Standard Deviation

The standard deviation concept was first introduced by Karl Pearson in 1823. It is  the most important and widely used measure of studying the dispersion. Its significance lies in the fact that it is free from those defects from which the earlier techniques suffer and satisfies most of the properties of a good measure of dispersion. The Standard deviation is also termed as root mean square deviation for the reason that it is the square root of the mean of the squared deviation from the arithmetic mean. The Standard deviation is represented by the small Greek letter Q (pronounce as sigma.)

The standard deviation measures the absolute dispersion (or variability of distribution), the greater the amount of dispersion or variability the greater is the standard deviation, the greater will be the magnitude of the deviations of the values form their mean. If we have two or more comparable series with identical or nearly identical means, it is the distribution with the smallest standard deviation that has the most representative mean. And hence standard deviation is extremely useful in judging the representativeness of the mean.

Deviations taken from actual mean: When deviations are taken from actual mean the following formula is applied:

Σ = √Σ x2 / N

X = (X - X)

Steps:

1) Calculate the actual mean of the series   ...X...

2) Take the deviations of the items from the mean and find (X-X). Denote these deviations by x.

3) Square these deviations and obtain the total Σ X2.

4)Divide Σ x2 by the total number of observations N and extract the square - root this will the value of standard deviations.

Deviations taken from assumed mean: When the actual mean is in fractions say it is 123.674 it would be too cumbersome to take deviations from it and then obtain the squares of these deviations. In such either the mean may be approximated or else the deviations be taken from an assumed mean and the necessary adjustment made in the value of the standard deviation. The former methods of approximation are less accurate and therefore invariably and in such cases deviations are taken from the assumed mean.

When deviations are taken from assumed mean the following formula is applied:

Σ = √ Σ d2 / N - (Σ d)2/N

Steps:

1) Take the deviations of the item from an assumed mean obtain (X - A) denote these deviations by d. Take the total of these deviations and obtain Σ.D.

2) Square these deviations and obtain the total Σd2

3) Substitute the values of Σd2 and N in the above formula.

Illustration: 

Blood serum cholesterol levels of 10 persons are as

24, 26,290,245, 255, 288, 272, 263, 277, 251

Calculate standard deviation with the help of assumed mean.

Solution:

Calculation of standard deviation by the Assumed mean method

Σ =  √ d² / N - (Σ d / N)²

Σ d² = 2689, Σ d = + 1 N = 10

Σ = √ 2689 / 10 - (1/10)²

= √ 268.9 - 0.01 = 16.398