Introduction

One aspect of most sets of data is that the values are not all alike; indeed, the extent to which they are unalike, or vary among themselves, is of basic importance in statistics. Consider the following examples:

In a hospital where each patient's pulse rate is taken three times a day, that of patient A is 72, 76, and 74, while that of patient B is 72, 91, and 59. The mean pulse rate of the two patients is the same, 74, but observe the difference in variability. Whereas patient A's pulse rate is stable, that of patient B fluctuates widely.

A supermarket stocks certain 1-pound bags of mixed nuts, which on the average contain 12 almonds per bag. If all the bags contain anywhere from 10 to 14 almonds, the product is consistent and satisfactory, but the situation is quite different if some of the bags have no almonds while others have 20 or more.

Measuring variability is of special importance in statistical inference. Suppose, for instance, that we have a coin that is slightly bent and we wonder whether there is still a fifty-fifty chance for heads. What If we toss the con 100 times and get 28 heads and 72 tails? Does the shortage of heads-only 28 where we might have expected 50-imply that the count is not "fair?" To answer such questions we must have some idea about the magnitude of the fluctuations, or variations, that are brought about by chance when coins are tossed 100 times.

We have given these three examples to show the need for measuring the extent to which data are dispersed, or spread out; the corresponding measures that provide this information are called measures of variation. In Sections 1 through 3 we present the most widely used measures of variation and some of their special applications. Some statistical descriptions other than measures of location and measures of variation are discussed in Section 4.5.

5.1 The Range

To introduce a simple way of measuring variability, let us refer to the first of the three examples cited previously, where the pulse rate of patient A varied from 72 to 76 while that of patient B varied from 59 to 91. These extreme (smallest and largest) values are indicative of the variability of the two sets of data, and just about the same information is conveyed if we take the differences between the respective extremes. So, let us make the following definition:

The range of 8 set of data is the difference between the largest value and the smallest.

For patient A the pulse rates had a range of 76 -72 = 4 and for patient B they had a range of 91 -59 = 32, and for the waiting times between eruptions of Old Faithful in Example 2.4, the range was 118 -33 = 85 minutes.

Conceptually, the range is easy to understood, its calculation is very easy, and there is a natural curiosity about the smallest and largest values. Nevertheless, it is not a very useful measure of variation - its main shortcoming being that it does not tell us anything about the dispersion of the values that fall between the two extremes. For example, each of the following three sets of data

Set A: 5    18    18    18    18     18    18    18    18    18

Set B: 5    5      5      5      5      18    18    18    18    18

Set C: 5    6      8      9      10    12    14    15    17    18

has a range of 18 -5 = 13, but their dispersions between the first and last values are totally different

In actual practice, the range is used mainly as a "quick and easy" measure of variability; for instance, in industrial quality control it is used to keep a close check on raw materials and products on the basis of small samples taken at regular intervals of time.

Whereas, the range covers all the values in a sample, a similar measure of variation covers (more or less) the middle 50 percent. It is the inter quartile range: Q₃ –Q₁, where Q₁ and Q₃ may be defined as before. For instance, for the twelve temperature readings in Example 3.16 we might use 88 -76 = 12 and for the grouped data in Example 3.24 we might use 87.58 -69.71 = 17.87. Some statisticians also use the semi-inter quartile range, which is sometimes referred to as the quartile deviation.

5.2 The Variance and the Standard Deviation

To define the standard deviation, by far the most generally useful measure of variation. Let us observe that the dispersion of a set of data is small if the values are closely bunched about their mean, and that it is large if the values are scattered widely about their mean. Therefore, it would seem reasonable to measure the variation of a set of data in terms of the amounts by which the values deviate from their mean. If a set of numbers

x₁,    x₂,    x₃,   … and    x_n

constitutes a sample with the mean , then the differences

are called the deviation from the mean, and we might use their average (that is, their mean) as a measure of the variability of the sample. Unfortunately, this will not do. Unless the x’s are all equal, some of the deviations from the mean will be positive, some will be negative, the sum of deviations from the mean, , and hence also their mean, is always equal to zero.

Since we are really interested in the magnitude of the deviations, and not in whether they are positive or negative, we might simply ignore the signs and define a measure of variation in terms of the absolute values of the deviations from the mean. Indeed, if we add the deviations from the mean as if they were all positive or zero and divide by n, we obtain the statistical measure that is called the mean deviation. This measure has intuitive appeal, but because the absolute values if leads to serious theoretical difficulties in problems of inference, and it is rarely used.

An alternative approach is to work with the squares of the deviations from the mean, as this will also eliminate the effect of signs. Squares of real numbers cannot be negative; in fact, squares of the deviations from a mean are all positive unless a value happens to coincide with the mean. Then, if we average the squared deviation from the mean and take the square root of the result (to compensate for the fact that the deviations were squared), we get

and this is how, traditionally, the standard deviation used to be defined. Expressing literally what we have done here mathematically, it is also called the root-mean-square deviation.

Nowadays, it is customary to modify this formula by dividing the sum of the squared deviations from the mean by n-1 instead of n. Following this practice, which will be explained later, let us define the sample standard deviation, denoted by s, as

And its square, the sample variance, as

These formulas for the standard deviation and the variance apply to samples, but if we substitute  for  and N for n, we obtain analogous formulas for the standard deviation and the variance of a population. It is customary to denote the population standard deviation by (sigma, the Greek letter for lower case) when dividing by N, and by S when dividing by N-1. Thus, for  we write

and the population variance is .

Ordinarily, the purpose of calculating a sample statistics (such as the mean, the standard deviation, or the variance) is to estimate the corresponding population parameter. If we actually took many samples from a population that has the mean, calculated the sample means, and then averaged all these estimated of, we should find that their average is very close to . However, if we calculated the variance of each sample by means of the formula  and then averaged all these supposed estimates of . Theoretically, it can be shown that we can compensate for this by dividing by n-1 instead of n in the formula for s². Estimators, having the desirable property that their values will, on the average, equal the quantity they are supposed to estimate are said to be unbiased; otherwise, they are said to be biased. So, we say that  is an unbiased estimator of the population mean and that s² is an unbiased estimator of the population variance. It does not follow from this that s is also an unbiased estimator of , but when n is large the bias is small and can usually be ignored.

In calculating the sample standard deviation using the formula by which it is defined, we must (1) find , (2) determine the n deviations from the mean , (3) square these deviations, (4) add all the squared deviations, (5) divide by n-1, and (6) take the square root of the result arrived at in step 5. In actual practice, this formula is rarely used – there are various shortcuts – but we shall illustrate it here to emphasize what is really measured by a standard deviation.

Example (1)

A bacteriologist found 8, 11, 7, 13, 10, 11, 7, and 9 microorganism of a certain kind in eight cultures. Calculate s.

Solution:

First calculating the mean, we get

and then the work required to find may be arranged as in the following table:

Finally, dividing 32.00 by 8 -1 = 7 and taking the square root (using a simple handheld calculator), we get

rounded to two decimals

Note in the preceding Table that the total for the middle column is zero; since this must always be the case; it provides a convenient check on the calculations.

It was easy to calculate s in this Example because the data were whole numbers and the mean was exact to one decimal. Otherwise, the calculations required by the formula defining s can be quite tedious, and, unless we can get s directly with a statistical calculator or a computer, it helps to use the formula

Example (2)

Use this computing formula to rework Example (1).

Solution:

First we calculate  and , getting

and

Then, substituting these totals and n = 8 into the formula for S_xx, and n-1 = 7 and the value obtained for S_xx into the formula for s, we get

and, hence, rounded to two decimals. This agrees, as it should, with the result obtained before.

As should have been apparent from these two examples, the advantage of the computing formula is that we got the result without having to determine  and work with the deviations from the mean. Incidentally, the computing formula can also be used to find with the n in the formula for S_xx and the n -1 in the formula for s replaced by N.

In the introduction to this chapter we gave three examples in which knowledge about the variability of the data was of special importance. This is also the case when we want to compare numbers belonging to different sets of data. To illustrate, suppose that the final examination in a French course consists of two parts, vocabulary and grammar, and that a certain student scored 66 points in the vocabulary part and 80 points in the grammar part. At first glance it would seem that the student did much better in grammar than in vocabulary, but suppose that all the students in the class averaged 51 points in the vocabulary part with a standard deviation of 12, and 72 points in the grammar part with a standard deviation of 16. Thus, we can argue that the student's score in the vocabulary part is  standard deviations above the average for the class, while her score in the grammar part is only  standard deviation above the average for the class. Whereas the original scores cannot be meaningfully compared, these new scores, expressed in terms of standard deviations, can. Clearly, the given student rates much higher on her command of French vocabulary than on her knowledge of French grammar, compared to the rest of the class.

What we have done here consists of converting the grades into standard units or z-scores. It general, if x is a measurement belonging to a set of data having the mean  (or) and the standard deviation s (or ), then its value in standard units, denoted by z, is

Depending on whether the data constitute a sample or a population. In these units, z tells us how many standard deviations a value lies above or below the mean of the set of data to which it belongs. Standard units will be used frequently in application.

Example (3)

Mrs. Clark belongs to an age group for which the mean weight is 112 pounds with a standard deviation of 11 pounds, and Mr. Clark, her husband, belongs to an age group for which the mean weight is 163 pounds with a standard deviation of 18 pounds. If Mrs. Clark weighs 132 pounds and Mr. Clark weighs 193 pounds, which of the two is relatively more overweight compared to his / her age group?

Solution:

Mr. Clark's weight is 193 -163 = 30 pounds above average while Mrs. Clark's weight is "only" 132 -112 = 20 pounds above average, yet in standard units we get for Mr. Clark and for Mrs. Clark.

Thus, relative to them age groups Mrs. Clark is somewhat more overweight than Mr. Clark.

A serious disadvantage of the standard deviation as a measure of variation is that it depends on the units of measurement. For instance, the weights of certain objects may have a standard deviation of 0.10 ounce, but this really does not tell us whether it reflects a great deal of variation or very little variation. If we are weighing the eggs of quails, a standard deviation of 0.10 ounce would reflect a considerable amount of variation, but this would not be the case if we are weighing, say, 100-pound bags of potatoes. What we need in a situation like this is a measure of relative variation such as the coefficient of variation, defined by the following formula:

The coefficient of variation expresses the standard deviation as a percentage of what is being measured, at least on the average.

Example (4)

Several measurements of the diameter of a ball bearing made with one micrometer had a mean of 2.49mm and a standard deviation of 0.012mm, and several measurements of the unstretched length of a spring made with another micrometer had a mean of 0.75 in. with a standard deviation of 0.002 in. Which of the two micrometers is relatively more precise?

Solution:

Calculating the two coefficients of variation, we get

Thus, the measurements of the length of the spring are relatively less variable, which means that the second micrometer is more precise.

5.3 The Description of Grouped Data

As we saw in before, the grouping of data entails some loss of information. Each item has lost its identity and we know only how many values there are in each class or in each category. To define the standard deviation of a distribution we shall have to be satisfied with an approximation and, as we did in connection with the mean, we shall treat our data as if all the values falling into a class were equal to the corresponding class mark. Thus, letting x₁, x_, ..., and x_k denote the class marks, and f₁, f₂, ..., and f_k the corresponding class frequencies, we approximate the actual sum of all the measurements or observations with

Sx.f = x₁f₁+ x₂f₂ + ….. x_kf_k and the sum of their squares with

Then, we write the computing formula for the standard deviation of grouped  sample data as

Which is very similar to the corresponding computing formula for s for ungrouped data. To obtain a corresponding computing formula for, we replace n by N in the formula for S_xx and n -1 by N in the formula  for s.

When the class marks are large numbers or given to several decimals, we can simplify things further by using the coding suggested below. When the class intervals are all equal, and only then, we replace the class marks with consecutive integers, preferably with 0 at or near the middle of the distribution. Denoting the coded class marks by the letter u, we then calculate S_xx and substitute into the formula

This kind of coding is illustrated by Figure 5.1, where we find that if u varies (is increased or decreased) by 1, the corresponding value of x varies (is increased or decreased) by the class interval c. Thus, to change s_u from the u-scale to the original scale of measurement, the x-scale, we multiply it by c.

Figure 5.1: Coding the class marks of a distribution

Example (5)

With reference to the distribution of the waiting times between eruptions of Old Faithful shown in before, calculate its standard deviation 

(a) Without coding;

(b) With coding.

Solution:

so that

and

so that

and

Finally, s = 10(1.435) = 1435, which agrees, as it should, with the result obtained in part (a). This clearly demonstrates how the coding simplified the calculations.

5.4 Some Further Descriptions

So far we have discussed only statistical descriptions that come under the general heading of measures of location or measures of variation. Actually, there is no limit to the number of ways in which statistical data can be described, and statisticians continually develop new methods of describing characteristics of numerical data that are of interest in particular problems. In this section we shall consider briefly the problem of describing the overall shape of a distribution.

Although frequency distributions can take on almost any shape or form, most of the distributions we meet in practice can be described fairly well by one or another of few standard types. Among these, foremost in importance is the aptly described symmetrical bell-shaped distribution. The two distributions shown in Figure 5.2 can, by a stretch of the imagination, be described as bell shaped, but they are not symmetrical. Distributions like these, having a "tail" on one side or the other, are said to be skewed; if the tail is on the left we say that they are negatively skewed and if the tail is on the right we say that they are positively skewed. Distributions of incomes or wages are often positively skewed because of the presence of some relatively high values that are not offset by correspondingly low values.

Figure 5.2: Skewed distributions.

The concepts of symmetry and skewness apply to any kind of data, not only distributions. Of course, for a large set of data we may just group the data and draw and study a histogram, but if that is not enough, we can use anyone of several statistical measures of skewness. A relatively easy one is based on the fact that when there is perfect symmetry, the mean and the median will coincide. When there is positive skewness and some of the high values are not offset by correspondingly low values, as shown in Figure 5.3, the mean will be greater than the median; when there is a negative skewness and some of the low values are not offset by correspondingly high values, the mean will be smaller than the median.

Figure 5.3: Mean and median of positively skewed distribution

This relationship between the median and the mean can be used to define a relatively simple measure of skewness, called the Pearsonian coefficient of skewness. It is given by

For a perfectly symmetrical distribution, such the mean and the median coincide and SK = 0. In general, values of the Pearsonian coefficient of skewness must fall between -3 and 3, and it should be noted that division by the standard deviation makes SK independent of the scale of measurement.

Example (6)

Calculate SK for the distribution of the waiting times between eruptions of Old Faithful, using the results of Examples 3.21, 3.22, and 4.7, where we showed , and s = 14.35.

Solution:

Substituting these values into the formula for SK, we get

Which shows that there is a definite, though modest, negative skewness. This is also apparent from the histogram of the distribution, shown originally and here again in Figure 5.4, reproduced from the display screen of a TI-83 graphing calculator.

Figure 5.4: Histogram of distribution of waiting times between eruptions of old faithful

When a set of data is so small that we cannot meaningfully construct a histogram, a good deal about its shape can be learned from a box plot (defined originally). Whereas the Pearsonian coefficient is based on the difference between the mean and the median, with a box plot we judge the symmetry or skewness of a set of data on the basis of the position of the median relative to the two quartiles, Q₁ and Q₃. In particular, if the line at the median is at or near the center of the box, this is an indication of the symmetry of the data; if it is appreciably to the left of center, this is an indication that the data are positively skewed; and if it is appreciably to the right of center, this is an indication that the data are negatively skewed. The relative length of the two "whiskers," extending from the smallest value to Q_I and from Q₃ to the largest value, can also be used as an indication of symmetry or skewness.

Example (7)

Following are the annual incomes of fifteen CPAs in thousands of dollars: 88, 77, 70, SO, 74, 82, 85, 96, 76, 67, 80, 75, 73, 93, and 72. Draw a box plot and use it to judge the symmetry or skewness of the data.

Solution:

Arranging the data according to size, we get

It can be seen that the smallest value is 67; the largest value is 96; the median is the eighth value from either side, which is 77; Q₁ is the fourth value from the left, which is 73; and Q₃ is the fourth value from the right, which is 85. All this information is summarized by the MINITAB printout of the box plot shown in Figure 5.5. As can be seen, there is a strong indication that the data are positively skewed. The line at the median is well to the left of the center of the box and the "wisker" on the right is quite a bit longer than the one on the left.

Figure 5.5: Box plot of incomes of the CPAs.

 Besides the distributions we have discussed in this section, two others sometimes met in practice are the reverse J-shaped and U-shaped distributions shown in Figure 5.6. As can be seen from this figure, the names of these distributions literally describe their shapes. Examples of such distribution may be found in real life situations.

Figure 5.6: Reverse J-shaped and U-shaped distributions