the next figure 7 is uncertain by 14 units. The next figure would be uncertain by at least 140 units and is discarded.

If the first significant figure of the deviation is 8 or 9, the figure in the corresponding position of the average and the figure following are retained but only one significant figure of the measure of deviation is retained.

The calculation of the standard deviation is simplified by means of the

dn +

с

metic mean A of a series of measurements with frequencies f1, ƒ2, fn, if d1, d2,

are the deviations from an estimated mean E, and if AE+c, then

T

FIG. 219.

where σ and σa are the standard deviations of the deviations from A and E respectively.

Since AE+c, we have, by the directed lines in the figure, for the first measurement, d1 = x1 + c.

Hence the sum of the squares of the deviations from E is

But fx

is zero,

or

Σfd2 = Σf(x + c)2 = Σƒx2 + 2cΣƒx + c2Zƒ.

0, since the sum of the deviations from the mean

and hence

Σfd2 = Σfx2 + c2Zƒ.

(1)

(2)

The value of the correction c is (Theorem, Section 139)

Corollary 1. The sum of the squares of the deviations from the arithmetic mean A is less than the sum of the squares of the deviations from any other number E.

For, from (1), Σfd2 is greater than fr2 by the positive num

ber c2Σf.

Corollary 2. The standard deviation of a series of measurements is a minimum when the deviations are measured from the arithmetic mean.

This follows from equation (2).

An application of this theorem is made in

EXAMPLE 2. Find the arithmetic mean and the standard deviation for the following distribution of grades in geometry.

Hence the average grade in the above distribution is 75.2% and the standard deviation is 12%.

The absolute value of two measures of variability may be the same and yet their significance be quite different, for instance, if two averages and their measures of variability are 255 and 2505, the first indicates greater relative variability than the second.

A measure of the relative variability of a set of measurements is obtained by dividing the measure of deviation d by

d

the average a. The quotient is called a coefficient of relative

a

variability. The quantity v = 100

100, which gives the ratio of the

standard deviation to the arithmetic mean expressed as a percentage, is called the coefficient of variation.

EXERCISES

1. (a) Find the median grade and the quartile deviation of the following distribution of grades in algebra, the given grades being at the lefthand ends of the intervals.

(b) Find the arithmetic mean and the standard deviation of the distribution.

(c) Find the mean deviation from the median.

(d) Find the mode and the median deviation from the mode.

(e) Draw the frequency curve of the distribution and show the graphical significance of each of the averages and the corresponding dispersion. Which pair of numbers best represents the distribution and variability? 2. By the use of directed lines show that the mean deviation from a point of reference of a set of seven points placed arbitrarily on the x-axis is least when the point of reference is the median point of the set.

3. Compare the coefficients of variability from the arithmetic mean of a group of men and a group of women measured with respect to the number of associations set up by a series of words.

4. The following table gives the errors in minutes in the predictions of high water at Portsmouth during three months in 1897. Find the median deviation, the mean deviation, and the standard deviation. Approximately what fractional part of the standard deviation is the median deviation? the mean deviation?

Errors

Frequencies

0-5, 6-10, 11-15, 16-20, 21-25, 26-30, 31-35, 52 4, 1

69,

50,

25,

10,

11,

7,

141. Equation of the Frequency Curve Representing a Symmetrical Distribution. The derivation of the equation is based on the theory of probability.

If a coin is tossed four times, the different ways in which it may fall are heads every time, heads three times and tails once, heads twice and tails twice, heads once and tails three times, and tails every time. The probabilities of the different ways in which it can fall are the terms of the expansion

(1 + 1)^ = (})1 + 4(1)3(1)1 + 6(1)2(1)2 + 4(1)1(1)3 + (1)1 1⁄2 (1 + 4 + 6 + 4 + 1)

=

Now consider the adjoined frequency table in which an arbitrary interval Ax is chosen, and in which the frequencies are the terms of the expansion above. The frequency polygon (page 405) plotted from this

m – 2▲x, – ▲x, 0, Ax, 2▲x

f

table is symmetrical with respect to the y-axis.

The standard deviation for this table is

Σfm2 (-2x)2 + 1(− Ax)2 + §·02 + †(Ax)2 + r2(2^x)2

Therefore σ = Ax.

1

=

(Ax)2.

6

Plotting the table, using Ax = σ as the unit on the x-axis we obtain the upper frequency polygon in the figure.

If the coin is tossed six times, the probabilities of the ways in which it can fall are given by the terms of the expansion

(3 + 1)° = 126 (1 + 6 + 15 + 20 + 15 + 6 + 1).

The standard deviation of the frequency table analogous to the above is found to be σ = VAx, so that Ax = √3σ = .80. Δι

The frequency polygon for this case is the middle one in the figure. It is assumed that σ has the same value in both cases, so that the value of Ax is smaller than in the preceding case.

If the coin is tossed eight times, a similar procedure gives the lowest polygon in the figure. In this case σ =

These three cases may be regarded as the particular cases obtained by tossing a coin an even number of times, 2n, for n = 2, 3, 4. In each case the frequency polygon is symmetrical with respect to the y-axis. As n increases and σ keeps the same value, the vertices of the polygon get closer together. A comparison of the values of σ for n = 2, 3, 4 shows that in each case 202 = n(Ax)2.

With these preliminaries, consider the expansion

(1

1+2nC1 + 2nC2 + ... + 2nCn + ... + 2nCntr