ately asymmetric group the mode, median and mean lie in the order named with the mean toward the longer branch of the curve, as in the figure.

Frequencies

A good approximation to the value of the mode is obtained by

the use of the formula

Mode Mean

Mode

Median--.
Mean-

FIG. 217.

This formula is based on the assumption, which has been ob

served to hold approximately in a large number of cases, that the median lies one third of the distance from the mean toward the mode. The application of this formula shows that the modal income in Examples 1 and 2 is

The best method of determining the mode is to find the equation of the smooth frequency curve that best fits the data,

500, 600, etc., as abscissas, the ordinates being respectively the number of families with incomes less than 500, 600, etc. For convenience in plotting $100 is taken as a unit so that the abscissas are 5, 6, etc.

Let M and N be the projections on the y-axis of A and B, the end points of the curve. Divide MN into four equal parts by the points D', E', F'. Let the perpendiculars to the y-axis at these points cut the curve at D, E, F, respectively. Then the abscissa of D is the median, M. = 820, approximately, and the abscissas of E and F are the quartiles, Q1 = 700 and Q3 = 960.

The mode is represented graphically by the abscissa of the point of inflection I. This point may be determined roughly by inspection, or by placing a ruler tangent to the curve and rolling it along the curve so that it remains tangent. The direction of rotation of the ruler changes when the point of contact coincides with the point of inflection. In this figure the mode, Mo, appears to be about 800.

The geometric mean is defined by the equation,

n

G

V mim 2

Mn•

It is most easily calculated by using logarithms, since

log G

log m1 + log m2 +

+ log mn

n

The geometric mean is used in averaging rates of increase such as arise in the study of the growth of population, growth of skill in an individual, and relative changes in the prices of commodities.

The trend of prices in a series of years is gauged by finding the ratio of the average price of a commodity in any year to that in a particular year which is chosen as a base and given the arbitrary value 100. Numbers which are determined for the purpose of showing the trend in prices are called index numbers.

If the index numbers of three commodities for one year are a, b, c, and for a second year are ar, bs, ct, then the ratios for the three commodities are r, s, t, and the geometric mean of the ratios is vrst.

3

The geometric mean of the index numbers for the first year is abc and for the second is Vabcrst. The ratio of these two numbers is Vrst.

Hence the geometric mean of the ratios of the index numbers of several commodities for two years is equal to the ratio of the

geometric mean of the index numbers for the second year to that

of the first.

This property of the geometric mean is the reason for its use in averaging index numbers.

The harmonic mean is defined by the equation

It is used in finding the average amount of work performed in a given time, and the average amount of a commodity purchased for a given price.

In the equation v =

s

ť

if s is constant and t varies, then the

average time for a given distance would be found by the har

The mode determined as in Example 3 can be found roughly more easily than the median or mean can be calculated. The calculation of the precise true mode is more difficult than that of any of the averages. In the case of discrete variation, as for instance the number of seeds in an apple, the mode may be the only average that will mean anything, as the values of the other averages are quite likely not to occur in the series. The mode is the most probable value of a distribution of the asymmetrical type, since it is the value that occurs most frequently in the distribution, and hence it is the best representative value of central tendency. It is the typical case. It is not useful if it is desirable to give weight to extreme variation, since all the items of the group do not enter into its determination.

The median ranks after the mode in ease of determination, and usually may be located more precisely.

If the unit of measure is difficult or impossible to determine and a ranking of the items is all that can be attained, as in the measurement of scholarship, the median is the best representative value. If the values of some of the items are given vaguely, as for instance if items of an upper class are given as greater than some value, the median can be determined more precisely than the mean.

All the items enter into the determination of the median but extreme cases affect its value but slightly. Changes in the values of extreme cases would affect the value of the mean without disturbing the value of the median.

The arithmetic mean is the average most generally employed. It is the most familiar of the averages, is the most precise when all the items are given, gives weight to extreme deviations, which is desirable in certain cases, and is affected by every item in the distribution. It may be determined if the aggregate and the number of items are known though knowledge of the values of the items is lacking, and conversely the aggregate may be determined if the mean and the number of items is known. This property is not possessed by the other averages.

The harmonic mean and geometric mean are used less frequently than the other averages as they are unfamiliar and more difficult to calculate. The arithmetic mean is sometimes incorrectly employed in place of the harmonic mean in averaging time rates, and sometimes incorrectly in place of the geometric mean in averaging rates of increase. If variations are measured by their ratio to, rather than by their difference from, the average, then the geometric mean is the best average to employ.

EXERCISES

1. Which average is meant in the following: average student, average wage in an industry, average daily temperature, average stature, average number of potatoes of a given species in a hill, average annual rainfall, average gain per year in height of a child, average ability in arithmetic,

average rate of increase of population, average time in doing a piece of work?

2. Find the mean, median, and mode of the distribution in the example in Section 138.

3. Ten men in a department can complete a piece of work in the following number of minutes respectively, 45, 50, 60, 60, 60, 65, 65, 70, 75, and 85. What is the average time for the work?

4. The population of a city increased in a decade from 185,000 to 260,000. What was the average annual rate of increase?

5. By the probable duration of life of a man m years of age is meant the number of years which he has an even chance of adding to his life. By the expectancy of life for a man of m years is meant the arithmetic mean of the number of additional years of life enjoyed by all men m years of age. Find the probable duration and the expectancy of life of a man 20 years old (use the table in Section 137).

6. The following table gives the distribution of wages per week of a group of laborers.

Wages (mid-values)
Frequency

1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23 5, 23, 50, 80, 105, 130, 160, 165, 148, 120, 36, 3

Find the mean, median and mode of the distribution. Which average represents the table best?

7. What is the average age at death for each disease in the tables in Exercise 6, page 387. What average should be used? Determine by inspection of the tables which are diseases of children and which of adults. Which warrants being called a vacation disease? Why?

8. The frequencies of distribution of budgets of a group of college students with a class interval of $50 starting at $350 and running to $1800 inclusive were 4, 15, 21, 26, 39, 46, 52, 32, 34, 24, 17, 17, 14, 11, 8, 10, 6, 6, 4, 4, 3, 3, 1, 1, 0, 2, 1, 0, 0, 3. Calculate the median and quartiles algebraically and graphically. Which average would best represent the distribution?

9. In 1913 the index numbers for steak, bacon, chickens, eggs, butter, milk, flour, potatoes, sugar were 94, 95, 97, 91, 108, 100, 100, 90, 100. In 1918 the numbers in the same commodities were respectively 131, 179, 170, 177, 151, 151, 200, 188, 193. Calculate the average index numbers in each year and the relative increase in prices.

140. Measures of Variability. Next in importance to selecting an average to represent a group of measurements is the determination of a measure of the extent to which the other values cluster about or are dispersed from the average chosen, that is, a measure of the variability of the group with respect to the average.