Question

Explain the process of computing Standard Deviation with the help of an imaginary example.

Answer 1

Standard deviation (SD) is the most widely used measure of dispersion. It is defined as the square root of the average of squares of deviations. It is always calculated around the mean. The standard deviation is the most stable measure of variability and is used in so many other statistical operations. The Greek character denotes it.

• Steps:

1. To obtain SD, the deviation of each score from the mean (x) is first squared (x²).
2. It makes all negative signs of deviations positive. It saves SD from the major criticism of mean deviation which uses modulus x. Then, all of the squared deviations are summed -x²
3. (care should be taken that these are not summed first and then squared).
4. This sum of the squared deviations (x²) is divided by the number of cases and then the square root is taken. Therefore, Standard Deviation is defined as the root mean square deviation.

Calculate the standard deviation for the following distribution:

Groups	120-130	130-140	140-150	150-160	160-170	170-180
ƒ	2	4	6	12	10	8

Solution:
The method of obtaining SD for grouped data has been explained in the table below. The initial steps up to column 4, are the same as those we followed in the computation of the mean for grouped data. We begin with a deviation value of zero that has been assigned to the group. Likewise, other deviations are determined. Values in column 4(fx’) are obtained by the multiplication of the values in the two previous columns. Values in column 5(fx’ 2) are obtained by multiplying the values given in columns 3 and 4. Then various columns have been summed.

(1) Group	(2) ƒ	(3) x'	(4) ƒx'	(5) ƒx'2
120-130	2	-3	-6	18
130-140	4	-2	-8	16
140-150	6	-1	-6 -20	6
150-160	12	0	0	0
160-170	10	1	10	10
170-180	6	2	12 22	24
	N = 40		∑ ƒx'=2	∑ ƒx'2= 74

The following formula is used to calculate the Standard Deviation:

SD = `"t"^2| sumfx^2 - (sumfx)/"N"`