Imágenes de páginas
PDF
EPUB

The average obviates the necessity of stating all the measurements from which it is derived, and a measure of variability is a single number characterizing the deviation from the average. A series of measurements can thus be summarized by two numbers which are usually written in the form ad, where the first number gives the average value chosen and the second gives a measure of the deviations of the items of the series from the average.

The principal measures of dispersion or variability of a distribution from an average are (a) the quartile deviation, (b) the mean deviation, (c) the median deviation or probable error, (d) the standard deviation.

The quartile deviation is defined by the equation Q

3

Q3 - Q1

2

where Q1 and Qs are the first and third quartiles respectively. It is the simplest measure of deviation to calculate.

The quartile deviation for the distribution of incomes in Example 2, Section 139, is

[blocks in formation]

Hence the group is summarized by the median value of the group and the quartile deviation in the form 822.6 ± 132.1. This means that approximately 50% of the incomes lie between $822.6 $132.1 = $690.5 and $822.6 + $132.1 = $954.7.

The mean deviation is the arithmetic mean of the numerical values of the deviations from an average. The method of calculation for a frequency distribution is shown in

EXAMPLE 1. Find the mean deviation from the arithmetic mean of the distribution of incomes in Example 1, Section 139.

In this example the estimated mean is E = 850, the numerical value of the correction is c = 1.82 units = 0.0182 class intervals and the true mean is A = 848.2.

The sum of the numerical values of the deviations from E is 306 +299 605 class intervals.

The deviation from A of each of the 176 items less than A is 0.0182 class intervals less than the deviation from E.

The deviation from A of each of the 73+ 135 208 items greater than

A is 0.0182 class intervals greater than the deviation from E. Hence the sum of the numerical values of the deviations from A is

605 + 208 × .0182 - 176 × .0182 = 606.

[merged small][merged small][merged small][merged small][merged small][ocr errors][merged small]

The median deviation or probable error. If all the deviations from some one of the averages are arranged in order of magnitude without regard to sign, the median deviation is calculated in the same way as the median of the distribution.

The middle 50% of the items come within the range of the median deviation if the median is the average used to represent central tendency.

Approximately 50% of the items come within the range of the quartile deviation, but these cases are not necessarily the middle 50% since the median does not lie exactly halfway between the quartiles except in a symmetrical distribution. The median deviation is usually called the probable error and will be discussed further in Section 142.

The standard deviation is defined by the equation

[merged small][merged small][merged small][merged small][ocr errors][ocr errors][ocr errors][merged small][ocr errors]

where d1, d2, . from the average frequencies fi, f2,

dn are the deviations of the measures chosen. If the deviations occur with the fn, then the standard deviation is

[merged small][ocr errors]

This measure of dispersion gives more weight to extreme cases than the other measures of variability, and while more tedious to calculate is more generally used.

In all deviation measures retain at most two significant figures. The reason for this rule is as follows. Consider a length l 324.57 cm. with a mean deviation of .14 cm. The mean deviation indicates that the figure 5 in 1 in the first position after the decimal point is uncertain by 1 unit and that

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

[ocr errors]

Xn are deviations from the arith

[ocr errors][ocr errors][merged small]

dn +

Theorem. If X1, X2, metic mean A of a series of measurements with frequencies f1, f2,..., fn, if d1, d2, . . ., dn are the deviations from an estimated mean E, and if A E+c, then

[merged small][ocr errors][merged small][ocr errors]
[ocr errors][merged small][merged small]

where σx and oa are the standard deviations of the deviations

[ocr errors]

from A and E respectively.

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

Similarly, d2 = x2 + C, . . ., dn = xn + c.

d2

Hence the sum of the squares of the deviations from E is Σƒd2 = Σƒ(x + c)2 = Σƒx2 + 2cZfx + c2Zƒ.

But Zfx

0, since the sum of the deviations from the mean is zero, and hence

Σfd2 = Σfx2 + c2Σf.

Transposing and dividing by Zƒ,

(1)

[merged small][ocr errors][ocr errors][merged small][merged small][merged small][merged small][ocr errors][subsumed][merged small]

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

[blocks in formation]

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), Zfd2 is greater than fr2 by the positive num

ber c2f.

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.

[merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors]

Let the estimated mean be E = 72.5 and let the class interval be taken as a unit.

49

49

+ class intervals

=

90

× 5% 90

2.7 %.

Zfd
The correction is c=-
Zf
Hence the mean is A = E + c = 72.5 +2.7 = 75.2.
The standard deviation of the deviations from E is

[merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small]

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

α

variability. The quantity v = 100 100, which gives the ratio of the

A'

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.

[blocks in formation]

(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.

[blocks in formation]
« AnteriorContinuar »