Introduction

When we are about to describe a set of data, it is a sound advice to say neither too little nor too much. Thus, depending on the nature of the data and the purpose we have in mind, statistical descriptions can be very brief or very elaborate. Sometimes we present data just as they are and let them speak for themselves; on other occasions we may just group the data and present their distribution in tabular or graphical form. Most of the time, though, we have to describe data in various other ways.

It is often appropriate to summarize data by means of a few well-chosen numbers that, in their way, are descriptive of the entire set. Exactly what sort of numbers we choose depends on the particular characteristics we want to describe. In one study we may be interested in a value that somehow describes the middle or the most typical of a set of data; in another we may be interested in the value that is exceeded only by 25% of the data; and in still another we may be interested in the length of the interval between the smallest and the largest values among the data. The statistical measures cited in the first two situations come under the heading of measures of location and the one cited in the third situation fits the definition of a measure of variation.

4.1 Populations and Samples

When we stated that the choice of a statistical description may depend on the nature of the data, we were referring among other things to the following distinction:

If a set of data consists of all conceivably possible (or hypothetically possible) observations of a given phenomenon, we call it a population; if a set of data consists of only a part of these observations, we call it a sample.

Here, we added the phrase "hypothetically possible" to take care of such clearly hypothetical situations as where we look at the outcomes (heads or tails) of 12 flips of a coin as a sample from the potentially unlimited number of flips of the coin, where we look at the weights of ten 30-day-old lambs as a sample of the weights of all (past, present, and future) 30-day-old lambs raised at a certain farm, or where we look at four determination of the uranium content of an ore as a sample of the many determinations that could conceivably be made. In fact, we often look at the results of an experiment as a sample of what we might get if, the experiment were repeated over and over again.

Originally, statistics dealt with the description of human populations, census, counts and the like, but as it grew in scope, the term "population" took on the much wider connotation given to it in the preceding distinction between populations and samples. Whether or not it sounds strange to refer to the heights of all the trees in a forest or the speeds of all the cars passing a checkpoint as populations is beside the point-in statistics, "population" is a technical term with a meaning of its own.

Although we are free to call any group of items a population, what we do in practice depends on the context in which the items are to be viewed. Suppose, for instance, that we are offered a lot of 400 ceramic tiles, which we may or may not buy depending on their strength. If we measure the breaking strength of 20 of these tiles in order to estimate the average breaking strength of all the tiles, these 20 measurements are a sample from the population that consists of the breaking strengths of the 400 tiles. In another context, however, if we consider entering into a long-term contract calling for the delivery of tens of thousands of such tiles, we would look upon the breaking strengths of the original 400 tiles only as a sample. Similarly, the complete figures for a recent year, giving the elapsed times between the filing and disposition of divorce suits in a County, can be looked upon as either a population or a sample. If we are interested only in a County and that particular year, we would look upon the data as a population; on the other hand, if we want to generalize about the time that is required for the disposition of divorce suits in: the entire Country, in some other County, or in some other year, we would look upon the data as a sample.

As we have used it here, the word "sample" has very much the same meaning as it has in everyday language. A newspaper considers the attitudes of 150 readers toward a proposed school bond to be a sample of the attitudes of all its readers toward the bond; and a consumer considers a box of Mrs. See's candy a sample of the firm's product. Later, we shall use the word "sample" only when referring to data that can reasonably serve as the basis for valid generalizations about the populations from which they came; in this more technical sense, many sets of data that are popularly called samples are not samples at all.

In this chapter and in the next one we shall describe things statistically without making any generalizations. For future reference, though, it is important to distinguish even here between populations and samples. Thus, we shall use different symbols depending on whether we are describing populations or samples.

4.2 The Mean

The most popular measure of central location is what the lay person calls an "average" and what the statistician calls an arithmetic mean, or simply a mean. It is defined as follows:

The mean of n numbers is their sum divided by n

It is all right to use the word "average," and on occasion we shall use it ourselves, but there are other kinds of averages in statistics and we cannot afford to speak loosely when there is any risk of ambiguity.

Example (1)

 From 1990 through 1994, the combined seizure of drugs the Drug Enforcement Administration, Custom's Service added up to 1,794, 3,030, 2,551, 3,514, and 2,824 pounds. Find the mean seizure of drugs for the given five-year period.

Solution:

The total for the five years is:

1,794 + 3,030 + 2,551 +3,514 + 2,824 = 13,713

Pounds, so that the mean is  pounds.

Example (2):

 In the 9th through 97th Congress of Egypt, there were, respectively, 67, 71, 78, 82, 96, 110, 104, and 92 Representatives at least 60 years old at the beginning of the first session. Find the mean.

Solution:

 The total of these figures is 67+ 71 + 78 + 82 + 96 + 110 + 104 + 92 = 700. Hence, the mean is   .

Since we shall have occasion to calculate the means of many different sets of sample data, it will be convenient to have a simple formula that is always applicable. This requires that we represent the figures to be averaged by some general symbol such as x, y, or z; the number of values in a sample, the sample size, is usually denoted by the letter n. Choosing the letter x, we can refer to the n values in a sample as x₁, x₂, …, and x_n (which read "x sub-one," "x sub-two," ..., and "x sub-n"), and write

Sample mean =

This formula will take care of any set of sample data, but it can be made more compact by assigning the sample mean the symbol  (which reads “x bar") and using the notation. The symbol is capital sigma, the Greek letter for S. In this notation we let ; stand for "the sum of the x's" (that is, ), and we can write

If we refer to the measurements as y's or z's, we write their mean as  or . In the formula for  the term does not state explicitly which values of x are added; let it be understood, however, that  always refers to the sum of all the x's under consideration in a given situation.

The number of values in a population, the population size, is usually denoted by N. The mean of a population of N items is defined in the same way as the mean of a sample. It is the sum of the N items,  or  divided by N.

Assigning the population mean the symbol  (mu, the Greek letter for lowercase m) we write

With the reminder that is now the sum of all N values of x that constitute the population.

Also, to distinguish between descriptions of populations and descriptions of samples, we not only use different symbols such as  and , but we refer to a description of a population as a parameter and a description of a sample as a statistic. Parameters are usually denoted by Greek letters.

To illustrate the terminology and notation just introduced, suppose that we are interested in the mean lifetime of a production lot of N = 40,000 light bulbs. Obviously, we cannot test all of the light bulbs for there would be none left to use or sell, so we take a sample, calculate X, and use this quantity as an estimate of .

Example (3)

 If n = 5 and the light bulbs in the sample last 967, 949, 952, 940, and 922 hours, what can we conclude about the mean lifetime of the 40,000 light bulbs in the production lot?

Solution:

The mean of this sample is

 hours

If we can assume that the data constitute a sample in the technical sense (namely, a set of data from which valid generalizations can be made), we estimate the mean of all 40,00 light bulbs as  = 946 hours.

For nonnegative data, the mean not only describes their middle, but it also puts some limitation on their size. If we multiply by n on both sides of the equation , we find that  and, hence, that no part, or subset of the data can exceed .

Example (4)

 If the mean salary paid to three NBA players for the 1998-1999 season is $2,450,000, can:

a.      Anyone of them receive an annual salary of $4,000,000;

b.      Any two of them receive an annual salary of $4,000,000?

Solution:

The combined salaries of the three players total 3(2,450,000) = $7,350,000.

a.     If one of them receives an annual salary of $4,000,000, this would leave 7,350,000 - 4,000,000 = $3,350,000 for the other two players, so this could be the case.

b.     For two of them to receive an annual salary of $4,000,000 would require 2(4,000,000 = $8,000,000, which exceeds the total paid to the three players. Hence, this cannot be the case.

Example (5)

If six high school juniors averaged 57 on the verbal part of the PSAT/MSQT test, at most how many of them could have scored 72 or better on the test?

Solution:

Since n = 6 and x = 57, it follows that their combined scores total 6(57) = 342. Since 342 = 4 x 72 + 54, we find that at most four of the six students could have scored 72 or more.

The popularity of the mean as a measure of the "middle" or "center" of a set of data is not accidental. Anytime we use a single number to describe some aspect of a set of data, there are certain requirements, or desirable features, that should be kept in mind. Aside from the fact that the mean is a simple and familiar measure, the following are some of it- noteworthy properties:

1-       The mean can be calculated for any set of numerical data, so it always exists.

2-       Any set of numerical data has one and only one mean, so it is always unique.

3-       The mean lends itself to further statistical treatment; for instance, as we shall see, the means of several sets of data can always be combined into the overall mean of all the data.

4-       The mean is relatively reliable in the sense that means of repeated samples drawn from the same population usually do not fluctuate, or vary, as widely as other statistical measures used to estimate the mean of a population.

Finally, let us consider another property of the mean that, on the surface, seems desirable.

5- The mean takes into account each item in a set of data.

Note, however, that samples may contain very small or very large values that are so far removed from the main body of the data that the appropriateness of including them in the sample is questionable. Such values may be due to chance, they may be due to gross errors in recording the data, gross errors in calculations, malfunctioning of equipment, or other identifiable sources of contamination. In any case, when such values are averaged in with the other values, they can affect the mean to such an extent that it is debatable whether it really provides a useful, or meaningful, description of the "middle" of the data.

Example (6)

 The editor of a book on nutritional values needs a figure for the calorie count of a slice of a l2-inch pepperoni pizza. Letting a laboratory with a calorimeter do the job, she gets the following figures for the pizza from six different fast- food chains: 265, 332, 340, 225, 238, and 346.

a)     Calculate the mean, which the editor will report in her book.

b)     Suppose that when calculating the mean, the editor makes the mistake of entering 832 instead of 238 in her calculator. How much of an error would this make in the

c)      Figure that she reports in her book?

Solution:

a) The correct mean is:

(b) The correct mean is:

So that her error would be a disastrous 390-291 = 99.

Example (7)

The ages of six students who went on a geology field trip are 16, 17, 15, 19, 16, and 17, and the age of the instructor who went with them is 54. Find the mean age of these seven persons.

Solution:

The mean is:

But any statement to the effect that the average age of the group is 22 could easily be misinterpreted. We might well infer incorrectly that most of the persons who went on the field trip are in their low twenties.

To avoid the possibility of being misled by a mean affected by a very small value or a very large value, we sometimes find it preferable to describe the middle or center of a set of data with a statistical measure other than the mean; perhaps, with the median, which we shall discuss.

4.3 The Weighted Mean

When we calculate a mean, we may be making a serious mistake if we overlook the fact that the quantities we are averaging are not all of equal importance with reference to the situation being described. Consider, for example, a cruise line that advertises the following fares for single-occupancy cabins on an 11-day cruise:

The mean of these five fares is

But we cannot very well say that the average fare for one of these single occupancy cabins is $5,626. To get that figure, we would also have to know how many cabins there are in each of the categories. Referring to the ship's deck plan, where the cabins are color-coded by category, we find that there are, respectively, 6, 4, 8, 13, and 22 cabins available in these five categories. If it can be assumed that these 53 cabins will all be occupied, the cruise line can expect to receive a total of:

6(7,870)+4(7,080)+8(5,470)+13(4,250)+22(3,460)=250,670

for the 53 cabins and, hence, on the average  per cabin.

To give quantities being averaged their proper degree of importance, it is necessary to assign them (relative importance) weights and then calculate a weighted mean. In general, the weighted mean  of a set of numbers x₁, x₂, x₃, … and x_n, whose relative importance is expressed numerically by a corresponding set of numbers w₁, w₂, w₃, … and w_n is given by:

Here is the sum of the products obtained by multiplying each x by the corresponding weight, and  is simply the sum of the weights. Note that when the weights are all equal, the formula for the weighted mean reduces to that for the ordinary (arithmetic) mean. ..

Example (8)

The following Table shows the number of households in the five Pacific states in 1990, and the corresponding percentage changes in the number of households 1990-1994:

Calculate the weighted mean of the percentage changes using the 1990 numbers of households as weights.

Solution:

 Substituting x₁ = 9.1, x₂ = 8.3, X₃ = 4.5, x₄ = 10.3, X_s = 7.1, W_I = 1,872, W₂ = 1,103, w₃ = 10,381, W₄ = 189, and W_s = 356 into the formula for the weighted mean, we get

Note that we used the symbol » to mean "approximately equal to." We use this symbol only for steps where numerical rounding occurs.

A special application of the formula for the weighted mean arises when we must find the overall mean, or grand mean, of k sets of data having the means  and  and consisting of , and measurements or observations. The result is given by:

Where the weights are the sizes of the samples, the numerator is the total of all the measurements or observations, and the denominator is the number of items ~ in the combined samples.

Example (9)

There are three sections of a course in European history, with 19 students in the 1^st section meeting MWF at 9 A.M., 27 in the 2^nd section meeting MWF at 11 A.M., and 24 in the 3^rd section meeting MWF at 1 P.M. If the students in the 9 A.M. section averaged 66 in the midterm examination, those in the 11 A.M. section averaged 71, and those in the 1 P.M. section averaged 63, what is the mean score for all three sections combined?

Solution:

Substituting  into the formula for the grand mean of combined data, we get

Or 67 rounded to the nearest integer.

4.4 The Median

To avoid the possibility of being misled by one or a few very small or very large values, we sometimes describe the "middle" or "center" of a set of data with statistical measures other than the mean. One of these, the median of n values requires that we first arrange the data according to size. Then it is defined as follows:

The median is the value of the middle item when n is odd, and the mean of the two middle items when n is even.

In either case, when no two values are alike, the median is exceeded by as many values as it exceeds. When some of the values are alike, this may not be the case.

Example (10)

In five recent weeks, a town reported 36, 29, 42, 25, and 29 burglaries. Find the median number of burglaries for these weeks.

Solution:

 The median is not 42, the third (or middle) item, because the data must first be arranged according to size. Thus, we get:

                                                                                                                        25        29        29        36       42

and it can be seen that the middle one, the median, is 29.

Note that in this Example there are two 29's among the data and that we did not refer to either of them as the median - the median is a number and not necessarily a particular measurement or observation.

Example (11)

In some cities, persons cited for minor traffic violations can attend a class in defensive driving in lieu of paying a fine. Given that 12 such classes in Phoenix, Arizona, were attended by 37, 32, 28, 40, 35, 38, 40, 24, 30, 37, 32, and 40 persons, find the median of these data.

Solution:

Ranking these attendance figures according to size, from low to high, we get

and we find that the median is the mean of the two values nearest the middle namely,  

Some of the values were alike in this example, but not affect the median, which exceeds six of the values and is exceeded by equally many. The situation is quite different, however, in the Example that follows.

Example (12)

On the seventh hole of a certain golf course, a par four, nine golfers scored par, birdie (one below par), par, par, bogey (one above par), eagle (two below par), par, birdie, birdie. Find the median.

Solution:

Ranking these figures according to size, from low to high, we get

                                                                                                               2       3         3         3         4         4                                                                                                                        4       4         5

and it can be seen that the fifth value, the median, is equal to par 4.

This time the median exceeds four of the values but is exceeded by only one, and it may well be misleading to think of the median, 4, as the middle of the nine scores. It is not exceeded by as many values as it exceeds, but by definition the median is 4.

The symbol that we use for the median of n sample values x₁, x₂, x₃, …, and x_n (and, hence, or if we refer to the values of y's or z's) is m. If a set of data constitutes a population, we denote its median by .

Thus, we have a symbol for the median, but no formula; there is only a formula for the median position. Referring again to data arranged according to size, usually ranked from low to high, we can write

 Example (13)

Find the median position for

                                                                      (a) n = 17;        (b) n = 41.

Solution:

With the data arranged according to size (and counting from either end)

(a)   and the median is the value of the 9th item;

(b)    and the median is the value of the 21st item.

Example (14)

Find the median position for

(a) n = 16;                                                                                                       (b) n = 50.

Solution:

With the data arranged according to size (and counting from either end)

(a)   and the median is the mean of the values of the 8^th and 9^th items;

(b)   and the median is the mean of the values of the 25^th and 26^th items.

It is important to remember that is the formula for the median position and not a formula for the median, itself. It is also worth mentioning that determining the median can usually be simplified, especially for large sets of data, by first presenting the data in the form of a stem-and-leaf display.

Example (15)

We gave data on the number of rooms occupied each day in a resort hotel during the month of June, and we displayed these data as follows:

2                                                                                                      3                      57

6                                                                                                      4                      0023

13                                                                                                   4                      5666899

(3)                                                                                                   5                      234

14                                                                                                   5                      56789

9                                                                                                      6                      1224

5                                                                                                      6                      9

4                                                                                                      7                      23

2                                                                                                      7                      8

1                                                                                                      8                      1

Use this double-stem display to find the median of these room-occupancy data.

Solution:

When we gave this display in earlier, we did not explain the significance of the figures in the column to the left of the stem labels. As can easily be verified, they are simply the accumulated numbers of leaves counted from either end. Furthermore, the parentheses around the 3 are meant to tell us that the median of the data are on that stem (or else are the mean of two values on that stem).

Since n = 30 for the given table, the median position is , so that the median is the mean of the fifteenth and sixteenth largest values among the data. Since 2 + 4 + 7 = 13 of the values are represented by leaves on the first three stems, the median is the mean of the values represented by the second and third leaves on the fourth stem. These are 53 and 54, and hence the median of the room-occupancy data is. Note that this illustrates why we said that it is generally advisable to arrange the leaves on each stem, so that they are ranked from low to high.

As a matter of interest, let us also mention that the mean of the room-occupancy data is 55.7. It really should not come as a surprise that the median does not equal the mean-it defines the middle of a set of data in a different way. The median is average in the sense that it splits the data into two parts so that, unless there are duplicates, there are equally many values above and below the median. The mean, on the other hand, is average in the sense that if each value is replaced by some constant k while the total remains unchanged, this number k will have to be the mean. (This follows directly from the relationship,  ) In this sense, the mean has also been likened to a center of gravity.

The median shares some, but not all, of the properties of the mean. Like the mean, the median always exists and it is unique for any set of data. Also like the mean, the median is simple enough to find once the data have been arranged according to size, but as we indicated earlier, sorting a set of data manually can be a surprisingly difficult task.

Unlike the mean, the medians of several sets of data cannot generally be combined into an overall median of all the data, and in problems of statistical inference the median is usually less reliable than the mean. This is meant to say that the medians of repeated samples from the same population will usually vary more widely than the corresponding means. On the other hand, sometimes the median may be preferable to the mean because it is not so easily, or not at all, affected by extreme (very small or very large) values. For instance, we showed that incorrectly entering 832 instead of 238 into a calculator caused an error of 99 in the mean. As the reader will be asked to verify, the corresponding error in the median would have been only 37.5.

Finally, also unlike the mean, the median can be used to define the middle of a number of objects, properties, or qualities that can be ranked, namely, when we deal with ordinal data. For instance, we might rank a number of tasks according to their difficulty and then describe the middle (or median) one as being of "average difficulty." Also, we might rank samples of chocolate fudge according to their consistency and then describe the middle (or median) one as having "average consistency."

Besides the median and the mean there are several other measures of central location; for example, the midrange described and the mid quartile. Each describes the "middle" or "center" of a set of data in its own way, and it should not come as a surprise that their values may well all be different. Then there is also the mode.

4.5 Other Fractiles

The median is but one of many fractiles that divide data into two or more parts, as nearly equal as they can be made. Among them we also find quartiles, deciles, and percentiles, which are intended to divide data into four, ten, and a hundred parts. Until recently, fractiles were determined mainly for distributions of large sets of data.

In this section, we shall concern ourselves mainly with a problem that has arisen in exploratory data analysis - in the preliminary analysis of relatively small sets of data. It is the problem of dividing such data into four nearly equal parts, where we say "nearly equal" because there is no way in which we can divide a set of data into four equal parts for, say, n = 27 or n = 33. Statistical measures designed for this purpose have traditionally been referred to as the three quartiles, Q₁, Q₂, and Q₃, and there is no argument about Q₂, which is simply the median. On the other hand, there is some disagreement about the definition of Q₁, and Q₃.

As we shall define them, the quartiles divide a set of data into four parts such that there are as many values less than Q₁ as there are between Q₁ and Q₂ between Q₂ and Q₃, and greater than Q₃. Assuming that no two values are alike, this is accomplished by letting

Q₁ be the median of all the values less than the median of the whole set of data, and Q₃ be the median of all the values greater than the median of the whole set of data.

Example (16)

Following are the high-temperature readings in twelve European capitals on a recent day in the month of June: 90, 75, 86, 77, 85,72,78,79,94,82,74, and 93. Find Q₁, Q₂ (the median), and Q₃.

Solution:

For n = 12 the median position is  and, after arranging the data according to size, we find that the sixth and seventh values among

are 79 and 82. Hence the median is. For the six values below 80.5 the median position is, and since the third and fourth values are 75 and 77,  Counting from the other end, the third and fourth values are 90 and 86, and . As can be seen from the data and also from figure 4.1, there are three values below 76, three values between 76 and 80.5, three values between 80.5 and 88, and three values above 88.

Figure 4.1: Three quartiles of Example 3.16

Everything worked nicely in this example, but n = 12 happened to be a multiple of 4, which raises the question whether our definition of Q₁ and Q₃ will work also when this is not the case.

Example (17)

Suppose that the city where the high temperature was 77 failed to report, so that we are left with the following 11 numbers arranged according to size:

Solution:

For n = 11 the median position is  and, referring to the preceding data, which are already arranged according to size, we find that the median is 82. For the five values below 82 the median position is , and Q₁, the third value, equals 75. Counting from the other end, Q₃, the third value, equals 90. As can be seen from the data and also from figure 4.2, there are two values below 75, two values between 75 and 82, two values between 82 and 90, and two values above 90. Again, this satisfies the requirement for the three quartiles, Q₁, Q₂, and Q₃.

Figure 4.2: Three quartiles of Example 3.17

If some of the values are alike, we modify the definitions of Q₁ and Q₃ by replacing "less than the median" by "to the left of the median position" and "greater than the median" by "to the right of the median position". For instance, for Example (12), we already showed that the median, the fifth value, equals 4. Now, the median of the four values to the left of the median position, Q₁, equals 3, and the median of the four values to the right of the median position, Q₃, equals 4.

Quartiles are not meant to be descriptive of the "middle" or "center" of a set of data, and we have given them here mainly because, like the median, they are fractiles and they are determined in more or less the same way. The midquartile has been used on occasion as another measure of central location.

The information provided by the median, the quartiles Q₁ and Q₃, and the smallest and largest values is sometimes presented in the form of a box plot. Originally referred to somewhat whimsically as a box-and-whisker plot, such a display consists of a rectangle that extends from Q₁ to Q₃, lines drawn from the smallest value to Q₁ and from Q₃ to the largest value, and a line at the median that divides the rectangle into two parts. In practice, box plots are sometimes embellished with other features, but the simple form shown here is adequate for most purposes.

Example (18)

In Example 15 we used the following double-stem display to show that the median of the room occupancy data, originally given before is 53.5:

            2                      3                     57

            6                      4                      0023

            13                    4                      5666899

            (3)                   5                      234

            14                    5                      56789

            9                      6                      1224

            5                      6                      9

            4                      7                      23

            2                      7                      8

            1                      8                      1

(a) Find the smallest and largest values.

(b) Find Q₁ and Q₃.

(c) Draw a box plot.

Solution:

a)     As can be seen by inspection the smallest value is 35 and the largest value is 81.

b)     For n = 30 the median position is and, hence, for the 15 values below 53.5 the median position is. It follows that Q₁ the eighth value, is 46. Similarly, Q₃, the eighth value from the other end, is 62.

c)      Combining all this information, we obtain the box plot shown in Figure 4.3.

  30                40              50              60                 70            80

Smallest value       Q₁          Median         Q₂                     largest

                                                                                  value     

Figure 4.3: Box plot of room occupancy data

Box plots can also be constructed with appropriate computer software or a graphing calculator. Using same data as in Example 18, we reproduced the one shown in Figure 4.4 from the display screen of a TI-83 graphing calculator.

Figure 4.4: Box plot of room occupancy data (TI-83 graphing)

4.6 The Mode

Another measure that is sometimes used to describe the middle or center of a set of data is the mode, which is defined simply as the value that occurs with the highest frequency and more than once. Its two main advantages are that it requires no calculations, only counting, and it can be determined for qualitative, or nominal, data.

Example (19)

The 20 meetings of a square dance club were attended by 22, 24, 23, 24, 27, 25, 20, 24, 26, 28, 26, 23, 21, 24, 24, 25, 23, 28, 26, and 25 of its members. Find the mode.

Solution:

Among these numbers, 20, 21, 22, and 27 each occurs once, 28 occurs twice, 23, -25, and 26 each occurs three times; and 24 occurs 5 times. Thus, the modal attendance is 24.

Example (20)

In Example 12, we gave the scores of nine golfers on a par-four hole as 2, 3, 3, 3, 4, 4, 4, 4, and 5. Find the mode.

Solution:

Since these data are already arranged according to size, it can easily be seen that 4, which occurs four times, is the modal score.

As we have seen in this chapter, there are various measures of central location that describe the middle of a set of data. What particular "average" should be used in any given situation can depend on many different things and the choice may be difficult to make. Since the selection of statistical descriptions often contains an element of arbitrariness, some persons believe that the magic of statistics can be used to prove nearly anything. Indeed, a famous nineteenth-century British statesman is often quoted as saying that there are three kinds of lies: lies, damned lies, and statistics.

4.7 The Description of Grouped Data

In the past, considerable attention was paid to the description of grouped data, because it usually simplified matters to group large sets of data before calculating various statistical measures. This is no longer the case, since the necessary calculations can now be made in a matter of seconds with the use of computers or even hand-held calculators. Nevertheless, we shall devote this section to the description of grouped data, since many kinds of data (for example, those reported in government publications) are available only in the form frequency distributions.

As we have already seen, the grouping of data entails some loss of information. Each item loses its identity, so to speak; we know only how many values there are in each class or in each category. This means that we shall have to be satisfied with approximations. Sometimes we treat our data as if all the values falling into a class were equal to the corresponding class mark, and we shall do so to define the mean of a frequency distribution. Sometimes we treat our data as if all the values falling into a class are spread evenly throughout the corresponding class interval, and we shall do so to define the median of a frequency distribution. In either case, we get good approximations since the resulting errors will tend to average out.

To give a general formula for the mean of a distribution with k classes, let us denote the successive class marks by x₁, x₂ ..., and x_k, and the corresponding class frequencies by f₁, f₂, …, and f_k. Then, the sum of all the measurements is approximated by:

 and the mean of the distribution is given by

Here n is the size of the sample, f₁ + f₂ + ...+ f₃, and to write a corresponding formula for the mean of a population we substitute  for  and N for n.

Example (21)

Find the mean for the distribution of the waiting times between eruptions of Old Faithful Geyser that was obtained in Example before.

Solution:

To get , we perform the calculations shown in the following table, where the first column contains the class marks, the second column consists of the class frequencies shown on page 24, and the third column contains the products x. f:  

Then, substitution into the formula yields  rounded to two decimals.

To check on the grouping error, namely, the error introduced by replacing each value within a class by the corresponding class mark, we can calculate  for the original data, or use the same computer software. Having already entered the data, we simply change the command to MEAN C1 and we get 78.273, or 78.27 rounded to two decimals. Thus, the grouping error is only 78.59 -78.27 = 0.32, which is fairly small.

When dealing with grouped data, we can determine most other statistical measures besides the mean, but we may have to make different assumptions and / or modify the definitions. For instance, for the median of a distribution we use the assumption (namely, the assumption that the values within a class are spread evenly throughout the corresponding class interval). Thus, with reference to a histogram

The median of a distribution is such that the total area of the rectangles to its left equals the total area of the rectangles to its right.

To find the dividing line between the two halves of a histogram (each of which represents  of the items grouped), we must count  of the items starting at either end of the distribution. How this is done is illustrated by the following Example and Figure 4.5.

Figure 4.5: Median of distribution of eruptions of old faithful

Example (22)

Find the median of the distribution of the waiting times between eruptions of Old Faithful.

Solution:

Since, we must count 55 of the items starting at either end. Starting at the bottom of the distribution (that is, beginning with the smallest values), we find that 2 + 2 + 4 + 19 + 24 = 51 of the values fall into the first five classes. Therefore, we must count 55 -51 = 4 more values from among the values in the sixth class. Based on the assumption that the 39 values in the sixth class are spread evenly throughout that class, we accomplish this by adding  of the class interval of 10 to 79.5, which is its lower class boundary. This yields:

Rounded to two decimals.

In general, if L is the lower boundary of the class into which the median must fall, f is its frequency, c is its class interval, and j is the number of items we still lack when we reach L, then the median of the distribution is given by

If we prefer, we can find the median of a distribution by starting to count at the other end (beginning with the largest values) and subtracting an appropriate fraction of the class interval from the upper boundary of the class into which the median must fall.

Example (23)

Use this alternative approach to find the median of the waiting times between eruptions of Old Faithful.

Solution:

Since 2 + 3 + 15 = 20 of the values fall above 89.5, we need 50 – 20 = 35 of the 39 values in the next class to reach the median. Thus, we write

 and the result is, of course, the same.

Note that the median of a distribution can be found regardless of whether the class intervals are all equal. In fact, it can be found even when either or both classes at the top and at the bottom of a distribution are open, so long as the median does not belong to either class.

The method by which we found the median of a distribution can be also used to determine other fractiles. For instance Q₁ and Q₃ are defined for grouped data so that 25% of the total area of the rectangles of the histogram lies to the left of Q₁ and 25% lies to the right of Q₃. Similarly, the nine deciles (which are intended to divide a set of data into ten equal parts) are defined for grouped data so that 10 percent of the total area of the rectangles of the histogram lies to the left of D₁, 10 percent lies between D₁ and D₂, …, and 10 percent lies to the right of D₉. And finally, the ninety-nine percentiles (which are intended to divide a set of data into a hundred equal parts) are defined for grouped data so that 1 percent of the total area of the rectangles of the histogram lies to the left of P₁, 1 percent lies between P₁ and P₂, … and 1 percent lies to the right of P₉₉. Note that D_s and P₅₀ are equal to the median and that P₂₅ equals Q₁ and P₇₅ equals Q3.

Example (24)

Find Q₁ and Q₃ for the distribution of the waiting times between eruptions of Old Faithful.

Solution:

To find Q₁ we must count  of the items starting at the bottom of the distribution. Since there are 2+2+4+19 = 27 values in the first four classes, we must count 27.5 – 27 = 0.5 of the 24 values in the fifth class to reach Q₁. This yields:

Since 2+3+15=20 of the values fall into the last three classes, we must count 27.5 - 20 = 7.5 of the 39 values in the next class to reach Q₃. Thus, we write

Example (25)

Find D₂ and P₈ for the distribution of the waiting times between eruptions of Old Faithful.

Solution:

To find D₂ we must count  of the items starting at the bottom of the distribution. Since there are 2+2+4=8 values in the first three classes, we must count 22-8= 14 of the 19 values of the fourth class to reach D₂. This yields

Since 2+3+15=20 of the values fall into the last three classes, we must count 22-20 = 2 of the 39 values in the next class to reach P₈. Thus, we write

Note that when we determine a fractile of a distribution, the number of items we have to count and the quantity j in the formula on page 73 need not be a whole number.