Variance and Standard Deviation

The standard deviation is the most used measure of dispersion.  The value of the standard deviation tells how closely the values of a data are clustered around the mean.

The standard deviation is obtained by taking the positive square root of the variance.

Notation: 

Population variance =
Sample variance  =

Variance and Standard Deviation for Ungrouped Data

Formulas:

Population Standard Deviation:
Sample Standard Deviation:

Example 4.1: Following are the 1993 earnings (in thousand kwacha) before taxes for all six employees of a small company.

29.50

16.20

35.45

21.35

49.70

24.60

Calculate the variance and standard deviation for these data.

Solution

Let x denote the 1993 earnings before taxes of employees of this company. The values of   and are calculated in Table 4.1.

Table 4.1

x	x2
29.50	870.2500
16.20	262.4400
35.45	1256.7025
21.35	455.8225
49.70	2470.0900
24.60	605.1600
176.80	5920.4650

Since the data on earnings are for all employees of this company, we will use the population formula to complete the variance.  Thus the variance is . . .

The standard deviation is . . .

Variance and Standard Deviation for Grouped Data

Formulas:

Where  is the population variance,  is the sample variance and x is the class midpoint.

The standard deviation is given by:

Population standard deviation:
Sample standard deviation:

Example 4.2: The following table gives the frequency distribution of the number of orders received each day during the past 50 days at the office of a mail-order company.

Number of Orders	Number of Days
10 – 12	4
13 – 15	12
16 – 18	20
19 – 21	14

Calculate the variance and standard deviation.

Solution:

All the information required for the calculation of the variance and standard deviation appears in Table 4.2.

Table 4.2

Number of Orders	f	x	fx	fx2
10 – 12	4	11	44	484
13 – 15	12	14	168	2352
16 – 18	20	17	340	5780
19 - 21	14	20	280	5600
Total	50	-	832	14216

Because the data set includes only 50 days, it represents a sample.  So we will use the sample formula.  By substitution, the sample variance is:

Hence the standard deviation is