercises' accuracy and approach, there is nothing left to chance, which gives the student a chance to come up with alternative solutions to the problems.
Relevance/Longevity rating: 5
There are several topics in this book that can actually be dealt with in the classroom, so the content is practical.
Clarity rating: 4
It is the way in which he explains every topic and every exercise that made me choose this book as a teacher. This is a product I will continue to recommend to students for their individual work; the emphasis is on active learning.
Consistency rating: 5
It has been shown that organization and clarity are interconnected. There is a conceptual organization in this book. This is so students are able to apply their old background knowledge to the newly introduced topics in the book, which are explained in detail in this book.
Modularity rating: 5
In order to explain something meaningful in a meaningful way, the emphasis and change of color are two factors that are incredibly meaningful since they force us to stop, think, and explain what they are explaining.
Organization/Structure/Flow rating: 5
The order in which the materials appear as explained above allows the student to connect the old knowledge with the new without having to work backwards from the beginning.
Interface rating: 5
As far as graphs, images, tables, and others are concerned, they do not contain any type of distortion or graphic visualization problems. Due to the fact that these pages offer options for engaging with books from a platform, they are a smart choice. This is because you provide us with options to work with books and respect their copyright. This is why I think that these pages provide us with options to work with books.
Grammatical Errors rating: 5
It is imperative to note that the author uses correct language and grammar. I was very surprised by the amount of synonyms and antonyms that were incorporated to make this book useful on a number of levels. Throughout the book, the author uses the scientific method to explain each topic in a rich, comprehensive, and explanatory manner.
Cultural Relevance rating: 5
The book tells us about all the cases. references so that those who wish to do more research on the subject will be able to do so by using further references. In my opinion, there is nothing in the document that is not inclusive. As far as this point is concerned, I do not have an opinion on it. The author seems to have no preference in relation to gender, race, social group, or any of the other categories.
I believe this book would be perfect if more emphasis was placed on active learning. I think that the author makes comparisons all the time, but I believe that it would be beneficial to put them in context.
Reviewed by Graham Rankin, Adjunct professor, Western Oregon University on 3/8/22
Although it is intended as a text for an analytical chemistry (wet chem!) course, it does briefly present some instrumental techniques. At over 1000 pages, I would have hoped it would cover those instrumental techniques in greater depth so that... read more
Comprehensiveness rating: 4 see less
Although it is intended as a text for an analytical chemistry (wet chem!) course, it does briefly present some instrumental techniques. At over 1000 pages, I would have hoped it would cover those instrumental techniques in greater depth so that this text could be used for both traditional (gravimetric, titrations, etc) and an instrumental chemistry course, especially given the title of the print version "Modern Analytical Chemistry"
No problems noted. This version (2.1) published in 2016 so starting to be a little dated. Version 2.2 may be coming out in the next year or so, based on time between 2.0 and 2.1 (7 years)
Relevance/Longevity rating: 3
as mentioned above I expect a new edition in the next year or so. While a good understanding of equilibrium concentrations is necessary, the continued emphasis on gravimetric and titrimetric methods when most real world applications are instrumental methods seems less relevant these days.
This appears to be just an electronic copy of a traditional print book. Some hyperlinks are given to references, but use of multimedia an animation is limited. Links to videos of actual titrations showing color change at endpoints might be helpful for students who will need to perform that exercise in lab.
Consistency rating: 4
As another reviewer noted, some instrumental techniques are introduced but not covered in detail until several chapters (and 100s or pages) later.
Modularity rating: 4
As it appears to just be an electronic copy of a traditional printed text, it is divided into chapters that are more or less covering a single technique.
Organization/Structure/Flow rating: 3
As mentioned above, some topics are touched on, but not covered in depth until much later in the text. Might be better to save those methods until later.
Interface rating: 3
Could have used more hyperlinks to web sources (videos, animations, etc.)
None noted.
Cultural Relevance rating: 3
Nothing to indicate this is just another text by a white male author.
I think I covered everything already
Reviewed by Patricia Flatt, Professor, Western Oregon University on 2/15/22
This textbook provides a great introduction to analytical chemistry for undergraduate chemistry majors. It focuses on topics such as experimental design, sampling, calibration strategies, standardization, optimization, statistics, and the... read more
This textbook provides a great introduction to analytical chemistry for undergraduate chemistry majors. It focuses on topics such as experimental design, sampling, calibration strategies, standardization, optimization, statistics, and the validation of experimental results. It is very comprehensive in the areas of analytical chemistry covered and includes standard method development and the use of common chemical instrumentation to solve problems. The chapters each include summaries with key words, and problem sets. The only thing lacking are slide decks.
Techniques and topics are covered in depth with a high level of accuracy. I could not find obvious errors in the content within chapters.
This textbook looks like it can be robust and long lasting. Many of the techniques in analytical chemistry are standard techniques used throughout the field and are highly applicable. They are also broken down into chapter sections. Should newer techniques need to be added to an existing section or require the creation of an additional chapter, this could be easily added at a later time.
The text within each chapter is well written and easy to follow. The chapters are also arranged in a logical order that follows easily for classroom discussion. Some of the textbook figures are a bit complicated or could be made in a way that would be more useable for a presentation.
The writing throughout the chapters is very consistent and follows the same format for the presentation of topics. Each chapter has content sub-sections followed by end of the chapter problems and then the chapter summary and key words.
The text for this book is divided into chapters that reflect smaller packets of analytical chemistry. Within each chapter, it is divided further into smaller subsections that break content into logical reading blocks for students.
This textbook is very logically designed with early chapters focused on basic analytical tools that provide a strong foundation for the techniques presented in later chapters. Topics include gravimetric, titrimetric, spectroscopic, electrochemical, chromatographic, and kinetic methods.
Interface rating: 4
The text appears to transition between sections easily and without a lot of delay or problems displaying figures. However, some of the figures are quite large and cannot be easily displayed in powerpoint slide format to be used in lecture.
I have not found any grammatical or technical errors within the text.
This textbook only focuses on chemical techniques, methods, and tools and does not really focus on the scientists that made discoveries. Thus, there is really no cultural materials presented at all.
Overall, I think this is a great option for use in my analytical chemistry classroom that is comprehensive and on par with commercial textbooks in this field. It will decrease student costs and increase textbook availability for my students. I'm really happy that it is available! It's one of the few OER textbooks available on this topic.
Reviewed by Jason Parsons, Associate Professor, University of Texas Rio Grande Valley on 11/19/20
The book provides a great alternative to comparable to the print textbooks I have used in the past for teaching Analytic chemistry. I have just started using Harvey this year and I highly recommend the book for the analytical chemistry. The text... read more
The book provides a great alternative to comparable to the print textbooks I have used in the past for teaching Analytic chemistry. I have just started using Harvey this year and I highly recommend the book for the analytical chemistry. The text covers all the content in an undergraduate analytical chemistry course and the necessary fundamental contention to make it an excellent book and is at an appropriate level. The book also does an excellent job introducing the spectroscopic and chromatographic topics.
Content Accuracy rating: 4
I have not noticed any errors and the content does align very well with the available print textbooks. In addition, the content is accurate and it does provide some great examples.
The topic covered is one of the most fundamental areas of chemistry which covers titrations, equilibrium, calculations among other calculations and statistics. The book will be valid for years to come. The examples are good and are reflective of analytic chemistry.
The book is written very clear and no excess jargon is used
Very consistent.
Like most textbooks in chemistry it can be taught in sequence or out of sequence without the previous chapter being necessary for the next chapter.
The content arrangement of any analytic textbook. I did not see any major issues in the organization of the content.
I have not had issues with the interface to textbook and none of my current students have complained.
Grammatical Errors rating: 4
I have not noticed any major grammatical errors in the text, it is very well written. There are some minor typos which are present in any book.
Cultural Relevance rating: 4
It is a chemistry textbook and provides great fundament information in analytical chemistry. It is a technical book and is culturally neutral
It is a great text for introduction analytical chemistry and provides great examples for the students. The book also a good foundation of knowledge for the students taking analytical chemistry.
Reviewed by Anna Cavinato, Professor, Eastern Oregon University on 1/5/20
The textbook provides an excellent online alternative to other analytical chemistry books. I have used this book and its previous edition for the last five years and I strongly recommend it. The book offers a comprehensive coverage of analytical... read more
The textbook provides an excellent online alternative to other analytical chemistry books. I have used this book and its previous edition for the last five years and I strongly recommend it. The book offers a comprehensive coverage of analytical topics with great emphasis on sampling and statistical analysis of data. It also includes coverage of spectroscopic and chromatographic topics which makes the book useful for instrumental analysis courses as well.
The book is very accurate.
The content is definitely relevant and reflects current practices in the discipline. I particularly like the use of examples of analysis type of problems derived from the scholarly literature.
The book is very clear and all terminology is adequately explained. However, I typically have to narrow down for students the coverage of a given topic as the author's discussion often goes beyond the scope of what I can cover in 10 weeks of instruction.
The book consistency is excellent.
The book is organized in chapters that can be individually downloaded as pdf files. Thus an instructor has great flexibility in choosing which parts of the book to utilize. Each chapter is further organized with headings and subheadings, making the reading very straightforward.
The author does an excellent job in organizing any topics in a logical fashion. My only word of caution is that there is a lot of information that the instructor needs to sort out ahead of time to guide students to most important outcomes.
The interface is flawless.
The book is very well written and free from grammatical errors.
The text is by no means insensitive or offensive. It is hard to be inclusive when discussing topics in analytical chemistry. There are no references to a variety of races, ethnicity or backgrounds.
This is an excellent textbook that provides a comprehensive coverage of any topic one would want to address in an analytical chemistry course. After using this textbook for several years, I am very thankful that my students can access such high quality publication at no cost!
Reviewed by Dane Scott, Assistant Professor, East Tennessee State University on 10/30/19
This textbook is excellent as it is extensive for an introductory or basic course in analytical chemistry. This text with more detail could also be a comprehensive reference text. The topics covered are designed to allow instructors to customize... read more
This textbook is excellent as it is extensive for an introductory or basic course in analytical chemistry. This text with more detail could also be a comprehensive reference text. The topics covered are designed to allow instructors to customize sections taught as covering all of this material in one semester as stated is not possible. Having said that, I chose the section on acid base equilibrium to review. I appreciate the author for covering strong acid and weak acid titrations graphically and showing how pH values are calculated. However, calculating pH for titrations involving polyprotic acids is not discussed. The titration curve is shown and only the important species before and after equivalence points are discussed. Also, pH for salts of diprotic acids is not discussed. While I feel that many areas are well explained in great detail with excellent real figures, I feel this topic is not as comprehensive as it could be. Finding the pH of H2A and NaHA systems needs to be included. It is also important to note that many key spectroscopy techniques are included. This is ideal as one text may be used for both Quantitative and Instrumental Analysis covering topics on the ACS Analytical exam in one semester. However, I do not see any sections devoted to discussing noise in instrumentation. As a specific example, I reviewed the atomic absorption section. This does discuss flame atomic absorption. However, there is no discussion on why the hollow cathode lamp is modulated to a higher frequency or a discussion on the importance of a lock in amplifier. To be comprehensive, I feel this material should be included. Many instrumental methods are employed with microscope applications using fiber optics. I feel discussion on the use of fiber optics in instrumentation will permit modernizing the traditional instrumental topics covered.
I do not see any issues with accuracy in the sections that I reviewed.
Relevance/Longevity rating: 4
The goal of this as a reference text seemed to present several topics and allow the instructor to select topics to cover. This text does that for introductory courses in Quantitative and Instrumental courses.
Clarity rating: 5
This text is very easy to read and understand. All individuals should appreciate that. The examples, practice exercises and problems are easy to recognize, follow understand meaning this is a clear text to learn from.
The text is consistent in format. It would be nice in the table of contents to see major headings consistently aligned.
As with Organization, I really think that for example, Chapter 6, should be divided into separate chapters. A lot of information and different titration methods are discussed in the same section making re-organization and realignment difficult.
Organization/Structure/Flow rating: 4
As I reviewed the chapter on titrations (6), I found the information clear and presents a lot of material in the same breath so to speak. This chapter is about 110 pages. Some topics, such as complexometric and redox titrations, need their own chapter. This will permit breaking up the material into sections that the instructor can either gloss over or not cover. While this can be done with the text in its current form, it makes the text more difficult to read from a student’s perspective following which sections may or may not be covered.
There are no interface issues with the version I looked at both on a computer and mobile device.
The grammar is well done with the exception of some verb tense issues as pointed out by other reviewers. This text is well written.
This text stays focused on scientific topics and is appropriate for any culture.
Reviewed by Alycia Palmer, Analytical Laboratory Instructor, Metropolitan State University of Denver on 7/19/19
This textbook is comprehensive for undergraduate courses in quantitative analysis and instrumental analysis. The chapter arrangement is logical and the linking feature in the PDF makes navigation to each page easy. There is no glossary to search... read more
This textbook is comprehensive for undergraduate courses in quantitative analysis and instrumental analysis. The chapter arrangement is logical and the linking feature in the PDF makes navigation to each page easy. There is no glossary to search for key terms, but this is easily done using the search feature in a PDF viewer.
The content is accurate and makes for an excellent reference book for students. Many primary articles containing actual data are used as examples and end-of-chapter problems.
I really like that the author includes how to do statistics and data analysis in Word and R. Because screen images of the software are not included, this information will not be obsolete and will be easy to update with new editions of Word.
The language of the text is very easy to read, even for non-scientists. When esoteric vocabulary is necessary, it is defined or linked to a location in the text where there is more information. There are ample graphs to explain statistics performed on sets of data. The figures of instruments and their components are not taken by a professional photographer, but they still clearly show each component and are labeled well.
Both the formatting and flow of adding new information is consistent. When new vocabulary is first presented, it is defined. When these terms are mentioned again, the text links the reader back to the original definition in case more information is needed. In the later chapters when new instruments are discussed, each is evaluated based on its precision, sensitivity, and selectivity as well as cost.
The text is easily divisible and does not contain large portions of text. Subheadings frequently introduce new themes and continue the flow of the chapter.
The content is laid out coherently. Content for a quantitative analysis course is covered in the first chapters, and content for instrumental analysis is toward the end.
Navigating a PDF is never as convenient as flipping the pages of a hard-copy book. With that said, this textbook could be easier to navigate if the table of contents included links to subsections.
There are very few grammatical errors and formatting of chemical equations and chemical formulas were all very consistent and of textbook quality.
This textbook appears to be inclusive to people of all cultural backgrounds.
Reviewed by Patrick McVey, Assistant Professor, Marian University on 3/15/19
Analytical chemistry 2.1 is exactly what it should be: a textbook for a first semester analytical chemistry course. It doesn't add extraneous details or information that would confuse the first-semester analytical student and punts these topics to... read more
Analytical chemistry 2.1 is exactly what it should be: a textbook for a first semester analytical chemistry course. It doesn't add extraneous details or information that would confuse the first-semester analytical student and punts these topics to an instrumental or advanced analytical course appropriately. For example, instrumental methods such as mass spectrometry are omitted from the text. I found this to be a good thing (even as a mass spectrometrist). The authors basically rewrote commonly used analytical texts, supplemented them with extra examples and descriptions of concepts, and made it more applicable to the introductory analytical chemist. A good example is the text's description of an interferometer, which can be a confusing topic for students. Other texts I have used have horrendous explanations of how an interferometer works, which only confuse students. This text supplies a succinct overview with all necessary information about constructive versus destructive interference and how we take advantage of that in FTIR. With that being said, a few things are overlooked. For example, the titrations chapter is inferior to other texts. The text seems to skip over weak base-strong acid titrations, and doesn't do a great job with examples for weak-with-strong titrations in general. Most other chapters seem sufficient in their comprehensiveness at an appropriate level for the intended audience.
No accuracy issues were found while reading the text.
Analytical methods are always advancing, especially when it comes to chemical instrumentation. Since the bulk of instrumentation is avoided in this text, and its focus is on foundational analytical chemistry concepts, its relevancy should hold true.
All terminology is defined appropriately, with a chapter dedicated to some analytical terms the student may be unfamiliar with. This is important, as the student is generally unfamiliar with these terms, even though many are introduced previously in courses such as general chemistry. The text doesn't assume the reader has previous knowledge of technical terminology and describes such things effectively.
The text provides chapters on a general chemistry review of topics, analytical vocabulary, and statistical methods that provide a foundation consistently used throughout the text. No issues here.
This is probably the best part of the text, and what makes it ideal for an undergraduate analytical chemistry course. The text is broken down logically with each topic comprehensive yet succinct. The chapters build within themselves effectively, and are organized into "bite-sized" pieces that are, once again, perfect for undergraduate consumption. If I could change one thing I would put the kinetic review in the actual chapter as opposed to an appendix. While yes this SHOULD be a review from general chemistry in my experience students will not access information that isn't directly put into the chapter. While I could certainly try and force them to go to back of the book, it just seems unnecessary to add a short appendix when the information could be put into the chapter and only add a couple of pages.
Overall the organization in the text makes a lot of sense, and I prefer it to other analytical chemistry texts. This is especially true within each chapter where the flow of topics build upon one another exceptionally. I like the foundation built with the first few chapters (typical in analytical chemistry texts) that then shifts to the general chemistry review/expansion and on to actual analytical chemistry methods of analysis. I will say that I don't understand why collecting/preparing samples isn't introduced earlier. It's possible the author wanted it to be right before the text transitions to methods of analysis, but I'd introduce it earlier in my class. You could make a similar argument for developing a standard method, that it should be introduced earlier in the text maybe before chapters 8-13, however the way it is written relies on knowledge from the previous chapters for student clarity and understanding. I really like the idea of the chapter, as well as the content, but wish it was written to be chapter 8 instead of 14. This will make it difficult to get to in the curriculum and may delegate it to a second semester course such as Instrumental.
Figures, images, and text were all crisp and clear. No issues navigating throughout the text. There are some minor editing issues (see chapter 4 in the table of contents) but they don't seem to be related to any display problems.
Grammatical Errors rating: 3
A score of "3" is probably a bit harsh, as the text overall is well-written, but there were a number of noticeable grammatical errors in the text. These ranged from misspelled words to subject-verb disagreements or the use of singular versus plural nouns. Most are minor and don't take away from the text, such as "...you might considering using Calc..." doesn't necessarily change the meaning of the sentence, but may impact a student's perception of the text's validity. This is especially true if it's a common theme throughout the text.
Not applicable to this type of text, but references a number of different authors from varying backgrounds.
I intend to use this textbook in my fall 2019 analytical chemistry course. I hope to submit a second review after that semester to give a more thorough account of its effectiveness, as a read-through without direct application in a course can only supply so much information in my opinion. One final thing I wish the authors would change is the front cover! My students thought the word art on the front was, in their words, hilarious. Just a thought!
Reviewed by Richard Lahti, Associate Professor and Chair, Minnesota State University Moorhead on 1/1/19
Analytical Chemistry 2.1 covers a number of important analytical chemistry topics. It does not, however, cover mass spectroscopy, IR nor NMR, nor does it claim to. In fact, it says on page 8: "Modern methods for qualitative analysis rely on... read more
Comprehensiveness rating: 3 see less
Analytical Chemistry 2.1 covers a number of important analytical chemistry topics. It does not, however, cover mass spectroscopy, IR nor NMR, nor does it claim to. In fact, it says on page 8: "Modern methods for qualitative analysis rely on instrumental techniques, such as infrared (IR) spectroscopy, nuclear magnetic resonance (NMR) spectroscopy, and mass spectrometry (MS). Because these qualitative applications are covered adequately elsewhere in the undergraduate curriculum, they receive no further consideration in this text." Thus, if students take analytical chemistry after an organic chemistry class where these topics are adequately covered, then this text would be complete and adequate. If not (or perhaps if your analytical class had a large number of transfers from a 2-year college where they did not have some of these, and you wished to have these as part of your references) you will need to select another book or supplement. In my case, MS was very important so my rating is lower - in truth, probably 3.5. If this was not necessary for you, then probably a 4 or even a 5 would be appropriate.
No meaningful errors found. Yes, it is accurate and unbiased.
Yes. It is relevant. It does a good job with the fundamental theory behind most of the analytical techniques and instrumentation, and these are timeless. It does cover the details of applications to the specific historical development of instruments and how newer instruments and techniques addressed issues with older approaches. As some areas will develop faster than others, it might be necessary to supplement with articles if a particular cutting edge approach in HP-LC was desired, for instance.
Yes, very easy to read. Students should be able to read it and learn from it. Yes, vocabulary words are defined, and there is no unexplained jargon.
Yes, there is a high degree of internal consistency. No problems noted, such as when one chapter seems to take after one approach and different chapter, another, due to differences in the way each sub-field handles a certain concept.
Maybe too much foreshadowing? For example, chapter 12A gives an overview of separations, and 12A4 gives an intro to electrophoresis. But electrophoresis is not actually handled until 60+ pages later. In my opinion, the intro in 12A4 was too long to be an intro given all of GC and LC was in between. But once you get used to it, it is not a problem.
Yes the topics are organized in an appropriate manner so that one topic builds off of the next. For example chromatography (chapter 12) follows and builds off of liquid-liquid extractions (chapter 7). Some chapters could be moved around as desired, but his order makes sense.
I viewed this as a PDF on a desktop and laptop PC. Everything displayed fine, even if I resized the window. I didn't (as I assume my students will) try to read this on my cellphone or smartwatch, and can make no guarantees about that.
Everything looked good to me. No noticeable errors, and certainly none that impacted understanding.
The text is culturally neutral. The text covers chemistry and chemical techniques. Most of the end of chapter questions deal strictly with analysis of data. Some of these are written in such a way to make it a-cultural (not "Jose wishes to monitor" but "You have been asked to monitor". Many of these questions directly reference literature. I suppose someone could argue that the literature is dominated by dead White European Males and so does not make adequate use of inclusive examples, and in that light I will give it a 4, but I prefer his approach and think it is far more appropriate.
I think the whole is greater than the sum of the parts. If I was asked to rate the book overall, I'd go a solid 4.3-4.5. The science is good, the writing is good, the graphics are good, the sample questions are good, and probably for most analytical chemistry classes, the content is appropriate because they cover the missing content (IR, MS, etc.) in a prior class. I do wish that some materials (PowerPoints, test bank, etc.) were available, but it is a solid textbook.
Reviewed by Andre Venter, Associate Professor , Western Michigan University on 12/10/18
The text includes all the relevant topics that are typically included in Introductory Analytical course work. Topics are discussed in enough detail to fill out the course without needing to add in too many additional sources. As a plus, an... read more
The text includes all the relevant topics that are typically included in Introductory Analytical course work. Topics are discussed in enough detail to fill out the course without needing to add in too many additional sources. As a plus, an extensive and useful list of Additional Resources were provided at the end of the book, arranged by chapter and topic. The text is relatively easy to navigate using either the Brief Table of Contents and the Detailed Table of Contents both of which is readily accessible from every other page using hyperlinks. While, no glossary or Index is provided, one could find easily specific terms using the Find function in Acrobat or other PDF reader.
I did not find any factual mistakes in the sections that i studied, however a few typos were present and scattered throughout the text.
For the classic part of the textbook there were few problems and modern recommendations by IUPAC and other governing bodies were implemented throughout. However, some of the technique descriptions could be updated. For example, no mention was made of UPLC, which is fast replacing HPLC as the method of choice for liquid chromatography. Such additions can easily be made though, or information can be supplemented by the instructor.
I found this book to be much more approachable and it made for a far more engaging read than our current textbook (Skoog). I liked the historical perspective and the practical introduction to each section, illustrated by using real life examples or problems.
No problems with consistency presented themselves to me.
While this textbook combines wider ranges of topics together in single chapters than I'm used to from comparable texts, it does make use of extensive subsections in each chapter. This actually serves to present the connections between techniques or applications that might otherwise become fragmented if they are locked away in separate chapters e.g. dealing in the same chapter with monoprotic, diprotic and complex formation equilibria.
The book is laid out in a fairly typical sequence, and follows my current lecture schedule very well. In fact, I would have to jump between chapters less than with our current textbook for this class.
On a computer screen the pdf works very well with Acrobat, and with a slightly bigger desktop-screen, single page view worked great. On a 13" laptop this was still possible although text then became a little strenuous to read. Navigation was easy using the Acrobat functionalities that were seamlessly integrated. I was easy to jump forward to a figure or other relevant section and then right back to where i had left off. The hyperlinks to the Table of Contents was also very useful to jump between sections with ease.
There were minor typos throughout, but not overly frequently. This, for the most part, did not detract from the book.
The book called on examples from a variety of possible application fields such as environmental, medicinal, industrial etc etc which will stimulate all students irrespective of their particular motivation to study chemistry.
Reviewed by Susan Marine, Professor (full), Miami University on 8/2/18
This book conveys many important aspects of analytical chemistry that are often glossed over in other texts (e.g., method development and validation, QA/QC, detailed statistical analysis). The author chose not to include interpretation of IR and... read more
This book conveys many important aspects of analytical chemistry that are often glossed over in other texts (e.g., method development and validation, QA/QC, detailed statistical analysis). The author chose not to include interpretation of IR and NMR because those topics are covered in detail in other courses (i.e., Organic Chemistry). Electrochemical methods are covered in greater detail than in other introductory analytical textbooks. Additional Resources are provided for all chapters, and 18 appendices provide required data and information.
The content is accurate, error-free, and unbiased. Several inconsequential typographical errors exist.
Most of the text covers time-honored content; explanations are clear and will not change with time. References to Excel and R software are vague enough to remain current; exact keystroke instructions would be quickly out of date.
The text is written clearly in understandable prose. Terms and abbreviations are defined. Examples are worked and explained in great detail, showing all mathematical steps, followed by practice exercises and detailed answers.
Several basic tools and techniques are provided in Chapter 2 and used throughout the textbook. The consistency aids understanding and facilitates problem solving. All chapters are written using the same format.
The text is divided into 15 chapters, each of which contains 8-10 major parts. Each part is further subdivided for ease of reading and finding material. The Table of Contents is detailed, very helpful, and links to the listed page. The pdf format with numbered pages helps access desired locations. The textbook contains more material than can be covered in one semester and is designed to be tailored by the instructor.
The topics are arranged in the textbook in a logical, clear fashion. The text modularity allows the instructor to change the order if desired.
Internet Explorer and Foxfire display the pdf text correctly; links work well; images are not distorted. That is not the case when Chrome was used: text ran off the page, text appeared overlapped, some links did not work, not all text appeared.
The text contains a few typographical errors but no grammatical errors.
The scientific text contains no cultural references and is not offensive in any way. There are no photographs or mention of people. Rating this topic is not applicable.
I look forward to using this textbook next semester.
Reviewed by Luke Miller, Instructor, Portland Community College on 6/19/18
Harvey’s Analytical Chemistry 2.1 is very thorough and extensive its scope of material covered. Also, the end of each chapter of the text has a list of key terms. There is not an index and/or glossary for the entire text, but it is a searchable... read more
Harvey’s Analytical Chemistry 2.1 is very thorough and extensive its scope of material covered. Also, the end of each chapter of the text has a list of key terms. There is not an index and/or glossary for the entire text, but it is a searchable PDF, which makes finding terms and topics easy.
The content is accurate, error-free, and unbiased.
The text is up to date an even includes methods for use with Excel and R. Harvey also includes his email address for readers to provide feedback and suggestions.
The text is very well written and assumes a sufficient background in general and organic chemistry. The margins of the text are filled with extra helpful information or reminders that go along with the body text.
The text is internally consistent in terms of terminology and framework.
The text has many divided sections that are each easily accessed through the hyperlinked table of contents. The exhaustive text is designed for instructors to easily pick and choose which specific topics they would like to cover.
The topics in the text are presented in a logical, clear fashion that is very similar to other analytical chemistry texts
All of the images, graphs, and tables display clearly. The text is a 1122 page pdf, which can sometimes be difficult to quickly and easily navigate. However, this problem is diminished by the hyperlinks in the table of contents and throughout the text.
The text contains no grammatical errors and is well written.
The author uses examples and references in the text from others with a variety of races, ethnicities, and backgrounds.
Harvey’s text is very in depth, thorough, and approachable. It would work very well as an undergraduate textbook. Additionally, the text has very well outlined lab procedures and spreadsheet methods that make this text an excellent companion resource for an analytical chemist in an industry position.
Table of Contents
Chapter 1: Introduction to Analytical Chemistry
Chapter 2: Basic Tools of Analytical Chemistry
Chapter 3: The Vocabulary of Analytical Chemistry
Chapter 4: Evaluating Analytical Data
Chapter 5: Standardizing Analytical Methods
Chapter 6: Equilibrium Chemistry
Chapter 7: Obtaining and Preparing Samples for Analysis
As currently taught in the United States, introductory courses in analytical chemistry emphasize quantitative (and sometimes qualitative) methods of analysis along with a heavy dose of equilibrium chemistry. Analytical chemistry, however, is much more than a collection of analytical methods and an understanding of equilibrium chemistry; it is an approach to solving chemical problems. Although equilibrium chemistry and analytical methods are important, their coverage should not come at the expense of other equally important topics.
The introductory course in analytical chemistry is the ideal place in the undergraduate chemistry curriculum for exploring topics such as experimental design, sampling, calibration strategies, standardization, optimization, statistics, and the validation of experimental results. Analytical methods come and go, but best practices for designing and validating analytical methods are universal. Because chemistry is an experimental science it is essential that all chemistry students understand the importance of making good measurements.
My goal in preparing this textbook is to find a more appropriate balance between theory and practice, between “classical” and “modern” analytical methods, between analyzing samples and collecting samples and preparing them for analysis, and between analytical methods and data analysis. There is more material here than anyone can cover in one semester; it is my hope that the diversity of topics will meet the needs of different instructors, while, perhaps, suggesting some new topics to cover.
About the Contributors
David Harvey , professor of chemistry and biochemistry at DePauw University, is the recipient of the 2016 American Chemical Society Division of Analytical Chemistry J. Calvin Giddings Award for Excellence in Education. The national award recognizes a scientist who has enhanced the professional development of analytical chemistry students, developed and published innovative experiments, designed and improved equipment or teaching labs and published influential textbooks or significant articles on teaching analytical chemistry.
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Errors in Chemical Analysis
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In science, the definition of "error" is not always the same as in normal language. An error in chemistry may still be a mistake, for example reading a scale incorrectly, but could also include normal unavoidable inaccuracies associated with measurements in an experiment in a lab.
Image Credit: Elnur/Shutterstock.com
Using this definition of error, there are many possible different sources of error in an experiment or scientific process. In fact, the result of a chemical experiment may never give a definitive answer due to the number of possible errors and the result is usually a statistical approximation of the absolute answer.
Another area where chemistry and general conversation differ is the use of accuracy and precision. In Normal conversation these two words are interchangeable but in analytical chemistry, they have different meanings:-
Accuracy is a measure of how close a measurement/measurements is to the true or accepted answer.
Precision is how closely multiple measurements agree with each other, it is actually a measure of consistency. In practice, precision is related to the standard deviation of the repeated measurements.
What is an error
“Error” in Chemistry is defined as the difference between the true result (or accepted true result) and the measured result. If the error in the analysis is large, serious consequences may result. As reliability, reproducibility, and accuracy are the basis of analytical chemistry.
Errors fall into two basic categories:-
Indeterminate (or random) errors are caused by uncontrollable or unknown fluctuations in variables that may affect experimental results. Indeterminate or accidental errors can arise from uncertainties in measurements.
Determinate errors are those errors that are known and controllable errors e.g instrument errors, personal errors, etc. Determinate or systemic errors are known and avoidable. They can be composed of two parts that have a constant value or a proportionate value.
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Accurate measurements have low Determinate Error. Precise measurements have low Indeterminate Error.
Types of error
Human Error: Inaccuracies or mistakes by a person undertaking an experiment could be the cause of some errors. The types of human error are limitless and could include things like incorrect reading of gauges, miscalculating when diluting ingredients or similar calculations, and spillage when handling chemicals during transfer or following the wrong instructions for the experiment. Depending on what type of mistake is made and where in the process it happens will have a significant impact on the magnitude of its influence on the final solution. Human error can be minimized or eliminated by careful attention to procedures and techniques.
Calibration Error: Inaccurate calibration of instruments can lead to errors in chemical experiments. Calibration is the process of adjusting or checking an instrument according to the manufacturer's instructions to ensure that the instrument gives accurate and reproducible readings. Ideally, instruments should be calibrated regularly or even every time they are used so that they do not produce errors. Some instruments or equipment will be more prone to error than others and the chemist should assess each instrument's requirement.
Estimated Measurement Error: Estimating a measurement could lead to the production of an error. Some estimations are hard to eliminate. When filling a beaker with water to a specific volume there is potential to be either marginally over the mark or marginally under the mark but the chemist will estimate when they think it is spot on the mark. In an experiment that involves a color change the moment of change will be estimated by different people at different shades of color depending on their eyesight.
M easurement Device Limitation Error: The limitations of lab equipment in a lab used to measure parameters will be a potential source of error. Every instrument or device, no matter how accurate, will have limitations on accuracy associated with it. For example, a measuring flask may be provided by the manufacturer with a built-in accuracy of plus/minus 1 to 5 percent. Using this measuring flask to make measurements in a lab, therefore, introduces an error of up to 5 percent in any volume measurement.
Interpretation of experimental results requires some mathematical knowledge and the sensitivity of instruments and processes. An understanding of significant figures is often required. For example, if you are measuring the length and your instrument can read to the nearest millimeter and a calculation gives a figure to micrometers the significant figure is millimeters as there is no way of measuring smaller distances accurately.
When interpreting multiple results the values can be reported as the mean or the median. The median is the middle value and the mean is the arithmetic average. An outlier value may cause an error where an outlier is a result that is significantly different from all the other results. This could lead to a gross error being reported. The relative error can be calculated by taking the actual value minus the measured value and dividing it by the actual value and is usually expressed as a percentage.
Multiple experimental results can also be analyzed us Gaussian standardized normal curves where one standard deviation will contain 68% of results and two standard deviations will include 95% of results.
In summary, it is very unlikely that a chemical experiment will give the absolute answer. There will always be some element of error in the result. If the error is determinate it can be minimized or eliminated but indeterminate errors will not necessarily be known. The job of the chemist is to minimize or compensate for the errors to get an answer that is accurate as possible using suitable mathematical and practical strategies.
amc technical brief Analytical Methods Committee Royal Society of Chemistry 2003 www.rsc.org/.../terminology-part-1-technical-brief-13_tcm18-214863.pdf
Dr. Deepak Analytical Procedure Errors and their Minimization lab-training.com/.../
Chemistry Libre Texts, Errors in Chemical Analysis chem.libretexts.org/.../5._Errors_in_chemical_analysis
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Last Updated: Oct 27, 2021
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Oliver is a graduate in Chemical Engineering from the University of Surrey and has 25 years of experience in industrial water treatment in the UK and abroad. He has worked extensively in steam system controls and energy management. Oliver writes on science, engineering, and the environment.
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Sources of Error in Science Experiments
Science labs usually ask you to compare your results against theoretical or known values. This helps you evaluate your results and compare them against other people’s values. The difference between your results and the expected or theoretical results is called error. The amount of error that is acceptable depends on the experiment, but a margin of error of 10% is generally considered acceptable. If there is a large margin of error, you’ll be asked to go over your procedure and identify any mistakes you may have made or places where error might have been introduced. So, you need to know the different types and sources of error and how to calculate them.
How to Calculate Absolute Error
One method of measuring error is by calculating absolute error , which is also called absolute uncertainty. This measure of accuracy is reported using the units of measurement. Absolute error is simply the difference between the measured value and either the true value or the average value of the data.
absolute error = measured value – true value
For example, if you measure gravity to be 9.6 m/s 2 and the true value is 9.8 m/s 2 , then the absolute error of the measurement is 0.2 m/s 2 . You could report the error with a sign, so the absolute error in this example could be -0.2 m/s 2 .
If you measure the length of a sample three times and get 1.1 cm, 1.5 cm, and 1.3 cm, then the absolute error is +/- 0.2 cm or you would say the length of the sample is 1.3 cm (the average) +/- 0.2 cm.
Some people consider absolute error to be a measure of how accurate your measuring instrument is. If you are using a ruler that reports length to the nearest millimeter, you might say the absolute error of any measurement taken with that ruler is to the nearest 1 mm or (if you feel confident you can see between one mark and the next) to the nearest 0.5 mm.
How to Calculate Relative Error
Relative error is based on the absolute error value. It compares how large the error is to the magnitude of the measurement. So, an error of 0.1 kg might be insignificant when weighing a person, but pretty terrible when weighing a apple. Relative error is a fraction, decimal value, or percent.
Relative Error = Absolute Error / Total Value
For example, if your speedometer says you are going 55 mph, when you’re really going 58 mph, the absolute error is 3 mph / 58 mph or 0.05, which you could multiple by 100% to give 5%. Relative error may be reported with a sign. In this case, the speedometer is off by -5% because the recorded value is lower than the true value.
Because the absolute error definition is ambiguous, most lab reports ask for percent error or percent difference.
How to Calculate Percent Error
The most common error calculation is percent error , which is used when comparing your results against a known, theoretical, or accepted value. As you probably guess from the name, percent error is expressed as a percentage. It is the absolute (no negative sign) difference between your value and the accepted value, divided by the accepted value, multiplied by 100% to give the percent:
Another common error calculation is called percent difference . It is used when you are comparing one experimental result to another. In this case, no result is necessarily better than another, so the percent difference is the absolute value (no negative sign) of the difference between the values, divided by the average of the two numbers, multiplied by 100% to give a percentage:
% difference = [experimental value – other value] / average x 100%
Sources and Types of Error
Every experimental measurement, no matter how carefully you take it, contains some amount of uncertainty or error. You are measuring against a standard, using an instrument that can never perfectly duplicate the standard, plus you’re human, so you might introduce errors based on your technique. The three main categories of errors are systematic errors, random errors , and personal errors. Here’s what these types of errors are and common examples.
Systematic Errors
Systematic error affects all the measurements you take. All of these errors will be in the same direction (greater than or less than the true value) and you can’t compensate for them by taking additional data. Examples of Systematic Errors
If you forget to calibrate a balance or you’re off a bit in the calibration, all mass measurements will be high/low by the same amount. Some instruments require periodic calibration throughout the course of an experiment , so it’s good to make a note in your lab notebook to see whether the calibrations appears to have affected the data.
Another example is measuring volume by reading a meniscus (parallax). You likely read a meniscus exactly the same way each time, but it’s never perfectly correct. Another person taking the reading may take the same reading, but view the meniscus from a different angle, thus getting a different result. Parallax can occur in other types of optical measurements, such as those taken with a microscope or telescope.
Instrument drift is a common source of error when using electronic instruments. As the instruments warm up, the measurements may change. Other common systematic errors include hysteresis or lag time, either relating to instrument response to a change in conditions or relating to fluctuations in an instrument that hasn’t reached equilibrium. Note some of these systematic errors are progressive, so data becomes better (or worse) over time, so it’s hard to compare data points taken at the beginning of an experiment with those taken at the end. This is why it’s a good idea to record data sequentially, so you can spot gradual trends if they occur. This is also why it’s good to take data starting with different specimens each time (if applicable), rather than always following the same sequence.
Not accounting for a variable that turns out to be important is usually a systematic error, although it could be a random error or a confounding variable. If you find an influencing factor, it’s worth noting in a report and may lead to further experimentation after isolating and controlling this variable.
Random Errors
Random errors are due to fluctuations in the experimental or measurement conditions. Usually these errors are small. Taking more data tends to reduce the effect of random errors. Examples of Random Errors
If your experiment requires stable conditions, but a large group of people stomp through the room during one data set, random error will be introduced. Drafts, temperature changes, light/dark differences, and electrical or magnetic noise are all examples of environmental factors that can introduce random errors.
Physical errors may also occur, since a sample is never completely homogeneous. For this reason, it’s best to test using different locations of a sample or take multiple measurements to reduce the amount of error.
Instrument resolution is also considered a type of random error because the measurement is equally likely higher or lower than the true value. An example of a resolution error is taking volume measurements with a beaker as opposed to a graduated cylinder. The beaker will have a greater amount of error than the cylinder.
Incomplete definition can be a systematic or random error, depending on the circumstances. What incomplete definition means is that it can be hard for two people to define the point at which the measurement is complete. For example, if you’re measuring length with an elastic string, you’ll need to decide with your peers when the string is tight enough without stretching it. During a titration, if you’re looking for a color change, it can be hard to tell when it actually occurs.
Personal Errors
When writing a lab report, you shouldn’t cite “human error” as a source of error. Rather, you should attempt to identify a specific mistake or problem. One common personal error is going into an experiment with a bias about whether a hypothesis will be supported or rejects. Another common personal error is lack of experience with a piece of equipment, where your measurements may become more accurate and reliable after you know what you’re doing. Another type of personal error is a simple mistake, where you might have used an incorrect quantity of a chemical, timed an experiment inconsistently, or skipped a step in a protocol.
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Published: 26 August 2022
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The chemistry of errors
Jacqueline M. Cole ORCID: orcid.org/0000-0002-1552-8743 1 , 2
The application of machine learning to big data, to make quantitative predictions about reaction outcomes, has been fraught with failure. This is because so many chemical-reaction data are not fit for purpose, but predictions would be less error-prone if synthetic chemists changed their reaction design and reporting practices.
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