Characterizing Measurements and Results
Let’s begin by choosing a simple quantitative problem requiring a single measure- ment. The question to be answered is—What is the mass of a penny? If you think about how we might answer this question experimentally, you will realize that this problem is too broad. Are we interested in the mass of United State pennies or Cana- dian pennies, or is the difference in country of importance? Since the composition of a penny probably differs from country to country, let’s limit our problem to pennies minted in the United States. There are other considerations. Pennies are minted at several locations in the United States (this is the meaning of the letter, or absence of a letter, below the date stamped on the lower right corner of the face of the coin). Since there is no reason to expect a difference between where the penny was minted, we will choose to ignore this consideration. Is there a reason to expect a difference between a newly minted penny not yet in circulation, and a penny that has been in circulation? The answer to this is not obvious. Let’s simplify the problem by narrow- ing the question to—What is the mass of an average United States penny in circula- tion? This is a problem that we might expect to be able to answer experimentally.
A good way to begin the analysis is to acquire some preliminary data. Table 4.1 shows experimentally measured masses for seven pennies from my change jar at home. Looking at these data, it is immediately apparent that our question has no simple answer. That is, we cannot use the mass of a single penny to draw a specific conclusion about the mass of any other penny (although we might conclude that all pennies weigh at least 3 g). We can, however, characterize these data by providing a measure of the spread of the individual measurements around a central value.
One way to characterize the data in Table 4.1 is to assume that the masses of indi- vidual pennies are scattered around a central value that provides the best estimate of a penny’s true mass. Two common ways to report this estimate of central tendency are the mean and the median.
The mean, X–, is the numerical average obtained by dividing the sum of the individual measurements by the number of measurements
where Xi is the ith measurement, and n is the number of independent measurements.
The mean is the most common estimator of central tendency. It is not consid- ered a robust estimator, however, because extreme measurements, those much larger or smaller than the remainder of the data, strongly influence the mean’s value.2 For example, mistakenly recording the mass of the fourth penny as 31.07 g instead of 3.107 g, changes the mean from 3.117 g to 7.112 g!
The median, Xmed, is the middle value when data are ordered from the smallest to the largest value. When the data include an odd number of measure- ments, the median is the middle value. For an even number of measurements, the median is the average of the n/2 and the (n/2) + 1 measurements, where n is the number of measurements.
As shown by Examples 4.1 and 4.2, the mean and median provide similar esti- mates of central tendency when all data are similar in magnitude. The median, however, provides a more robust estimate of central tendency since it is less sensi- tive to measurements with extreme values. For example, introducing the transcrip- tion error discussed earlier for the mean only changes the median’s value from 3.107 g to 3.112 g.
If the mean or median provides an estimate of a penny’s true mass, then the spread of the individual measurements must provide an estimate of the variability in the masses of individual pennies. Although spread is often defined relative to a specific measure of central tendency, its magnitude is independent of the central value. Changing all measurements in the same direction, by adding or subtracting a constant value, changes the mean or median, but will not change the magnitude of the spread. Three common measures of spread are range, standard deviation, and variance.
The range, w, is the difference between the largest and smallest values in the data set.
The range provides information about the total variability in the data set, but does not provide any information about the distribution of individual measurements. The range for the data in Table 4.1 is the difference between 3.198 g and 3.056 g; thus
The absolute standard deviation, s, describes the spread of individual measurements about the mean and is given as
where Xi is one of n individual measurements, and X is the mean. Frequently, the
relative standard deviation, sr, is reported.
The percent relative standard deviation is obtained by multiplying sr by 100%.
Another common measure of spread is the square of the standard devia- tion, or the variance. The standard deviation, rather than the variance, is usually re- ported because the units for standard deviation are the same as that for the mean value.