Standard Deviation :
In calculating mean deviation we ignored the algebraic signs, which is mathematically illogical. This drawback is removed in calculating standard deviation, usually denoted by ‘ σ ’ (read as sigma)
Definition : Standard deviation is the square root of the arithmetic average of the squares of all the deviations from the mean. In short, it may be defined as root-mean-square deviation from the mean.
If x is the mean of x1 , x2 , ……., xn , then σ is defined by
Different formulae for computing s. d. (a) For simple observations or variates.
(b) For simple or group frequency distribution For the variates x1 , x2 , x3 , ……., xn , if corresponding frequencies are f1 , f2 , f3 , ….., fn
Note : If variates are all equal (say K), then
Example 49: For observations 4, 4, 4, 4
Short cut method for calculating s.d. If x (A. M.) is not an integer, in case (1), (2) ; then the calculation is lengthy and time consuming. In such case, we shall follow the following formulae for finding s.d.
Note : Formula (3) may be written as, for step deviation where
Computation for Standard Deviation :
(A) For individual observations computation may be done in two ways :
(a) by taking deviations from actual mean. Steps to follow––
(1) Find the actual mean, i.e. .
(2) Find the deviations from the mean, i.e., d.
(3) Make squares of the deviations, and add up, i.e. .
(4) Divide the addition by total number of items, i.e., find d2 / n and hence make square root of it.
(b) by taking deviations from assumed mean. Steps to follow––
by taking deviations from assumed mean. Steps to follow––
(1) Find the deviations of the items from an assumed mean and denote it by d find also .
(2) Square the deviations, find .
(3) Apply the following formula to find standard deviation
Example 50 : Find s.d. of 7, 9, 16, 24, 26. Calculation of s.d. by methods (a) Taking deviations of Sum (b) Taking deviations from Assumed Mean
Here the average or A.M. 16.40 and the variates deviate on an average from the A.M. by ` 7.66. For method (b) : Let A (assumed mean) = 16
Note : If the actual mean is in fraction, then it is better to take deviations from an assumed mean, for avoiding too much calculations. (B) For discrete series (or Simple Frequency Distribution). There are three methods, given below for computing Standard Deviation. (a) Actual Mean, (b) Assumed Mean, (c) Step Deviation. For (a) the following formula are used. This method is used rarely because if the actual mean is in fractions, calculations take much time.
(In general, application of this formula is less) For (b), the following steps are to be used :–
(i) Find the deviations (from assumed mean), denote it by d.
Example 51 : Find the Standard deviation of the following series :
For (c) the following formula is used.
The idea will be clear from the example shown below :
Formula is, where d′ = step deviation, i = common factor.
Example 52: Find the standard deviation for the following distribution :
(C) For Continuous Series (or group distribution) :
Any method discussed above (for discrete series) can be used in this case. Of course, step deviation method is convenient to use. From the following example, procedure of calculation will be clear
. Example 53 : Find the standard deviation from the following frequency distribution.
Let A (assumed mean) = 49
MATHEMATICAL PROPERTIES OF STANDARD DEVIATION :
Combined Standard Deviation.
We can also calculate the combined standard deviation for two or more groups, similar to mean of composite group. The required formula is as follows
where σ12 = combined standard deviation of two groups.
σ1 = standard deviation of 1st group.
σ2 = standard deviation of 2nd group.
For Three Groups
Example 54 : Two samples of sizes 40 and 50 respectively have the same mean 53, but different standard deviations 19 and 8 respectively. Find the Standard Deviations of the combined sample of size 90.