How to recognize errors: systematic, random and gross
In a most idealized sense, a value which has been determined experimentally is one element of a population with a normal distribution. If the method produces a result which is an acceptable reflection of reality, that is, if it really works and gives result which reflects that which is known by some other independent analysis, then one says that this method does not produce a significant systematic error. A method not so reliable and one which routinely produces an error on the low or high side is said to exhibit systematic error. Any value obtained using this second method could be considered, again in a most idealized sense, to be one element in a population with a normal distribution. One could say at the outset that the first method produces an accurate mean, or µA , whereas the second method produces a lousy mean, or µL . One then could, graphically, illustrate the two populations superimposed as shown on the left.
If the first method through repeated comparison with independent methods of analysis becomes a preferred method because it can be shown to be (1) as precise as could be expected by the instrumentation used and (2) that it yields values as close to the true value as the precision of the method, then one might argue that µ A is indistinguishable from the true value. The bias produced by the second method is said to be
Where might one encounter such systematic error? With the use of glassware such as a buret, poor bore would produce systematic error. In our laboratory we use mostly Class A burettes as opposed to Class B burettes. The two classes are based on criteria set up by the American Society for Testing and Materials (ASTM). It is instructive to examine a table of tolerances established by the ASTM. The table below shows the Class A designation and accuracy specification for six different forms of volumetric glassware.
Accuracy specifications in ±mL for Class A volumetric glassware
(Class B volumetric glassware has ±mL tolerances twice those of Class A glassware)
|Capacity (mL)||Transfer pipets (E969)||Micro- volumetric vessels (E237)||Measuring pipets (E1293)||Volumetric flasks (E288)||Burets (E287)||Graduated cylinders (E1272)|
Even with this stringent standard, during the period 1995-1999 we encountered one buret with an internal defect. It appeared to be a small fragment of unmelted glass which produced an internal bump in the bore. During the calibration process one would expect such a defect to cause a serious systematic error in the delivered volume.
Any glassware used for quantitative measurements is a potential source of systematic error. Pipettes, burets, graduated cylinders and even graduated beakers fall into this category. From time to time one observes mislabeled graduations on burets which could lead the technician to erroneous procedures.
Where more sophisticated equipment is used, namely electronic measuring apparatus, systematic errors can come about as the result of low batteries, poor contacts within the device, sensitivity to temperature and humidity and even mechanical defects in the case of meter movements.
One's methods are threatened with systematic errors as well. A reaction which comes to completion slowly, an indicator whose color change occurs well before or after the equivalence point of a reaction, a step which is particularly cumbersome or requires meticulous attention to detail (the transfer of the barium sulfate precipitate in the gravimetric analysis experiment), a step which one might wish to perform to a fault (excessively washing the barium sulfate with water until peptization and subsequent loss of precipitate through the filter paper) all carry the peril of systematic error.
Consider some of the areas prone to systematic errors in our other experiments:
|Carbonate determination||Copper in Brass||Manganese in Steel|
|Failure to boil the solution at the final end point||Failure to dissolve all of the copper in the brass||Failure to dissolve all of the steel sample|
|Failure to do a blank reading||Adding too much 3M sulfuric acid subsequently producing a pH which is too low||Not filtering out unwanted particles|
|Use of an indicator with an end point well before or well after the equivalence point.||Not removing all of the nitrogen oxides after solution of the brass||Not oxidizing all carbon granules.|
A more serious systematic error often comes about because the method is not universally applicable. The Kjeldahl method for the determination of nitrogen in plant and animal tissues depends on a digestion by concentrated H2SO4. Nitrogens which yield to such a digestion are -NH2 and -NH- groups. But fully bound nitrogen atoms, in the form of =N-, are often incompletely digested. Thus there is a tendency for the determination to give a low result.
The investigator often contributes to systematic errors by routinely reading instrument scales high or low or developing preconceived notions of an anticipated result and reading the instrument scales so as to improve the results.
An investigator who is asked to weigh three samples to ±0.0001 grams and to make the weight in the vicinity of 3.0 g might consciously try to make each sample as close to 3.0000 g as can be managed by scooping off excess sample or adding more in tiny increments. Such a technique risks errors in weight due to spillage on the pan but outside the container holding the sample and also by the absorption of moisture from the air. A buret reading based upon a starting point of 0.00 mL can lead to systematic errors if the same convention for the relative location of .10, .20, .30 mL and so on is not the same as that used for 0.00. One ought to decide on the top, the middle or the bottom of the width of the calibration mark. Finally, when interpolating the volume reading of a buret between 0.30 and 0.40, for example, some people tend to favor 0.30, 0.35 and 0.40 mL over some other number which on close examination might be better than any of these three.
Errors which are constant vs. those which are proportional
In the gravimetric determination of sulfate, a precipitate of BaSO4 is developed in a 400 mL (approx.) solution. One would expect in the transfer process that the same perils for loss of sample are present at all stages. Thus one would expect that approximately the same amount of barium sulfate might be lost regardless of the mass of the sample. A constant error is one which does not change with the size of the sample. It stands to reason then that a large sample weight would be preferred over a small sample weight, because such a constant error will produce a smaller relative error when using a larger sample.
If 0.8 mg barium sulfate is lost during an average transfer of the precipitate, compare the relative errors which would be realized if the precipitate weighed
(a) 0.8000 g and
(b) 0.4000 g.
(To be solved in class with attention paid to the magnitude of the two relative errors)
The amount of titrant necessary to produce a color change is another example of a constant error.
If no blank correction is made in a typical carbonate titration, what is the relative error in parts per thousand of a titration requiring (a) 10.00 mL, (b) 20.00 mL, (c) 30.00 mL? Assume that the blank correction would be -0.05 mL.
(To be solved in class)
Instrumental errors often are systematic in nature. If a constant error is suspected in an instrument which produces a reading that is directly proportional to some concentration of an analyte, a plot can reveal such a constant error, as the following example attempts to demonstrate:
Five concentrations of potassium permanganate are prepared and the absorbances read at a wavelength of 525 mµ. Beer's Law predicts that A= epsilon x c, or that absorbance is directly proportional to concentration, and is a constant called the molar absorptivity. Plot the values of absorbance vs. concentration in ppm and determine the constant error in these measurements.
|ppm Mn as KMnO4(aq)||Absorbance|
Question: What are the possible explanations for the error observed in the plot of this group of measurements?
A proportional error is any error which is proportional to the quantity of sample. Contaminants which interfere with the reaction to be used in analysis are prime candidates for the production of proportional errors because the absolute size of the error increases with the size of the sample. The presence of iron in a sample of brass can interfere with the reduction of copper as shown in the following two equations:
It is for this reason that precautions must be taken to prevent this interference. Fe3+ complexes with the phosphate ion, PO43- , in a manner that removes it from availability for reduction.
Finally, there are personal errors against which, sadly, there is no vaccination. Care, self-discipline and a meticulous attention to detail need to be exercised at all times in the laboratory to protect oneself against personal errors.
Detection of Systematic Errors Germaine to the Method.
Systematic errors characteristic of the method used may be revealed by using the method to analyze a standard sample. The National Institute of Standards and Technology maintains a site at
http://ts.nist.gov. Within that site is located the entire catalog of Standard Reference Materials. It is located at
You ought to visit this site at least once to take some measure of the level of precision attempted by the NIST to offer the highest quality reference materials. For example, one of the links shows a list of meticulously prepared Single Element Standard Solutions. They are intended as standard solutions for use in calibrating instruments used in atomic spectroscopy as well as in conjunction with any other analytical technique or procedure where aqueous standard solutions are required. A certificate of analysis accompanies each standard solution.
The certificate one receives starts out as follows:
For this lot the certificate includes the information that the certified value of arsenic is
8.44 mg/g±0.03 mg/gm, the method of preparation, the impurities and their amounts (65 mg/kg total metallic impurities and 375 mg/kg dissolved gases). There is an expiration date approximately two years in advance for this particular certificate and a promise to notify the purchaser should there be a change in the conditions of certification.
The reputation of the NIST in preparing Standard Reference Materials remains the highest in the world. Each of the materials is prepared and/or analyzed using one or more of the following strategies:
- the use of a previously validated reference method,
- an analysis using two or more independently reliable measurement methods, and
- multiple analyses by a network of cooperating laboratories, each with a record of technical competence and reliability.
Should standard samples for a particular target material not be available, laboratories quite often enter into joint agreements to spend some fraction of their time in independent sample analysis using sufficiently different methods for the same sample so as to diminish the chances that the same potential mistakes will be repeated.
Statistical Tests for the Evaluation of the Reliability of Experimental Data
Not infrequently one is faced with a small number of experimental values which, owing to some gross error, don't belong with the others. It is interesting to reflect on the fact that the results obtained by an undetected loss of a portion of one's sample would be indistinguishable from some malicious altering of a sample so as to contain less of an analyte than the sample originally set to be analyzed. Both would result in an analysis previously shown to be reliable in a measured quantity considerably less than the other samples in the group. The population is defined as all possible samples analyzed by the same method without externally imposed systematic errors, either loss of sample or maliciousness. Any such a perturbation of the analysis would place the affected sample in another population. But is there a method to assist the investigator in going the extra distance to obtain an objective reason to exclude a sample from consideration? There are two strategies which may be followed. First, it is generally agreed upon that if a known error was made in an analysis, the result ought to be discarded, regardless of its relative size in comparison to the other results. Secondly, there is a widely used statistical test, the Q-test to aid the investigator in deciding what may safely be discarded within certain confidence limits.
One finds widely divergent opinions about the rejection of data. Deming, an authority in the industrial application of statistics has said that, "a point is never to be excluded on statistical grounds alone." Others agree with Parratt, who writes, "rejection on the basis of a hunch or of general fear is not at all satisfactory, and some sort of objective criterion is better than none." The objective criterion we use is referred to as a Confidence Level, or Confidence Interval. A conservative confidence level of 99% means that of all experimentally determined values within a population which lie within a Gaussian Distribution, only 1% of all legitimate values would be expected to fall outside this level of confidence. That is, one tends to consider practically all points as legitimate, hence a conservative approach to keeping data -- the conservative approach is to keep practically everything, whereas a confidence level of 90% would be one in which 10% of all legitimate values would lie outside this level. Since in the minds of many investigators 10% represents a significant percentage of all values determined and the rejection of this large number of otherwise carefully determined values is viewed to be excessive, even rash, radical, revolutionary, perhaps hare-brained, a compromise is made at the 95% confidence level. We shall predominately use this confidence level though our tables offer a number of other confidence levels. Even here, some values which are suspicious may not be rejected. Natrella has noted, with no little amusement, "the only sure way to avoid publishing any 'bad' results is to throw away all results."(1) Before presenting the formula to be used for the simplest case, it is instructive to consider the situations one might encounter in an analysis. In the diagram below, the asterisks represent experimentally determined values on an arbitrary scale of 5.00 to 5.50, which could represent "percent" anything, carbonate, sulfate, copper, etc.
Most of the time we shall focus on the simplest of the six tests (the r10 case) commonly available to quantitative chemists. The tables used for all six tests are included here should the student wish to pursue the matter further, and because examples taken from real situations in class often force us to consider the other cases. The formulas used to compare a Q exp with one of Dixon's Q Parameters precede each of the tables for which that formula is to be used. The tables are listed at the end of this chapter.
The Q test
The simplest Q test is that in which there appears to be single outlier, that is, a data point which does not belong to the population of the rest of the data points. This "r10" Q-test draws on the spread, w, between the extreme values and the difference between the doubtful data point and its nearest neighbor (when the values are arranged from lowest to highest). If this calculated Q "experimental", Qexp is greater than the value found in the table, then at the level of confidence indicated, the value may be discarded.
In this first variation on the r10 formula it is assumed that there are n data elements in the group, that x2 is the value of the nearest neighbor to the questionable value and that x1is the questionable value. Should the questionable value be on the high side, as in the second variation on the right, xn is the questionable result, xn-1 is its nearest neighbor and the denominator is the spread of all values.
Exercise 6-4. Read a partially filled graduated cylinder. Write down the result on a scrap of paper and turn in to the instructor. Don't let any of your fellow students see what you have written. The instructor will write the results on the board, arranged from lowest to highest. Which case does this fit? With what level of confidence can you reject any of the extreme values?
(To be performed in class)
Exercise 6-5. Here are similar results for one recent semester of CHE 230, written exactly as they were reported by students:
1995: 5.09, 5.4, 5.5, 5.57, 5.58, 5.59, 5.61
Which case does this collection of data fit? With what level of confidence can you reject the extreme value? (To be performed in class)
Exercise 6-6a. Here are some results obtained the following semester:
1996: 6.3, 6.385, 6.4, 6.61
Exercise 6-6b. Look at Exercise 5-12, the determination of %Cu in brass. With what level of confidence can you reject the two outliers in this student's data?
Which case does this collection of data fit? With what level of confidence can you reject the extreme value? (To be performed in class).
The confidence interval when the standard deviation of your sample is a good approximation of sigma or when your sample comes from a known population.
To recap a point about the normal distribution, the area 1 on either side of this distribution contains 68% of the values determined. 2 contains 96% of all values. The function you see is continuous and is a characteristic distribution to which one extrapolates if the population contains an infinite number of elements, but as we have seen before, even if the total number of events is only 10000, the shape of a normal distribution begins to be revealed.
Exercise 6-7. Consider again the plot first shown in Chapter 5. Does it appear here that 68% of all events will be enclosed within an interval of ±1? That 96% of all values will be enclosed within ±2? Taking the list of values given to you in class or which you have calculated with one of the programs made available to you, determine the point on either side of the mean which encompasses 68 and 96%. Are they sufficiently near to 1 sigma and 2 sigma to satisfy you?
There is a strategy to determine confidence limits of the inclusion of the mean of the population if a small sample is used but the method is well known. Before we get to that strategy, let's for a moment consider the "method." The "method" is often thought to be a chemical technique, like the gravimetric determination of sulfate by precipitating barium sulfate, or the volumetric determination of carbonate. But where statistical tests are concerned there is a subtlety to the method which includes the technician who does the test. Often, if the technician is well trained, maintains always a meticulous attention to detail, has an exemplary background and years of experience in analytical chemistry, then this added subtlety is minimal. It is fair then to propose that any new data point reported by such a technician belongs to the same population. At least that is the assumption. So, we talk about the Confidence Interval for µ, the mean for that chemist's population:
The value of z is related to the confidence limit by the area under a normal distribution at ±z:
Confidence levels for various values of z
|Confidence Levels, %||z|
Exercise 6-8. Turn to the appendix in chapter 5. If he remembers, the instructor will pass out cut-up segments of that appendix so as to assure greater randomness for the following process. Use a pencil for this exercise. Close your eyes, circle the pencil around above your desk top and put the point down on the paper showing the array of 10000 events each one of which represents flipping a coin 100 times. Remembering the value of the standard deviation for this collection of events, determine the 90, the 95 and the 99% confidence limits that the mean of the population lies within these limits. (To be solved in class).
Exercise 6-9. Extract the values found by two of your classmates. Determine the 90, the 95 and the 99% confidence limits that the mean of the population lies with these limits. (To be solved in class).
Exercise 6-10. Taking all values determined by all members of the class, determine the 90, the 95 and the 99% confidence interval that the mean of the population lies within these limits. (To be solved in class).
For your consideration:
- A confidence interval is based on probability. We might be unlucky and by the luck of the draw have values that would put us outside the envelope.
- Note that as the confidence limit increases, so does the envelope. The width of the envelope is linked to the confidence limit.
- As the certainty of a mean increases by an increase in the number of reported values, the width of the envelope decreases (the square root of N is in the denominator). This illustrates that the greater number of samples increases precision.
- The mean of a population may not be known, but its standard deviation may be available. That is, the "population" may be a determination of the same analyte done by the same chemist hundreds of times. The pooled standard deviation would exist as a reflection of the precision of the method used, but there would be no "population mean." So the chemist's pooled value of s would be indistinguishable from sigma but one wouldn't speak of the population "mean" when the samples come from many different sources.
Another way to put a problem like that above:
Exercise 6-11: How many values would you need to decrease the 80% confidence limit to ±2? (To be performed in class).
The Student T Test
But what if the "method" ISN'T well known? What happens if your bench chemist of 35 years has just retired and you've had to go to a temporary employment agency to find a replacement. Your new chemist and the technique used is a part of the "method." And you don't know very much about the scatter of the method this new guy is going to offer you. So here is a situation in which sigma is not known and all you have to go by is a small number of samples which he analyzes. The technique to be used here is called Student's T Test.
Shortly after the turn of the twentieth century, a paper was published by "A. Student" which showed how some knowledge of a population mean could be gained if only a small sample of experimentally determined results was available and nothing was known about the scatter of a large number of determinations which would be more characteristic of the population of results by some given method. The real name of the author -- A. Student was an obvious pseudonym -- wasn't known until the 1950s when it was revealed that W.S. Gossett had as a young man been employed by the Guinness Brewery. He had evidently been denied permission to publish the paper under his own name so he did so anonymously, giving the world the Student T-Test. Some observers have suggested that Guinness had begun to use statistical techniques such as this to improve the company's quality control and the company did not want its competitors to know its strategy, thus the denial to Gossett.
Here's another way of saying the same thing: Your new chemist does a small number of analyses. You don't know to which population his results belong. The Student T-test allows you to glean some knowledge from his experimental standard deviation.
The method goes like this: (1) Choose a value of "t" from the table for a given confidence level. (2) Determine the mean and standard deviation for the small sample (3) Calculate the confidence limit for that level of confidence from this formula:
The "N" in the equation above is the number of reported values, but the value of t in the table below is found in the row showing the number of degrees of freedom = N-1.
Values of t for various levels of probability
|Deg. of freedom||80%||90%||95%||99%||99.8%|
Exercise 6-12. Consider that you and two of your classmates each flip a coin 100 times (well, why not?). Each of you gets some value for the total number of heads. Come to think of it, each of you has already determined such a number in the random choice exercise above. If each of you had actually flipped a coin a hundred times and put together your results without any knowledge of the plot of 10000 identical events, trying to find the interval for a given level of confidence that the population mean would be found within those limits would be a job for Student's T test. (A) Apply Student's t test to this situation and determine with 95% confidence how far away the population mean might lie from the mean of your three-event sample. (B) But let's say that suddenly, voila!, we're told the standard deviation of the normal distribution of the 10000 events each event of which is a coin flip 100 times. That can be used as the sigma. The number of degrees of freedom in such a population is infinite our Student T-test reverts to the case in which sigma is known. Use the standard deviation of the population to calculate the 95% confidence interval for the mean. (To be solved in class).
Exercise 6-13a. Consider the results reported by Student 5 in the soda ash unknown:
|Student||Sample 1||Sample 2||Sample 3||mean||s|
With 95% confidence how far away might be the mean of this student's unknown population?
Exercise 6-13b. What is the effect in the case of Student 5? An old professor hobbles in and says, "Well, it's all well and good for Student 5 to have had three samples which showed a standard deviation of 0.09 for values having a mean of 20.89%, but I had her in my class last semester and I know with certainty that her work is routinely good to 2 parts per thousand, or 20.89±0.04. That being the case, calculate the 95% confidence interval for the mean. (To be performed in class).
Please note that Exercise 6-12 demands some consideration of the following two points. Since the same random number generator was used to generate the three values picked by students and their pencils as was used to generate all 10000, we know that there is no systematic error inherent in the three numbers manually picked (notwithstanding the argument that arranging the numbers in a rectangle with the student perhaps favoring the center over the edges might prejudice the results). Still, by the luck of the draw, the three numbers 36, 37 and 38 might have been chosen. Such a small standard deviation among the three would have predicted a mean far from 50. Any confidence limit is based on probability and it is good always to say to oneself, "Although there is a 95% probability that the mean lies within these limits, there is a 5% probability that it does not." Secondly, notice that once the population standard deviation is known, there is a narrowing of the interval for the same confidence level.
The probability of bias in an experimental result where a true value is known.
We've seen that there is a way to calculate the probability of finding the mean within a certain confidence interval if the value of sigma for a population is known (use of the table with z values). We've seen that if a few samples are determined and the value of s is calculated, we can determine the extremity of location of the mean of an unknown population within certain confidence limits (the Student T Test). In the case where we have a known value of the mean of a population, or for a chemist that might mean a true value which will be µ for a reliable method of analysis, the question is for an unknown method (read "generally untried technique," or "new analytical chemist" or "Carrot Top on his first day at work.") is "does it produce results which have bias?" This test is nearly identical to the t-test described earlier, but the statement one uses after all calculations have been made deserves some study and reflection. The statement is based on a "null" hypothesis. But first the method:
(1) On the basis of the number of analyses reported, find a t for a given level of confidence.
(2) Calculate the mean and s for the reported analyses.
(3) Calculate (the mean - µ) and compare it with
Exercise 6-14. Consider the following three reported values for the percent copper in brass during one recent semester:
80.47,80.62,80.32. The true value is known from the analytical laboratory which prepared this sample to be 82.10% Cu. Do the test for bias at the 95%, the 99% and the 99.8% confidence levels. (To be performed in class and get ±0.37 for 95%, ±0.859 for 99%, and ±1.931 for 99.8%
But the mean - µ = 1.63.
Here are the statements: If there were no bias,
fewer than 5 times in a hundred (95% confidence level) will an experimental mean deviate from the true mean by 0.37 or more.
fewer than 1 time in a hundred (99% confidence level) will an experimental mean deviate from the true mean by 0.859 or more.
fewer than 2 times in one thousand (99.8% confidence level) will an experimental mean deviate from the true mean by 1.931 or more.
And the zinger:
If we say that 1.63 is significant and that there is systematic error we would be wrong less than 5 times in a hundred, less than 1 time in a hundred, but more than 2 times in 1000.
Exercise 6-15. In 1997 seven CHE230 students read the volume of water in a 10 mL graduated cylinder and report the following values: 6.78,6.79,6.8,6.80,6.800,6.82,6.82 mL.
The instructor had previously read the volume and decided it to be 6.78 mL (the "true" value). With what level of confidence can one establish bias in these results? (To be performed in class)
On the other hand, if it turns out that the method used is well-known (or alternatively, that one has confidence that the experimental results belong to the same population which has a known standard deviation) then the solution is reduced to the case of the known sigma; z replaces t and sigma replaces s.
Exercise 6-16. The instructor says, "It's reasonable to assume that these fine students can read a 10 mL graduated cylinder in a manner which for many readings will give them a standard deviation of ±0.01 mL." Taking 6.78 to be "equal" to µ and replacing t with z, now with what level of confidence can bias be established in these results?
The comparison of two experimental means
Finally, you are presented with two sets of results which are sufficiently far from each other to suggest any of the following:
(A) they come from different sources, that is the % analyte is clearly different in each.
(B) they are analyzed by two different technicians one or both of which produce a systematic error,
(C) or more generally, the two sets of samples come from different populations.
(1) Calculate each mean.
(2) determine a pooled standard deviation
(3) Calculate the absolute value of the difference between the two means and compare it with
Consider the carbonate reports from students 9 and 12:
|Student||Sample 1||Sample 2||Sample 3||mean||s|
Do a determination of bias at the 95%, 99% and 99.8% levels of confidence and make a concluding statement consistent with the reasoning used for a comparison of experimental values with a true value. (To be performed in class)
Carry out the same procedure as in Exercise 6-17 with the results of students 3 and 5:
|Student||Sample 1||Sample 2||Sample 3||mean||s|
Least Squares Linear Regression
Exercise 6-19. Consider the following table of data from the determination of Mn in steel:
|4.00 x 10-6||0.181|
|6.00 x 10-6||0.255|
|10.00 x 10-6||0.438|
|14.00 x 10-6||0.623|
|16.00 x 10-6||0.689|
Referring to the instructions on carrying out least squares linear regression in your laboratory manual, determine the best slope m and y-intercept b for these data, for a plot of concentration along x and Absorbance along y so that the following linear relation between concentration and Absorbance is predicted:
(To be solved in class).
Dixon's Q Parameters
for various arrangements of doubtful results
and for various levles of confidence from 80% to 99%(2)
r10 Q Parameter, based on one doubtful result (one outlier). If Qexp (in the formula below) > Q in the table, then the outlier may be rejected with that level of confidence.
r11 Q Parameter, where one has two doubtful results on opposite ends and one is being tested. If Qexp (in the formula below) > Q in the table, then the outlier may be rejected with that level of confidence.
r12 Q Parameter where one has three doubtful results distributed unevenly and the lone one is tested. If Qexp (in the formula below) > Q in the table, then the outlier may be rejected with that level of confidence.
r20 Q Parameter, where one has two doubtful results both located at the low or high end. If Qexp (in the formula below) > Q in the table, then the outlier may be rejected with that level of confidence.
r21 Q Parameter, (three doubtful results distributed unevenly; the furthest of the extreme pair is tested). If Qexp (in the formula below) > Q in the table, then the outlier may be rejected with that level of confidence.
r22 Q Parameter, where one has four doubtful results distributed evenly on either side and one is tested. If Qexp (in the formula below) > Q in the table, then the outlier may be rejected with that level of confidence.
1. All quotations in this paragraph and the tables of Dixon's Q parameters at the end of the chapter come from D.B. Rorabacher, Anal. Chem., 1991, 63, 139.
2. Extracted from D.B. Rorabacher, Anal. Chem., 1991, 63, 139.
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