## 2. MEASURES OF ERROR
The rules for significant digits given in the last chapter are too crude for a serious study of experimental error. A better analysis of error should include: (1) A more precise way to measure and express the size of uncertainties in measurements. (2) Rules to predict how the uncertainties in results depend on the uncertainties in the data. Error analysis is an essential part of the experimental process. It allows us to make meaningful quantitative estimates of the reliability of results. A laboratory investigation done without concern for error analysis can not properly be called a scientific experiment. In this chapter we take the first step toward development of useful measures of size of uncertainties. In this text the word "error" is reserved for measurements or estimates of the
However, recent books
In Sec. 1.4 we discussed indeterminate errors, those which are "random" in size and sign. Consider again the data set introduced in that section: 3.69 3.68 3.67 3.69 3.68 3.69 3.66 3.67 We'd like an estimate of the "true" value of this measurement, the value which is
somewhat obscured by randomness in the measurement process. Common sense suggests that
the "true" value probably lies somewhere between the extreme values 3.66 and 3.69, though
it From this limited information we might say that the true value "lies between 3.66 and 3.69." This expresses the range of variation actually observed, and is a rather conservative statement. Or we might quote the arithmetic mean (average) of the measurements as the best value. Then we could specify a maximum range of variation from that average: 3.68 ± 0.02 This is a standard way to express data and results. The first number is the experimenter's best estimate of the true value. The last number is a measure of the "maximum error."
There are many ways to express sizes of errors. This example used a conservative "maximum error" estimate. Other, more realistic, measures take advantage of the mathematical methods of statistics. These will be taken up more fully in chapter 5. Quotes were placed around "true" in the previous discussion to warn the reader that we have not yet properly defined "true" values, and are using the word in a colloquial or intuitive sense. A really satisfying definition cannot be given at this level, but we can clarify the idea somewhat. One can think of the "true" value in two equivalent ways: (1) The true value is the value one would measure if all sources of error were completely absent. (2) The true value is the average of an infinite number of repeated measurements. Since uncertainties can never be completely eliminated, and we haven't time to take an infinite set of measurements, so we never obtain "true" values. With better equipment and more care we can narrow the range of scatter, and therefore improve our estimates of average values, confident that we are approaching "true" values more and more closely. More importantly, we can make useful estimates of how close the values are likely to be to the true value. Note that we have introduced a special scientific meaning for the word "error." Colloquially the word means "mistake" or "blunder." But in science we use "error" to name estimates of the size of scatter of data. Two other words, "deviation" and "discrepancy" are also used in science with very specific meanings:
Discrepancies may also be expressed as percents. The term "discrepancy" is usually used when several independent experimental determinations of the same quantity are compared. When a student compares a lab measurement or result with the value given in the textbook, the difference is called the "experimental discrepancy." Never make the mistake of calling this comparison an "error." Textbook or handbook values are not "true" values, they are merely someone else's experimental values. If several experimenters quote values of the same quantity, and have estimated their errors properly, the discrepancy between their values should be no larger than their estimated maximum errors.
The determinate (random) errors we have been discussing are unbiased in sign. They are as likely to make a measurement "too high" as they are to make it "too low". Our assertion that the average value is an appropriate approximation to the "true" value was based on this assumption. But there are also influences which affect the data so as to make values consistently too
large or too small. These are called (1) The measuring instrument is miscalibrated. (2) The observation technique has a consistent bias. (3) There are unnoticed outside influences on the apparatus, or on the quantity being measured. Such influences may be determinate or indeterminate or both. One common determinate error in elementary lab work is the miscalibrated scale (an example of cause 1). This can happen even with such a simple instrument as a meter stick. The millimeter divisions may vary in size. The end of the stick may have been sawed off inaccurately (or be worn from use) so that a fraction of the first millimeter is "lost." This error is easily eliminated by avoiding making measurements from the end of the stick; start at 10 cm instead, and subtract 10 cm from the reading. All determinate errors may be eliminated, More will be said of determinate errors in chapter 8.
We strongly recommend that several independent measurements be made of each data quantity, preferably by different observers. This procedure has several advantages: (1) Bias of a single observer is eliminated. (2) Blunders may be discovered. (3) The values will indicate whether the data is scale limited. There is no other way to determine this, and unless it is determined you will have no idea of the size of the uncertainty. There It is
The goal of most experiments is to calculate numerical results from measured data. The goal of error analysis is to determine the reliability of these results. Just as we must take data from which to calculate a result, so we must also know the uncertainties in the data to calculate the uncertainty of a result. The first step in any error analysis is to determine the error in each data quantity. In some cases this information may be known from past experience, or it may be supplied by the manufacturer of the measuring device. The experimenter confronted with an unfamiliar measuring device must experimentally
determine its reliability. This will often occupy a large fraction of the laboratory time.
Laboratory manuals usually do not spell out this procedure, but it must be done anyway. Do
One of the best ways to estimate the precision of a measurement is to make a number of independent measurements. As we have illustrated, in the example of section 2.1, these values will show a distribution, or scatter. The distribution may be graphed. If only a few measurements are made, a bar-graph like Fig. 2.1 may be appropriate. If very many measurements are made, the distribution usually approaches a smooth curve. Figs. 2.2, 2.3, and 2.4 illustrate only a few of the many possible distribution curves. In all cases the graphs are a plot of the number of occurrences of each value (on the vertical axis) plotted against the values (on the horizontal axis.) Error distributions like that of Fig. 2. aren't often encountered in science. Many of the distributions will resemble Fig. 2.4. This curve is the famous "normal" or Gaussian curve. Most of the mathematics of statistics, and of error theory, is based upon this curve.
Analysis of a sample set of repeated measurements allows us to calculate several important things: (1) An "average" or "best" value representative of the set. If the distribution is normal or near-normal, the arithmetic mean is the best value. (2) A measure of the dispersion (width or spread) of the distribution. This is also a measure of the "average" error of a typical measurement. (3) An estimate of the error in the mean itself.
A simple and useful measure of scatter is the
Where:
The average of the
DATA DEVIATIONS MAGNITUDES OF SET FROM THE MEAN THE DEVIATIONS 3.69 +0.01 0.01 3.68 0.0 0.0 3.67 -0.01 0.01 3.69 +0.01 0.01 3.68 0.0 0.0 3.69 +0.01 0.01 3.66 -0.02 0.02 3.67 -0.01 0.01 _______ ______ 8 ) 29.43 = 3.67875 7 ) 0.07 = 0.01 , average deviation Mean = 3.68 of the data set. Result: 3.68 ± 0.01 Note that when calculating error estimates we are satisfied with values rounded off to about two significant digits. To calculate an error to as great a precision as the quantity itself is wasted effort.
The mean (average) is often taken as the best representative value for a set of measurements. How good is the mean for this purpose? How far is it likely to deviate from the "true" value? You may have an intuitive feeling that the mean will be "better" if you average a
larger number of independently determined values. We will later show (in chapter 6)
that this is true. It turns out that for each method of measuring dispersion of the
measurements there is a corresponding estimate of error measure of data dispersion estimated error where n is the number of values which were averaged. Therefore we may define:
For the data set of Example 3, the average deviation 0.01 0.01 ———— = ———— = 0.0042 √8 2.38
When the "average deviation" is quoted as a measure of error of a result obtained
from averaging, of from graphical curve fitting, it is the "average deviation 3.68 ± 0.004
The 0.004 tells us the uncertainty In other words, the
The
We have included this definition here, for completeness, but will postpone discussion of it until chapter 5.
Words and their meanings can cause difficulty in scientific discussion. This is
especially true when words taken from common usage are given specialized scientific
meanings. Such words as "... Measuring processes never yield perfect precision or perfect accuracy, so absolutely correct or true results are not attainable. We can only speak of degrees of precision or degrees of accuracy expressed as numerical amounts or percents. The following statements supply the essential scientific meanings of these terms:
A These words refer more to how we make the measurements than to the results of the measurements. True values are never known, so the accuracy of results can never really be determined. When a result is said to be accurate, this means that analysis of the experimental procedure has shown that all sources of error known to the experimenter have been kept small. Statements of experimental accuracy should be supported by description of experimental equipment and technique, to inform others of just what precautions were taken. Quite often in the history of science, results thought accurate were later found inaccurate because of unrecognized determinate error. "Correct" is used in science primarily to indicate absence of mathematical error or blunder.
(2.1) Two wooden meter sticks, picked at random from the supply room, are laid with their scales adjacent and their zero marks coinciding. It is observed that the markings line up well near the zero ends of the sticks, but as we go to larger readings they do not coincide, and when we reach the 100 cm. end of the stick, the markings are a full millimeter off. On the basis of this limited information, how would you express the reliability you would expect from a meter stick? Would it be better to express the error in millimeters, or as a percent? Why? (2.2) A student measures a quantity four times, getting 4.86, 4.99, 4.80, and 5.02. What is the average value? The text book value for this measurement was 5.01. What is the student's maximum error? What is the percent maximum error? What is the discrepancy? What is the percent discrepancy? (2.3) When averaging deviations, we always insist that it is the (2.4) Six measurements of a quantity were found to be: 14.68, 14.23, 13.91, 14.44, 13.85, 14.16. Calculate the average, the deviation of each measurement, the maximum deviation, and the average deviation. Round off the average and state it, with its error, in standard form.
The descriptive term "exact science" is a misnomer, for science deals with measurements, and the measurement processes can never be perfect. We can only try to improve the precision of measurements by reducing experimental uncertainties. Repeated measurements of a quantity show a scatter (dispersion) throughout a range of
values. A mathematical average of the values (for example, the arithmetic mean) is often taken
as the best representative value of the set. But how good is that mean value? A clue is provided
by the "width" of the scatter distribution of the original measurements about the mean. The
average deviation of the measurements is a measure of this width. But we are more interested
in how much the sample mean deviates from the "true" mean, and that the "goodness" of the
mean is better estimated by the average deviation
where n is the number of values of x. We suspect that the mean value is closer to "truth," and more reliable than any single measurement. The mathematical study of statistics provides tools which are very helpful for estimating just how good mean values are. We used the term " The standard form for measured values and results is: (value) ± (est. error in the value) for example: 3.68 ± 0.02 seconds Errors and discrepancies may also be expressed in fractional form, or as percents. Always specify clearly what kind of error estimate you use (maximum, average deviation, standard deviation, etc.). The proper style for scientific notation is: (6.35 ± 0.003) x 106 © 1996, 2004 by Donald E. Simanek |