TABLE I

TABLE GIVING DATA FOR COMPUTING THE ARITHMETIC MEAN BY THE "STEP-DEVIATION" METHOD FOR FREQUENCY GROUPS FROM AN ASSUMED AVERAGE

1502 ÷ 434

the group) +(-$3.46)

=

=

22

= - 3.46. - 3.46 X $1.00 (the size of $3.46. $12.50 (the assumed average) = the true average.

$9.04

Where groups are not uniform in size, this method cannot be employed without considerable difficulty. When they are uniform, however, much trouble in multiplying is avoided by computing the deviations in round numbers and subsequently by converting them back into terms of the size of the "step." The following table illustrates the method when groups are of unequal size.1 In such cases it is far simpler to proceed in the regular manner by multiplying through in the first instance.

1 This method involves "averaging averages and is of doubtful value.

TABLE J

TABLE GIVING DATA FOR COMPUTING THE ARITHMETIC MEAN BY THE "STEP-DEVIATION" METHOD FROM AN ASSUMED AVERAGE WHEN THE GRoups Are of Unequal Size 1

1 Data taken from Report of the Tariff Board on Schedule "K," Vol. IV.,

Part 5. House Doc. 342, 62d Congress, 2d Session, p. 997.

Notes, 2, 3, 4, 5, and 6 on following page.

In summarizing the discussion of the arithmetic mean, attention should be called to the fact that it is easily understood, is readily calculated, is in everyday use, and is affected by all the items in a series. Indeed, when nothing more is wanted, as a summarizing expression, than the total divided by the sum of the parts, it thoroughly meets the need. But in statistical analysis of economic problems the needs generally run far beyond this. It is frequently the detail which is of most importance and which is so often concealed by the arithmetic mean. It is too susceptible to the extraordinary, too much affected by the exceptional, to serve all purposes equally well. Various checks may be imposed in order to test its validity for a definite purpose. The details themselves may be submitted. But this is often impossible, since the employment of an average is an indication of a desire or of a necessity to be free from detail. Other averages may be computed for purposes of comparison, and it is to a discussion of these to which we now turn.

2 Width of group assumed to be the same as that of the class to which it belongs.

.0509 X 2 (the width of the group) = $.1310 (average of the first group).

31266 24,885 = .0509. $.001018. $.13+ $.001018 = 4 2136 5076 == .421.

=

.421 X 5¢ (the width of the group) $.254 (average of the second group).

5 $.40 is the average of the third group. 6- 76 ÷ 202 = - .376.

--

=

=

.376 X 15¢ (the width of the fourth ($.05640) $.6186 (average of the

=

IV. THE MEDIAN

1. What the Median Is

The median has been defined as the item in a series, when arranged consecutively, which divides the distribution into equal parts. While it is generally called an average it is more accurately a measure of partition or distribution. It can be said to be characteristic of the other members of a series only in case they are uniformly dispersed around it. It divides frequencies into equal parts and not the units to which they apply. Indeed, the exact size of an item measured need not be known. The only thing necessary is to be able to place it in a distribution so that the order of arrangement is consecutive. Unlike the arithmetic mean, it is not primarily a mathematical concept, since it may be used where numerical significance is not attributed to the factors averaged, as, for instance, in the grading of pupils, salesmen, etc., simply by placing them in their order of excellence. This, of course, means nothing more than that relative rank is established. The middle position is then determinable. Yet it is like the arithmetic mean in the fact that the middle or median quantity itself need not be represented in a series. How accurately a distribution is characterized by the median alone depends almost entirely upon its nature. Perhaps we can get a clearer view of its meaning if we compute it for a variety of distributions. Remembering that it is that item which divides a series, consecutively arranged, into equal parts, and substituting n for the number of items in the series, n+ 1 the expression

putation.

2

may be used as a basis for its com

2. How the Median is Computed

Using the data in Table A, p. 242, but rearranging the units in an ascending order (a thing unnecessary in the computation of the arithmetic mean), we get the following series:

9 +1

=

2

n+ 1

when n = 9, we get

2.

5, i.e. the fifth item in the series divides it into

equal parts. Counting down from the smallest item, or up from the largest one a matter of indifference $4.00 is found to be the median. It should be noticed, however, that the thing which is really divided into two equal parts is the total frequency, and not the items to which the frequencies apply. That is, $4.00 is only $2.00 away from the first item, but $4.00 away from the last. Moreover, the $2.00 in