Chapter 5. Errors in chemical analysis

Any measurement is limited by the precision of the measuring instruments and the technique and the skill of the observer. Where a measurement consists of a single reading on a simple piece of laboratory equipment, for example a burette or a thermometer, one would expect the number of variables contributing to uncertainties in that measurement to be fewer than a measurement which is the result of a multi-step process consisting of two or more weight measurements, a titration and the use of a variety of reagents.

It is important to be able to estimate the uncertainty in any measurement because not doing so leaves the investigator as ignorant as though there were no measurement at all. The phrase "not doing so" perpetuates the myth that somehow a person can make a measurement and not know anything about the variability of the measurement. That doesn't happen very often. A needle swings back and forth or a digital output shows a slight instability, so the investigator can estimate the uncertainty, but what if a gross error is made in judgment, leading one to estimate an unrealistic "safe" envelope of uncertainty in the measurement? Consider the anecdote offered by Richard Feynman about one of his experiences while working on the Manhattan Project during World War II. Although this example doesn't address the uncertainty of a particular measurement it touches on problems which can arise when there is complete ignorance of parameter boundaries:

Some of the special problems I had at Los Alamos were rather interesting. One thing had to do with the safety of the plant at Oak Ridge, Tennessee. Los Alamos was going to make the [atomic] bomb, but at Oak Ridge they were trying to separate the isotopes of uranium -- uranium 238 and uranium 235, the explosive one. They were just beginning to get infinitesimal amounts from an experimental thing [isotope separation] of 235, and at the same time they were practicing the chemistry. There was going to be a big plant, they were going to have vats of the stuff, and then they were going to take the purified stuff and repurify and get it ready for the next stage. (You have to purify it in several stages.) So they were practicing on the one hand, and they were just getting a little bit of U235 from one of the pieces of apparatus experimentally on the other hand. And they were trying to learn how to assay it, to determine how much uranium 235 there is in it. Though we would send them instructions, they never got it right.

So finally Emil Segrè said that the only possible way to get it right was for him to go down there and see what they were doing. The army people said, "No, it is our policy to keep all the information of Los Alamos at one place."

The people in Oak Ridge didn't know any thing about what it was to be used for (that is, they didn't have knowledge of its range of safety -- O.S.); they just knew what they were trying to do. I mean the higher people knew they were separating uranium, but they didn't know how powerful the bomb was, or exactly how it worked or anything. The people underneath didn't know at all what they were doing. And the army wanted to keep it that way. There was no information going back and forth. But Segrè insisted they'd never get the assays right, and the whole thing would go up in smoke. So he finally went down to see what they were doing, and as he was walking through he saw them wheeling a tank carboy of water, green water -- which is uranium nitrate solution.

He said, "Uh, you're going to handle it like that when it's purified too? Is that what you're going to do?"

They said, "sure -- why not?"

"Won't it explode?" he said.

Huh! Explode?

Then the army said, "You see! We shouldn't have let any information get to them! Now they are all upset."

It turned out that the army had realized how much stuff we needed to make a bomb -- twenty kilograms or whatever it was -- and they realized that this much material, purified, would never be in the plant, so there was no danger. But they did not know that the neutrons were enormously more effective when they are slowed down in water. In water it takes less than a tenth -- no, a hundredth -- as much material to make a reaction that makes radioactivity. It kills people around and so on. It was very dangerous, and they had not paid any attention to the safety at all.(1)

Feynman's example illustrates that although there were individuals who knew something about the boundary of parameters in which one could safely work with these materials in the solid state, both they and the people who were doing the work had no knowledge of the parameters or boundaries of safety for the conditions under which the work was being done (in aqueous solution). That ignorance rendered their knowledge useless. That the nuclear accident in Japan in 1999 came about because of the ignorance of this same characteristic of neutrons after having been documented in many lay accounts of the history of nuclear energy testifies to the continuing illusion over the "justification of secrecy" even in the face of imminent danger to workers, equipment and the population at large.

A second recent case of knowledge rendered useless without recognizing the presence of a fatal uncertainty involved the loss of the Mars Climate Orbiter on September 23, 1999. As reported in the L.A. Times, "the $125 million spacecraft was lost because NASA navigators mistakenly thought a contractor used metric measurements. The contractor had used English units, and the probe burned up in the Martian atmosphere Sept. 23."

Information is useless if there is no knowledge of the precision of that information. Such uselessness may be the result of

1) the presentation of a single number as a statement of the information, but lacking an estimated error.

2) gross error in calculation or data collection.

A gross error is not necessarily one in which the investigator fails to report a precision if it is known which equipment was used, say an analytical balance with a precision of ±0.0001 g or a 50 mL buret with a precision of ±0.01 mL. Data presented to a number of significant figures less than that justifiable by the equipment certainly demonstrates carelessness but doesn't, in this writer's opinion, rise to the level demonstrated by a student doing a titration using the wrong solution, NASA engineers effecting a mid-course maneuver of a deep space probe based on the wrong system of units or Oak Ridge technicians carrying out purification procedures of 235U thinking that their margin of safety is 100 times greater than it is. The latter examples illustrate the very dangerous situation of investigators not knowing what they think they know, that is, some window of confidence in their data. That's when the data become useless.

Another and shorter way of saying the same thing is that without some knowledge of the uncertainty of a measurement, the reported value could be anything. A piece of jewelry could have a weight % gold of 0% or 100%. It wouldn't matter what one were to report if there is no knowledge of the experimental uncertainty.

Once experimental uncertainty is revealed, it is a forewarning of the boundaries beyond which there may be no experimental confidence. Without that knowledge all bets are off. The engineers of the Mars Climate Orbiter didn't have any boundaries beyond which lay potential disaster. The disaster was everywhere and nowhere. They were reduced to crossing their fingers as a Plan A for saving the mission. Science ought not to work that way. With the knowledge of experimental uncertainty intelligent decisions which were impossible before can be made.

In this chapter the important concepts of precision and accuracy will be introduced. Moreover, we will be concerned with the spread or range of a series of readings, and of decisions connected with removing outliers from a data set. The median and arithmetic mean will be discussed in the context of reporting a best value from a data set exhibiting random errors. Our discussion of accuracy as regards the closeness of a reported result to some true value and how precision and accuracy may differ due to systematic errors will be discussed.

Often an analytical chemist is faced with a choice between time constraints and accuracy; hence the question of maximum error tolerance must be asked. The determination of quantities of DNA sequences in a forensic lab may rest heavily on the "quick and dirty" comparison of intensities of "blips" of unknown concentrations on an electrophoresis gel with those of known concentrations. On the other hand, the establishment of conspiracy in a murder case might rest on careful analysis of trace elements in a series of bullets fired from different guns to show if they came from the same batch.

The Shroud of Turin is a celebrated case of time being available to develop adequate means of analysis to establish the necessary values beyond reasonable doubt. There was no deadline to be met before some decision had to be made. Some people claimed that the Shroud had been used to wrap the body of Jesus, the prophet of Christianity, after his crucifixion, though no one disputed that its history was not known before the 12th century, when it had become the property of the cathedral at Turin, Italy. It was not an official Relic of the Church, but its reputation over the centuries had grown and it probably was responsible for many pilgrimages to the cathedral among the faithful. When radiocarbon dating was proposed as a way to determine its age, the proposal was at first rejected because such a sizeable amount of material would have to be used to carry out the determination (perhaps as much as 10 cm2 for each sample, and at least 3 samples must be taken to assure reproducibility). The fear was that if its age could be traced to the beginning of the first millennium, then it might well be named a Church Relic -- but one that would have to be mutilated to gain that stature. So identification of the image had to rest on non-destructive methods of analysis, such as microscopic observations of the surface. Some said the "blood stains" were composed of a common pigment used by 13th century artists, others claimed the discoloration to be human blood. Meanwhile, back at the lab, techniques continued to improve, until reliable radiocarbon dating could finally be done with considerably smaller samples (in the case of the shroud, just a few short strands were needed for each sample). Such small sample sizes were judged by Church authorities not to constitute mutilation and the analysis went forward. Samples were taken from the shroud and sent to several laboratories along with other samples of fabrics of known ages. The laboratories were not told which was which. The reported values showed close agreement between shroud samples and none suggested an age of the fabric having been harvested from plants before the 12th century A.D. The committee which had taken on the task of judging the validity of the analysis was sufficiently satisfied to convince local Church authorities to retire the claim that it was a Holy Shroud.

Definitions

The arithmetic mean, or average, is defined as











The arithmetic mean is used to report a best value among a series of N replicate measurements.

The median is the value which divides a set of replicate measurements when the set is arranged in order from the smallest to the largest. In the set of titration volumes

23.45, 23.45, 23.47, 23.49, 23.50, 23.51, 23.55,

the arithmetic mean is found by

(23.45 + 23.45 + 23.47 + 23.49 + 23.50 + 23.51 + 23.55)÷7 = 23.489

Since there is uncertainty in the measurements at the hundredths place, it would be best to report this value no further. The value 23.49 would suffice except possibly in the rare case where the set showed an average deviation or standard deviation somewhat less than ±0.01

The median of this set is 23.49. If there is an even number of readings in the set, the median is the mean of the middle pair.

Range or Scatter

Any group of readings may be expected to extend over a range or to show some scatter. Members of our class are routinely asked to measure the volume reading of water contained in a burette. The instructor establishes the "true" value in advance by positioning the upper black boundary of a burette card just under the silhouette of the meniscus. Correct use of a buret is mandatory if the student is to do well in this class. For some images and instructions on the use of a buret, go to

http://www.csudh.edu/oliver/demos/buretuse/buretuse.htm

Over the course of a semester, it is not unusual to observe four types of scatter in student readings of a burette volume. Scatter is assumed to be the result of random error, influences caused by limitations in the equipment used and the limited skill of the observer. There is also the possibility of prejudice on the part of the observer and the counterpart to prejudice on the part of the instrument used: miscalibration. This second error is referred to as systematic error. The various types of scatter one might expect to find among a group of Quantitative Analysis students reading a burette are illustrated in the chart at the right.

(1) Low accuracy and low precision. Note that not only are the readings scattered widely about the true value of 8.38 mL, but several students have misconstrued the direction of the scale, reading upward from the 9.00 mL rather than downward from the 8.00 mL mark. This kind of scatter is often observed of student readings at the beginning of the semester. Case (2) which illustrates low precision and high accuracy. Although this case shows that the mean value of all the readings is close to the true value, it could be argued that it is by virtue of luck more than anything else that the students arrived at such a good mean value. Case (3) Low accuracy and high precision. Note in the example here that although the readings have a narrower range of scatter than in (1) and (2), there seems to be a systematic error in the low direction. If a group of readings are made without the use of a burette card, the bottom of the meniscus will appear to be higher in the burette (lower volume) and thus produce such an error. Although burette readings are corrected by subtracting the beginning volume from the ending volume, and such systematic errors would tend to cancel each other out, a burette card is necessary to produce a common, repeatable background. Finally, around the 14th week of the semester, students have had enough experience reading volumetric scales to present a set of readings like that shown in Case (4), high precision and high accuracy, where one sees a narrow range of scatter on both sides of the true value.

Example 5-1. One troy ounce = 31.103486 g. Three students weigh a Krugerand on a laboratory analytical balance and get 31.1033, 31.1033 and 31.1035 g. What kind of error does this represent, random, systematic or gross?

Example 5-2. A student weighs a mL of water at 20 oC three times and gets 0.9842, 0.9846 and 0.9844 g. What kind of error does this represent, random, systematic or gross? The density of water at 20 oC is 0.99823 g/cc.

Example 5-3. A student drops a dry sample of Na2CO3 on the floor and scoops it up before titrating it with HCl. What kind of error does this represent, random, systematic or gross?

Precision and accuracy

Precision is a measure of the extent to which the values in a series of readings vary from the mean. One speaks of deviations: average deviation and standard deviation are two expressions commonly used. The term precision ought not to be used in the context of the agreement of one's average value with some "true" value. The term to be used in that case is accuracy, or the extent to which the mean of a series of readings varies from the "true" value.

The absolute error is the difference between any particular reading xi and the true value xt:

absolute error = xi - xt

Note that the formula is set up so that a low value produces a negative error and a high value produces a positive one. Sometimes one speaks of the absolute error of a mean:





It is often more useful to speak in terms of the relative error which relates the absolute error to the value of measurement:







The percent relative error would then be given by







Percent is of course "parts per hundred." It is useful to be able to make conversions between percent and parts per thousand (multiply by 10) or even to be able to determine the precision to parts in some other number when the other number may be close to some integral power of 10 (100, 1000 or 10000).

Where a number is expressed as 4.372±0.006, the value 0.006 is the "deviation", "uncertainty" or "precision" expressed in the same units as the measured value, that is, in the same context as the absolute error above, the 0.006 would represent an "absolute deviation." or "absolute precision". So, for 4.372±0.006 g one would estimate an envelope of uncertainty limited to between 4.366 g and 4.378 g but not greater. One ought to make clear if the 0.006 is an average deviation or a standard deviation. Relative deviations are then calculated in the same manner as relative errors above except that the numerator is the absolute deviation by itself. So the relative deviation or relative precision in parts per thousand of this measured value would be (0.006/4.372) x 1000 = 1.4 ppt.

Example 5-4. A mixture of magnetite and limestone is found to have 23.72±0.05 % Fe. Determine the relative precision in parts per 100 (percent) and parts per 1000.

(To be solved in class with attention paid to the importance of being able mentally to convert values of relative precision to a needed normalization)

Example 5-5. A Chemistry 230 student weighs her last chance (gasp) sample of anhydrous sodium carbonate and finds it to be 0.0842±0.0001 g. How many parts per thousand is her precision and is it good enough to standardize her HCl solution, based on the precision of the equipment we use for this experiment?

(To be solved in class with attention paid to the maximum precision we might expect to attain in this class)

Example 5-6. A Chemistry 230 student finds that he has sodium carbonate in his unknown sample to the extent of 35.3±0.4% What do you think about the relative precision of this result based on the precision of the equipment we use?

(To be solved in class, again with attention paid to the maximum precision we might expect to attain in this class)

Significant figures and round off

The use of significant figures is ubiquitous in the language of science by virtue of its convenience. Its use will be encouraged if for no other reason than that such use provides an easily conveyed message, a verbal and written shorthand actually, the alternative for which is a mite more cumbersome. Yet the use of significant figures holds one glaring fault which we must state up front. Consider the two absorbances 0.109 and 0.901 taken from a visible absorption spectrophotometer. They both convey three significant figures because the rule says that the last digit shall be the one for which there is some uncertainty in the reading, usually the interpolated digit. Yet, at the most optimum, both would have an uncertainty of ±0.001 so that the relative uncertainty of the first would be almost nine times larger than that of the second:





and




That having been said, let's take a look at some examples of typical reported values and our responsibility as scientists in offering reported values.

If you as a scientist report that a soluble sulfate unknown contains 21% sulfate, that report conveys to the recipient the understanding that the determination is in error by at least 1%, that is 21±1%. It might be off by 2% or 3%, but the last digit in the reported value is an indication of where the uncertainty lies. The fact that we DON'T KNOW what the uncertainty actually is, unless it is explicitly stated, represents a second defect of the use of significant figures. On the other hand, a student in Quantitative Analysis ought not to report 21.0% or 21.000% if the value was known only to ±1%. This would convey some serious confusion to the recipient of the report, not to mention a perceptual error on the part of the student. If the student knows the percent sulfate to a hundredth of a percent, that is, if the calculations with uncertainty taken into account yielded the value of 21.37±0.04% sulfate, the student certainly ought not to report 21% because the value is known much more precisely than 21% and the report ought to reflect that.

That's fine for the investigator making the report. How about the recipient? As stated above, the recipient is to assume that any value presented will be offered according to the same rules, that it will be reported to the first uncertain digit.

With that as a jumping off point, the fundamental rule of significant figures is to report any value to the first digit for which there is some uncertainty and that uncertainty must be reported. If it is not, then one is left rather helplessly to assume some unknown uncertainty in the last digit.

The numbers 0.237, 4.38, 8.70 and 1.47 × 1023 all have 3 significant figures.

The expressions of the number 2.67 × 10-3, 0.267 × 10-2, 0.0267 × 10-1, or 0.00267 all have 3 significant figures because, without actually saying it, the use of significant figures is a way of reducing the statement of precision to the level of relative error but without rigor. That is to say, if we assume for the sake of argument that the uncertainty in any of these values is ±1 for the digit "7" then 2.67±0.01, 0.267±0.001, 0.0267±0.0001 and 0.00267±0.00001 all exhibit the same value for the ratio of the error to the value:





By way of review and emphasis, 3 significant figures means that there is a minimum uncertainty of 1 in a number that extends from 100 to 999. So the uncertainty could represent anywhere from 0.1% (1/999 x 100) to 1% (1/100 x 100) of the value. But it gets worse. Saying, "My value is good to three significant figures" doesn't state the level of uncertainty in the last figure. In principle it could be anywhere from 1 to 9. That being the case, three significant figures could show a range of uncertainty from 0.1% (1/999 x 100) to 9% (9/100 x 100). That is the primary reason always to state your values with the added qualifier of the uncertainty itself, as 547±6. That, then, nails down the extent to which a reported value can be trusted.

There is one other rule regarding significant figures which must be mentioned here. That deals with integers ending in one or more zeros. The rule of thumb is that these numbers are precise only to the last non-zero integer. Saying "The distance from the earth to the moon is 239,000 miles can be assumed to mean that the reported distance of 239,000 miles may have an uncertainty of 1000 to 9000 miles. If stated as 239,200 miles, one would take that to mean an uncertainty between 100 and 900 miles. If a writer (for example, a newspaper journalist) is forced to use integer notation to express a large whole number, then the trailing zeros must be there to establish the magnitude of the number, not necessarily its significant figures. It is for that reason that large integers ought always to be reported in scientific notation where there is little room for doubt: 2.39 x 105 miles leaves no room for doubt that the uncertainty of this figure starts at the position of the digit "9". The best way to report a number would be in any case to include the uncertainty, for example, 2.39 ±0.02 x 105 miles. The rule of thumb means "for most practical purposes." Rules of thumb always have exceptions. First of all, there are the definitions of sizes of units. That there are 1000 mL in a liter is a definition. The relationship is exact. There is no uncertainty. Secondly, there is always the case where some experimental value in the form of a large integer will come out with trailing zeros -- a vote count, for example. If the total number of voters turning out in one precinct is determined to be 23000 simply by the luck of the draw, one would not be justified in saying it might have been 24,000 or then again perhaps 22,000. If the total was the result of three counts one could assume that it is either a valid exact number, or at the very most unreliable to ±1 or ±2.

When you participate in some of our Web exercises, make sure that you follow the rule of thumb above to determine your answers, but store the exceptions somewhere in the back of your mind.

The Distribution of Experimental Data

The model we use to explain the tendency for values in a set of data to regress toward the mean is that in any set of readings there are multiple influences which may lead to error, some within the instrument itself, some on the part of the skill of the observer. The observed regression toward the mean, or the amassing of results somewhere toward the center of extreme readings is said to be due to the partial cancellation of some error effects against others. A simplified model to describe why there is this regression effect is that of flipping a coin. The coin flip example is not exactly the same as errors which can go either way in a scientific reading, but it does lead to a result which is self-consistent with the model, and the example leads to our understanding of the origin of the normal or Gaussian Distribution.

If you flip a coin once, it can be heads or tails. What happens if you flip a coin two times and you call this double flip the reading or the event or the outcome? There are four possibilities for the outcome: HH, HT, TH and TT. There is a probability of 25% that both flips will end up as heads, 25% that the two will be tails, but 50% that one will be a head and one will be a tail. If two heads are considered to be one extreme and two tails the other, then an even combination of heads and tails falls in the middle. When graphed accordingly, one gets:



x

x x x

_____|______

HH HT TT







This gives you an opening which leads to the study of statistics and the normal distribution, because it is the influences, sometimes perceptible but often imperceptible, which lead to variations in any reading or taking of a measurement which leads to scatter around some mean -- a mean often populated more frequently than either extreme because of the greater probability that the imperceptible influences will counteract each other.

Remember that if you actually flip a coin twice, that double flip is defined as the event, and if you carry out that event 4 times, you won't be guaranteed to get all of the combinations above in the frequencies indicated. Those represent probabilities only. Here is a link to an executable file which you ought to run to see this effect.

Go to

http://proton.csudh.edu/lecture_help/startcoinflip.html

This program allows you to simulate the flip of a coin up to 100 times for a given trial and up to 1000 trials. Give it a try and vary the parameters to your liking. Use your wordprocessor to read and/or print the output from the file you define. Notice that although there is a clear regression to a 50/50 mix of heads and tails, there is random variance of the mean, back and forth.

Exercise 5-7. Answer the following questions:

1. Execute the program so that the event (called a "dataset" in the program) is defined as 100 flips of a coin and define the number of events equal to 1000.

2. Draw a histogram containing the data produced by the computer program. The x axis should be the number of heads per event and the y axis should be the number of events.

3. Determine the mean number of heads. (There is a shortcut which you must use to calculate the mean. Otherwise you'll be adding numbers of heads all night.)

4. Determine the standard deviation of the number of heads. (In this calculation there is a shortcut which you must use; it is similar in concept to the shortcut in 3, above.)

5. It is often said in books on applied statistics that the probable difference between x-bar (the mean of a small sample) and mu (the mean of a population) decreases rapidly as the number of measurements in a sample increases and that by the time the total number of observations in the sample reaches 20 to 30, this difference is negligible. There's nothing like an experimental approach to test this claim. First of all, we might ask, just what is meant by negligible? Here are 25 values for the number of heads per event independently generated by the program:

52,53,55,48,55,53,52,54,52,51,46,52,49,51,52,46,52,49,51,50,47,51,46,48,50

Determine the mean number of heads for this smaller sample and the standard deviation. Would you agree that the difference between the mean for the 1000 generated events and the smaller sample of 25 events is "negligible?" For the purpose of this exercise, "negligible" is defined as less than one standard deviation.

This exercise gives you data clearly exhibiting the beginnings of a normal curve which illustrates the scatter of an infinite number of readings over a finite range in which there is a significant frequency of events; within this range is a smaller segment which includes 68% of all events and is bounded by the location of the inflection points of the population distribution. This defined range is called the standard deviation of the population, or (sigma) and its value on either side of the mean encompasses 68% of all readings. A distance of two standard deviations or 2 sigma, encompasses 96% of all readings.

The population mean,mu , and the sample mean, x bar

Here we have three figures. The first, on the left, shows a plot of the normal distribution function with a population mean, mu , equal to 50. The figure on the right shows the same distribution function except with the abscissa in units of z=(x-mu)/sigma . By normalizing, or dividing by the value of , each unit along the abscissa is equivalent to one standard deviation of the population. The third figure, on the left shows the results of 10000 events, each event the flip of a coin 100 times. Note that the figure on the left above has been adjusted so that the standard deviation is roughly equivalent to that shown in the figure below it.

















The population standard deviation, sigma and the sample standard deviation, s.

The population standard deviation which is an accepted measure of the precision of a population of data is given as







A small sample of data has a measure of precision given by the standard deviation, s, and uses a divisor of N-1 which is called the number of degrees of freedom. It represents the number of independent data points in the calculation of the standard deviation, s. If mu (the population mean) is unknown, then x-bar, or the mean of the sample, must be calculated. It is said that the number of degrees of freedom, originally N, is diminished by one by the calculation of x-bar because with the knowledge of x-bar there are only N-1 independent data points. Given x-bar, the Nth data point could be calculated from x-bar and the other N-1 data points.







There is an alternative designation for the standard deviation, s, of a small sample, for students who do not have calculators with the s function. It allows one to calculate the standard deviation without first having to calculate x-bar:







Exercise 5-x1. Prove that the previous two formulas for the standard deviation s are equivalent.

Exercise 5-8. For the whole group. Each student is to flip a coin ten times. Fill in the table. The frequency is the number of students who get that result.

No. Of heads 0 1 2 3 4 5 6 7 8 9 10
Frequency


Exercise 5-x2. The equation for the Gaussian distribution (the normal bell curve) can take the form







Prove that the inflection point of this function occurs when x=µ±sigma.

Standard error of a mean

The standard error of a mean is given the symbol sigmam in books on statistics and is related to the "scatter" of the means of small samples drawn from a larger collection. For the example below we shall use sm as a substitute for that symbol. The means of small groups taken from a large collection will show a characteristic standard deviation. That standard deviation of the means will diminish as the sample size increases in proportion to the function






that is, if the standard deviation sm of the group means is plotted vs 1/(square root N), the values ought to decrease linearly and approach zero as 1/(square root N) approaches zero. Let's see if there's any truth to the claim of linearity.



Exercise 5-9. Here is a table showing what happens if the file coinout.10k with 10000 events, each event being the flipping of a coin 100 times is read in a manner to calculate means and standard deviations of small groups of those events. Since the outcome of the events was the result of randomness and the file itself was not sorted, the groups are chosen in sequence. The following size of the groups was chosen: 3,4,5,10,50,100,250,500,1000,2500,5000,10000. For groups of size 3 there are 3333 groups, for groups of size 4 there are 2500, for 5, 2000 and so on. The mean for all groups of the same size (the mean of the means), and the standard deviation produced by the individual means within each collection of groups were calculated.

For the class. On the graph paper below, label two convenient scales, putting standard deviation along the vertical axis and 1/(square root N) along the horizontal axis. Then plot the standard deviation of group means from the mean for all groups against the values of 1/(square root N). How does this agree with statistical theory?

Group size

(N)

SQRT(N) 1/SQRT(N) Mean for all groups Std. deviation of group means.
1 1 1 50.039 4.99
2 1.414 0.707 50.039 3.53
3 1.732 0.577 50.039 2.93
4 2 0.500 50.039 2.49
5 2.236 0.447 50.039 2.25
10 3.162 0.316 50.039 1.58
50 7.071 0.141 50.039 0.68
100 10 0.10 50.039 0.48
250 15.81 0.063 50.039 0.31
500 22.36 0.045 50.039 0.22
1000 31.62 0.032 50.039 0.17
2500 50 0.020 50.039 0.13
5000 70.71 0.014 50.039 0.11
10000 100 0.010 50.039 ----




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+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +

The results of several experimental determinations and the subsequent calculation of the standard deviation says nothing about the true value. It is a measure solely of the reliability of the method being used. There is nevertheless what appears to be an irresistible tendency following some determination in analytical chemistry to link the standard deviation between individual determinations with their "trueness," but that tendency must be resisted. The standard deviation between individual determinations says nothing about the possibility of systematic error in the determination. It is the reliability of the method which is reflected by s.

Standard deviation from pooled data

Rather than simply relying on some average standard deviation for several small samples of data, statistical theory tells us that we can go one even better. We can determine a standard deviation from pooled data even though those data may represent determinations on different unknowns. Consider the samples of carbonate analyzed by CHE230 students during the fall semester, 1999. Four unknowns were used, so there were four different % sodium carbonate values to be determined. But the same method was used for each. The level of precision would be expected to be the same for each unknown, even though their percent sodium carbonate values were widely separated. So the mean percent sodium carbonate is of no great concern, but the precision, in principle, ought to remain the same. The formula which allows us to determine a more characteristic standard deviation of the method, from pooled data, is







Nalpha is the number of elements in group alpha, Nbeta is the number of elements in group beta, etc. and Ng is the number of groups that are pooled.

Exercise 5-10. During the fall of 1999, the following results were obtained from students carrying out the determination of sodium carbonate in samples of soda ash:

Student Sample 1 Sample 2 Sample 3 x-bar s SUM(xi - xbar )2
1(alpha) 39.46 39.50 39.32 39.43 0.09 0.0179
2(beta) 29.60 29.60 29.49 29.56 0.06 0.0081
3(gamma) 22.09 21.74 21.98 21.94 0.18 0.0641
4(delta) 40.15 40.27 40.22 40.21 0.06 0.0073
5(epsilon) 20.88 20.98 20.81 20.89 0.09 0.0146
6(zeta) 38.91 38.94 38.90 38.92 0.02 0.0009
7(eta) 32.38 31.84 32.88 32.37 0.52 0.5411
8(theta) 38.19 38.18 38.18 38.18 0.01 0.0001
9(iota) 48.88 48.83 48.27 48.66 0.34 0.2294
10(kappa) 28.74 28.77 28.76 28.76 0.02 0.0005
11(lambda) 49.90 49.81 49.79 49.83 0.06 0.0069
12(µ) 50.42 50.38 50.45 50.42 0.04 0.0025


Using the equation above, determine the pooled standard deviation. Note that the mean for each set is used only to determine the square of the sum of the deviations of each result and that the overall mean is of no importance where a pooled standard deviation is to be calculated.

Exercise 5-10x. Write a computer program to determine the pooled standard deviation of the data in the file coinout.10k. With group sizes set at 2, 3, 4, 5, 10, 50, 100, 250, 500, 1000, 2500 and 5000. Comment on the values you receive and comment on differences you observe with the standard deviation of the means offered in the table earlier in this chapter.

There are some alternative terms used for expressing the precision of sets or groups of data elements. They are important to know.



The Variance, s2









The Relative Standard Deviation

The RSD is







The Coefficient of Variation, CV is simply the RSD in percent:








The spread or range, w, is simply the difference between the lowest and highest values in a data set. There is a convenient table to estimate the standard deviation using the value of w. It is suprisingly in agreement with the calculated value for many applications:







N 2 3 4 5 6 7 8 9 10 11 12
k 0.89 0.59 0.49 0.43 0.39 0.37 0.35 0.34 0.32 0.32 0.31


Exercise 5-11. A water well known to contain particularly high levels of iron has five samples drawn for spectrophotometric analysis. The results show the following levels of iron in parts per million:

134, 147, 125, 131, 152

Determine the mean, the standard deviation, the variance, the RSD, the CV the spread and the estimated standard deviation from the table above.

Exercise 5-12. A Quantitative Analysis student determines Cu in brass and obtains the following percent copper: 87.85, 87.70, 87.95, 87.78 and 87.65. Calculate the same quantities requested in Exercise 5-11 above.

Determining the calculated uncertainty from individual values

The model used for the determination of the calculated uncertainty from individual values comes to us from vector algebra and is based on the assumption that if two values are to be added together, the factors resulting in small uncertainties are not linked. That is to say, a positive error in some mass reading would not somehow lead to a positive error in a later volume reading. Moreover, this model applies an estimate that there is a "most probable" error in which there might be some cancellation of effects which bring about the uncertainties. For example, in determining the anticipated error in four truckloads of oranges, each with an uncertainty of ±10 oranges, one would not assume the most probable error for the total load to be ±40 oranges because of the effects of probable cancellation of the total uncertainty between truckloads. In the final analysis, the two formulas used, one for addition and subtraction and the other for multiplication and division ought to be considered to be "best estimates" of anticipated calculated errors:

For addition and subtraction: Consider the operation y=a+b-c+d. Since uncertainties are considered to work in either direction symmetrically, the sign of the operation is unimportant and the function giving the uncertainty in the operation, vy is (we shall use v here instead of s because the uncertainty expressed in each parameter may or may not be a standard deviation).







Note that this formula is the diagonal of a 4-dimensional rectangular solid. It is felt that such a function gives a more probable estimate of the uncertainty owing to some cancellation of error effects rather than that which would be achieved simply by adding all of the uncertainties together. In other words, it would be overkill on error estimation to state that vy = va + vb + vc + vd , because of the presumption of partial cancellation.

For multiplication and division, the formula comes to us from both a vector algebra approach as above and a differential calculus model (be forewarned that the calculus model is something of a finesse to get to the rationalization used above for addition and subtraction calculations):

Consider the operation







Taking the natural logarithm of both sides, one gets







The derivative of ln y yields







This form is then estimated to be the relative uncertainties in each case:







But since the relative uncertainties are assumed to be symmetric around the determined values and can work in either direction, these relative uncertainties are added together regardless of the sign in the derivative. Moreover, since the most probable error would result in some cancellation between the errors, following the model used in addition and subtraction operations above, the form for estimating the error in a process of multiplication and division is (substituting v for in each case)







When a problem involving all four operations is to be solved, the uncertainty in the final result ought to be calculated in the same order as the calculation of the result.

Some exercises in significant figures

For the exercises below, consider each number presented to be precise to ±1 in the last digit.

Exercise 5-13. The sum of 3.4 + 0.020 + 7.31. To how many significant figures ought the result be reported and what is the calculated uncertainty?

Exercise 5-14a. Consider the operation (consider all factors to be experimentally determined)

(38.5 x 27)/252.3. To how many significant figures ought the result be reported and what is the calculated uncertainty?

Exercise 5-14b. Consider the operation (again, all factors have been determined experimentally) (24 x 4.52)/100.0. To how many significant figures ought the result be reported?

The rule of thumb for multiplication and division is to report the result to the same number of significant figures as the smallest number of significant figures in any of the original factors, but problems arise if the result just moves beyond 1xxx.xxx or drops just below 9xx.xxx. (To be explained in class)

Rounding rules

If the digit following the digit to be rounded is 6,7,8 or 9, round to the next highest digit. If the digit following the digit to be rounded is 0,1,2,3 or 4, do not change the rounding digit. If the digit following the digit to be rounded is 5, round to the even integer.



Exercise 5-15. Consider the following experimental values. An experimental value might be a direct observation or it might be a calculated value based on experimental observations. Next consider the estimated uncertainties. Report each experimental value as one ought to report it based on the uncertainties. State the number of significant figures indicated by the reported value. Calculate the relative uncertainty in percent in each case.

Experimental value Uncertainty Reported value Sig. Figures Relative uncertainty
3.827 ±0.04
0.08831 ±0.02
0.0243 ±0.003
2000 ±10
3.85 ±0.02
8.735 ±0.01


Significant Figure Rules with Logarithms

Two rules to remember here. (1) The logarithm ought to be reported to the same number of digits after the decimal point as there are digits in the original number, and (2) when taking an antilogarithm, the antilogarithm ought to be reported to a number of digits equal to the number of digits after the decimal point in the logarithm.

These rules work in most cases. When in doubt consider the relationship y=ln x. The question is what uncertainty in y ought to be reported, knowing the uncertainty in x? Consider the calculus notation:







How does that work in real life?

Consider ln 2475 = 3.3935752 with x = 2475 ±1.

dx/x is approximately equal to (delta x)/x=1/2475 = .0004040, which is approximately equal to y, so ln 2475 = 3.3935752±0.0004040 or

ln 2475 = 3.3936±0.0004. The rule above is followed.

APPENDIX A. The contents of Coinout.10k. If you don't like the idea of running an unfamiliar program to select a certain number of data points then tack this sheet up on your wall and throw darts at it the number of times equal to the number of data points you need. Pick the number closest to each point where the dart hits. Be careful not to hit your roommate.

49 59 47 49 45 48 51 51 59 58 49 45 58 57 50 56 43 40 52 47 49 53 57 51 52 42 48 41 50 41 50 54 48 59 51 51 50 47 49 47 44 46 48 49 46 50 56 49 53 57 52 46 46 54 49 50 54 47 64 50 51 54 57 55 41 49 44 47 53 42 49 58 54 51 41 45 52 54 49 50 46 49 52 47 47 60 45 56 50 46 50 46 51 48 54 54 53 54 44 49 52 41 57 50 55 56 51 46 51 56 53 53 50 50 52 54 52 61 45 40 54 53 53 53 60 59 54 49 49 51 56 46 53 60 52 47 52 48 50 53 47 60 48 53 44 44 55 43 53 48 48 45 56 43 55 48 45 47 49 48 42 54 49 47 44 42 52 38 56 43 50 43 51 57 44 48 57 55 51 48 52 49 48 40 45 53 50 45 51 52 52 44 52 48 52 52 47 46 42 54 49 51 50 56 50 47 51 53 55 51 43 47 52 39 52 53 60 45 47 48 46 52 40 56 50 51 50 62 59 41 48 55 53 52 51 45 47 51 42 44 50 52 48 63 52 50 48 52 51 42 57 40 59 47 43 48 56 53 51 41 41 48 54 51 51 52 49 50 49 49 47 48 45 44 52 45 49 49 38 56 42 43 43 38 52 51 60 52 58 56 54 52 54 42 54 48 54 48 52 36 54 50 59 52 48 46 57 42 44 57 50 56 47 55 55 51 53 47 55 48 52 50 54 59 50 46 54 43 49 55 55 52 51 52 53 58 54 52 45 51 48 53 56 49 55 51 55 53 46 51 52 49 52 40 46 53 49 66 56 42 50 61 57 49 45 46 50 49 53 50 51 48 57 51 61 44 55 44 62 51 51 49 59 46 49 50 47 51 50 54 50 49 53 47 53 49 58 43 45 45 50 39 46 62 50 51 49 57 49 57 52 52 53 49 53 45 57 60 47 54 49 44 52 50 49 48 54 58 47 54 53 44 54 57 49 56 51 52 49 49 46 50 53 53 43 58 57 54 58 54 41 45 43 47 57 46 47 54 50 54 52 49 45 54 45 49 54 40 55 54 53 47 48 39 48 44 58 42 49 53 50 54 55 54 44 51 50 55 53 53 58 50 55 46 62 48 51 44 46 57 47 45 46 51 46 53 68 35 52 47 56 51 53 57 49 61 58 52 53 43 55 58 46 47 48 54 51 40 52 52 53 47 49 51 53 42 51 48 54 55 43 44 40 58 50 47 48 56 55 46 50 51 50 42 50 52 57 55 49 45 48 55 54 52 35 41 58 49 42 53 48 62 51 43 54 50 50 45 40 53 49 46 51 48 47 47 50 49 53 49 47 53 50 52 51 55 47 52 53 46 50 56 57 49 52 56 41 45 49 44 52 46 57 55 44 41 54 49 51 55 58 49 49 52 52 43 54 45 48 52 45 46 60 42 42 53 52 54 61 48 38 59 45 56 51 50 49 59 43 48 59 53 59 61 52 51 46 49 58 58 55 44 45 47 55 53 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53 55 39 52 54 54 56 45 49 45 47 39 54 52 57 48 54 58 52 42 49 50 47 56 52 49 49 47 55 43 58 53 41 55 42 46 60 59 51 46 51 53 44 39 51 44 48 50 52 47 53 50 63 49 55 57 48 48 52 47 40 60 48 57 44 51 47 51 49 56 42 58 49 51 52 52 44 56 53 54 47 46 46 56 52 60 56 48 53 53 52 49 50 54 48 50 52 47 53 44 49 46 48 45 43 51 47 60 45 49 46 42 53 52 56 41 42 45 50 52 45 44 48 51 55 58 53 58 50 56 55 50 51 49 45 48 47 52 48 47 59 45 49 56 61 46 40 43 47 50 50 54 54 46 45 51 47 45 58 40 47 48 49 37 49 47 43 49 49 53 47 46 49 41 46 47 48 51 55 44 44 57 43 52 48 57 53 48 45 53 47 45 48 49 55 44 53 55 61 41 59 56 52 53 49 53 55 49 38 52 55 53 41 51 42 54 51 44 47 53 61 44 50 50 43 52 45 47 64 54 48 46 48 39 47 54 45 50 53 56 45 50 54 53 50 61 53 45 47 55 56 64 47 48 44 52 50 50 49 44 50 59 58 42 49 50 44 46 45 42 53 52 48 49 50 52 61 49 50 58 58 56 51 50 49 54 52 58 51 50 36 48 47 50 42 40 49 55 49 52 41 52 60 53 43 55 46 49 47 54 45 46 54 44 49 48 41 55 52 60 53 52 47 46 47 44 46 56 51 55 41 53 47 58 56 49 50 48 53 63 47 45 54 43 51 45 58 52 48 50 59 57 51 61 51 60 48 52 44 46 54 51 51 47 46 53 46 55 48 43 59 50 60 46 51 40 52 38 44 55 52 45 45 51 47 56 42 51 48 47 57 55 47 49 54 48 56 57 52 48 46 55 53 52 50 48 50 58 44 53 49 56 56 43 46 52 52 44 54 49 45 47 52 56 51 49 50 56 47 48 53 42 45 52 49 48 58 54 46 57 42 57 47 52 56 39 42 49 51 46 53 51 50 56 49 56 55 39 48 56 45 49 45 42 49 48 42 47 45 46 50 46 48 53 48 49 50 61 46 47 55 47 51 55 47 50 63 54 48 48 46 50 48 47 59 50 43 51 39 47 48 55 55 49 49 51 50 46 45 47 54 48 48 48 52 50 50 52 50 47 55 52 53 42 45 55 43 44 42 38 48 51 44 42 52 46 43 38 51 45 48 46 43 46 45 45 50 52 48 44 42 49 60 50 58 50 45 44 46 46 56 44 54 47 47 56 44 39 48 47 53 49 49 60 49 50 50 49 54 48 59 48 54 48 47 53 47 46 51 51 45 45 52 52 51 53 50 52 50 39 48 43 49 48 59 50 52 50 55 54 43 44 57 54 46 45 40 53 49 51 46 52 46 53 60 47 46 49 50 53 46 49 52 46 47 50 52 51 50 52 50 53 53 44 48 51 44 54 42 55 46 52 49 54 49 53 53 46 45 51 50 51 45 46 44 46 55 47 50 47 56 53 57 47 52 47 43 52 47 45 52 46 53 47 42 40 47 49 45 46 49 44 53 41 47 46 56 58 43 55 55 50 47 48 47 46 45 45 44 53 44 49 56 47 49 50 50 47 45 46 51 54 44 47 56 47 51 57 48 49 39 52 54 50 40 51 52 48 46 48 59 55 39 56 49 59 53 55 46 51 53 56 62 53 45 48 46 60 49 54 45 51 49 53 60 57 54 53 39 43 46 49 42 61 51 50 46 39 43 50 44 51 49 52 54 64 47 51 56 43 49 52 42 49 46 45 48 44 46 47 52 49 54 52 49 58 49 53 50 53 54 55 54 58 51 55 59 43 48 49 57 49 41 52 50 50 60 43 47 52 50 54 48 46 53 50 54 51 51 43 50 62 55 56 42 55 51 52 55 52 43 53 55 56 53 52 44 46 50 56 55 50 48 47 38 43 51 50 55 47 48 43 46 59 52 51 51 58 54 56 54 46 49 54 51 55 48 60 43 63 49 53 42 42 54 45 57 51 56 52 51 52 47 55 50 39 52 47 54 59 49 55 46 56 45 52 52 56 56 52 56 53 54 52 54 60 52 60 66 60 53 42 52 40 46 42 43 50 46 53 47 49 58 43 49 56 52 60 45 63 41 43 61 57 51 51 55 50 57 51 51 49 44 44 48 43 53 54 55 54 54 52 44 62 52 62 48 51 45 45 41 44 45 55 49 51 49 54 60 54 65 48 56 46 45 51 53 46 46 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1. Feynman, Richard P., "Surely You're Joking Mr. Feynman!", Bantam Books, New York, N.Y., 1985, pp 103-104.