Standard Deviation - Measures of Dispersion, Business Mathematics & Statistics B Com Notes | EduRev

Business Mathematics and Statistics

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B Com : Standard Deviation - Measures of Dispersion, Business Mathematics & Statistics B Com Notes | EduRev

The document Standard Deviation - Measures of Dispersion, Business Mathematics & Statistics B Com Notes | EduRev is a part of the B Com Course Business Mathematics and Statistics.
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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

Standard Deviation - Measures of Dispersion, Business Mathematics & Statistics B Com Notes | EduRev

Different formulae for computing s. d. (a) For simple observations or variates.

Standard Deviation - Measures of Dispersion, Business Mathematics & Statistics B Com Notes | EduRev

(b) For simple or group frequency distribution For the variates x1 , x2 , x3 , ……., xn , if corresponding frequencies are f1 , f2 , f3 , ….., fn

Standard Deviation - Measures of Dispersion, Business Mathematics & Statistics B Com Notes | EduRev

Note : If variates are all equal (say K), then  Standard Deviation - Measures of Dispersion, Business Mathematics & Statistics B Com Notes | EduRev

Example 49: For observations 4, 4, 4, 4  Standard Deviation - Measures of Dispersion, Business Mathematics & Statistics B Com Notes | EduRev

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.

Standard Deviation - Measures of Dispersion, Business Mathematics & Statistics B Com Notes | EduRev

Standard Deviation - Measures of Dispersion, Business Mathematics & Statistics B Com Notes | EduRev

Note : Formula (3) may be written as, for step deviation where  Standard Deviation - Measures of Dispersion, Business Mathematics & Statistics B Com Notes | EduRev

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. Standard Deviation - Measures of Dispersion, Business Mathematics & Statistics B Com Notes | EduRev .
(2) Find the deviations from the mean, i.e., d.
(3) Make squares of the deviations, and add up, i.e. Standard Deviation - Measures of Dispersion, Business Mathematics & Statistics B Com Notes | EduRev .
(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 Standard Deviation - Measures of Dispersion, Business Mathematics & Statistics B Com Notes | EduRev.
(2) Square the deviations, find Standard Deviation - Measures of Dispersion, Business Mathematics & Statistics B Com Notes | EduRev .
(3) Apply the following formula to find standard deviation

Standard Deviation - Measures of Dispersion, Business Mathematics & Statistics B Com Notes | EduRev

Example 50 : Find s.d. of Standard Deviation - Measures of Dispersion, Business Mathematics & Statistics B Com Notes | EduRev 7, 9, 16, 24, 26. Calculation of s.d. by methods (a) Taking deviations of Sum (b) Taking deviations from Assumed Mean

Standard Deviation - Measures of Dispersion, Business Mathematics & Statistics B Com Notes | EduRev

Standard Deviation - Measures of Dispersion, Business Mathematics & Statistics B Com Notes | EduRev

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

Standard Deviation - Measures of Dispersion, Business Mathematics & Statistics B Com Notes | EduRev

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.

Standard Deviation - Measures of Dispersion, Business Mathematics & Statistics B Com Notes | EduRev

(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.

Standard Deviation - Measures of Dispersion, Business Mathematics & Statistics B Com Notes | EduRev

Standard Deviation - Measures of Dispersion, Business Mathematics & Statistics B Com Notes | EduRev

Example 51 : Find the Standard deviation of the following series :

Standard Deviation - Measures of Dispersion, Business Mathematics & Statistics B Com Notes | EduRev

Solution :

Standard Deviation - Measures of Dispersion, Business Mathematics & Statistics B Com Notes | EduRev

Standard Deviation - Measures of Dispersion, Business Mathematics & Statistics B Com Notes | EduRev

For (c) the following formula is used.
The idea will be clear from the example shown below :

Formula is,  Standard Deviation - Measures of Dispersion, Business Mathematics & Statistics B Com Notes | EduRev where d′ = step deviation, i = common factor.

Example 52: Find the standard deviation for the following distribution :

 

Standard Deviation - Measures of Dispersion, Business Mathematics & Statistics B Com Notes | EduRev

Solution :

Standard Deviation - Measures of Dispersion, Business Mathematics & Statistics B Com Notes | EduRev

Standard Deviation - Measures of Dispersion, Business Mathematics & Statistics B Com Notes | EduRev

(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.

Standard Deviation - Measures of Dispersion, Business Mathematics & Statistics B Com Notes | EduRev

Standard Deviation - Measures of Dispersion, Business Mathematics & Statistics B Com Notes | EduRev

Let A (assumed mean) = 49

Standard Deviation - Measures of Dispersion, Business Mathematics & Statistics B Com Notes | EduRev

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

 

Standard Deviation - Measures of Dispersion, Business Mathematics & Statistics B Com Notes | EduRev

 

where σ12 = combined standard deviation of two groups.
σ1 = standard deviation of 1st group.
σ2 = standard deviation of 2nd group.

Standard Deviation - Measures of Dispersion, Business Mathematics & Statistics B Com Notes | EduRev

For Three Groups

Standard Deviation - Measures of Dispersion, Business Mathematics & Statistics B Com Notes | EduRev

Standard Deviation - Measures of Dispersion, Business Mathematics & Statistics B Com Notes | EduRev

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.

Solution:

Standard Deviation - Measures of Dispersion, Business Mathematics & Statistics B Com Notes | EduRev

Standard Deviation - Measures of Dispersion, Business Mathematics & Statistics B Com Notes | EduRev

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