THE PURPOSE of per cent figures is to indicate more clearly the relative size of two or more numbers. They achieve this clarification in two ways. First, they reduce all numbers to the range of easy multiplication and division: percentages are usually numbers smaller than 100. Second, they translate one of the numbers, the base, into the figure 100, which is easily divided into and by other numbers, thus making it less difficult to see the precise relationship of the part to the total.

REDUCING TO A COMMON BASE

Figure 1 shows the principle of the transformation from Table 1 to Table 2. By equalizing the base of both number columns to 100 and by reducing the other figures proportionately, direct comparison becomes easy. Thus we can see that General Motors is considerably stronger in New York than it is in North Carolina, while for the Ford cars the situation is reversed; the share of Chrysler and American Motors is approximately the same in both areas.

Fifty-three per cent of the voters declared themselves for the Democratic candidate and 47 per cent for the Republican. To be sure, mathematically the expressions 1,592/3,000 and 53 per cent are equivalent, but 53 per cent is the simpler and, therefore. the preferable expression.

COMPARING INCREASE OR DECREASE

Suppose company A increased its sales volume from one year to the next from $1 million to $2 million. Suppose Company B, a larger competitor of A, increased its sales during that same year from $4 million to $7 million, and one is asked to compare the sales progress of the two companies. In this case the following two comparisons can be made:

1. A increased its sales by $1 million.
2. B increased its sales by three times that amount, that is, by $3 million.

1. A increased its sales by 100 per cent.
2. B increased its sales by only 75 per cent.

Both comparisons are correct in the sense that they reflect true Information. But Comparison I gives the impression that Company B performed better than Company A, while Comparison II gives the opposite impression. Strictly interpreted, the two comparisons do not contradict each other. However, the fact remains that the percentage comparison (II) suggests a development opposite to that expressed by the absolute figures.

It is in this realm of comparisons that the problem of per cent computation acquires a new aspect. As long as per cent figures are nothing more than a simplified description of a set of numbers, this problem does not arise. On the other hand, if increase or decrease is meant to suggest--explicitly or implicitly--the underlying causes of these variations, it becomes necessary in each case to decide whether per cent figures are to be used at all and, if so, on what base they are to be computed.

ABSOLUTE OR PER CENT COMPARISON

To be sure, such decisions are not always made on scientific grounds. Those who want to color the picture in favor of Company B will use the first reading; those who want to show how well A did will use the second.

1. The initial advantage of each company, such as the size of its sales organization before the expansion, the number of its customers who serve as potential sources of recommendation, and so on.
2. The quality of the company's operation during the year to which the comparison refers.

By expressing the increase as a percentage of the company's sales volume at the beginning of the observation period, we project this thought: It would not be fair to compare the dollar volume increase of a big company with that of a small company; the proper comparison is the relative, per cent increase. The handicap resulting from Company A's smaller size is neutralized by expressing each company's increase as a percentage of the company's sales volume at the beginning of the period.

REJECTING THE PER CENT COMPUTATION

In passing, we might note that a firm's statement that its sales had doubled from $1 million to $2 million can never be in itself a measure of the quality of the firm's operations. If, for instance, every other firm in this field at least trebled its sales in that period, and none increased its sales by less than $2 million, the increase by $1 million would reflect the lowest increase of all

THE LOGIC BEHIND THE DECISION

One can readily think of other situations to which this reasoning applies. Suppose we wanted to compare the growth of two cities M and N. as in Table 3.

Should we say: "City M increased by 20 per cent and City N by 30 per cent," or should we say "City M increased by 200,000 and City N by 150,000''? If this growth is a normal increase, caused by an increasing birth rate and a declining mortality, then it will be fair to use the per cent computation, thereby indicating that population size at the beginning of the observation period is a crucial determinant of the absolute growth. If on the other hand, the population growthis not a normal increase, but one due to population shifts in connection with manpower needs, we might argue that the size of the city in 1950 will only slightly affect the number of newcomers. The larger city will not necessarily show the greater increase in population; the city with more new industries and new job opportunities will. Hence, we will say that City M showed a "greater" increase (200,000) then City N (150,000).

THE CEILING EFFECT

Suppose we go fishing with a net in a lake that is fairly full of fish. The first sweep will bring us X number of fish. On the second sweep we will expect to catch fewer fish simply because we have taken some out on our first sweep and their relative density in the lake will have decreased. On the third sweep we shall expect to catch even fewer, and so forth.

Readership of an advertisement is usually measured by the number of people who have read it as a percentage of all people who opened the magazine or newspaper in which the advertisement appeared. This yardstick has been used not only to measure individual advertisements, but to derive more general conclusions about the readership of different types of advertisements. For example, it seemed desirable to learn how readership of an advertisement is affected by switching from black and white to color. In Table 4 are three sets of data which are expected to, provide such a, generalization.

There are, to begin with, two obvious ways of measuring the increase in readership if the advertisement is printed in color instead of in black and white, as illustrated in Table 5.

Advertisement A, for instance, reached in black and white 42 per cent of all readers, and in color, 52 per cent, a difference of 10 percentage points; or of 24 per cent as against the starting point of 42 per cent (= 100 per cent).

Suppose, however, following the fishing paradigm, we express these 10 percentage points as the per cent of all readers who had not seen the black and white advertisement, that is, 100Ä 42 = 58 per cent. Taking this 58 per cent figure as the base ( 100 per cent), the 10 per cent increase amounts to 17 per cent. Analogous computations for all three advertisements yield the results shown in Table 6.

The general solution is that the percentage-point increase following the move to color is the greater the smaller the black and white readership of the advertisement was to begin with. We arrive at an approximately equal increase if we measure the increase in readers as a percentage of those potential readers who had not seen the black and white advertisement.

PER CENTS OFFER ONLY APPROXIMATE SOLUTIONS

In this sense, per cent comparisons will offer at times only a crude measure. This is the price paid for the great simplicity of per cent comparisons. To the trained statistician, it will be clear that the per cent computation is often only a simple substitute for a multiple correlation analysis. The latter, however, is available only if empirical data are at hand to which correlation analysis can be applied. In our example, we have no statistical data to prove or disprove the assumption that the success of a company is, on the average, proportionate to its size. Our decision to express success in terms of a percentage of the preceding year's business volume would have to be arrived at on the basis of impressionistic reasoning, not through precise statistical data.

SUMMARY

Special problems arise when increases or decreases of various kinds are compared. Whether or not to use per cent figures, and if so, on what base they are to be computed, can be decided only after careful scrutiny of the problem involved. Per cent figures are used here as a crude substitute for regression analysis and other, more sophisticated, statistical techniques.