Chapter 05Measures of Variation

Afjal HossainAssistant ProfessorDepartment of Marketing, PSTU

Meaning and Definition

Measures of Variation:

The statistical techniques through which the variation is

measured are called Measures of Variation.

Significance of Variation:

To determine the reliability of an average

To serve as a basis for the control of the variability

To compare two or more series with regard to their variability and

To facilitate the use of other statistical measures

Measurements of Variation

Range

Average Deviation

Standard Deviation

Variance

Measurements of Variation Range

In case of Ungrouped Data, Range is thedifference between the largest and smallestobservations in a distribution. In case of GroupedData, Range is called co-efficient of range.

Here,

minmax XXRRange/

minmax

minmax

XX

XXRange oft Coefficien

set data theof valueMinimumSmallest/ X

set data theof valueMaximumLargest/ X

min

max

Exercise Calculate the value of range, standard deviation for the blood

serum cholesterol levels of 10 persons of the following: 240,

260, 290, 245, 255, 288, 272, 263, 277 and 250.

Compute the value of range, average deviation and standard

deviation for the following:

Weekly Income No. of Employees

1300-1399 30

1400-1499 46

1500-1599 58

1600-1699 76

1700-1799 60

1800-1899 50

1900-1999 20

Measurements of Variation

Average Deviation

Average Deviation is obtained by calculating the absolutedeviations of each observation from median (mean) andthen averaging these deviations by taking theirarithmetic mean. In case of Grouped Data, the averagedeviation is called co-efficient of average deviation.

(for ungrouped data)

(for grouped data)

n

x-xA.D. x

n

x-xA.D. x

f

Measurements of Variation Standard Deviation

Karl Pearson in 1893 introduced this concept. It is ameasure of how much “spread” or “variability” is presentin the sample. If all the numbers in the sample are veryclose to each other the standard deviation is close to zero.It is the square root of the means of square deviationsfrom the arithmetic mean. It is symbolized by σ.Sometimes it is called as Root Mean Square Deviation.

(for ungrouped data) (for grouped data)

n

x-x2

n

x-x2

f

Uses of Standard Deviation:

The standard deviation enables us to determine with agreat deal of accuracy, where the values of a frequencydistributions are located in relation to the mean.

Measurements of Variation Variance

Variance is the square of the standard deviation.Variance is the average of the squared distances ofthe observations from the mean. It is symbolizedby . The corresponding relative measure is co-efficient of variation.

(for ungrouped data) (for grouped data)

2

n

x-x2

2

n

x-x2

2

f

%x100x

σ(C.V.)Variation oft Coefficien

Exercise From two firms A and B belonging to same industry, the following

details are available:

1. Which firm pays out larger amount as wages?

2. Which firm shows greater variability in the distribution of wages?

3. Find average monthly wages and the standard deviation of the wages of

all employees in both the firms.

Problem 57, Page 171.

Firm A Firm B

No. of Employees 100 200

Average Monthly Wage Tk. 4,800 Tk. 5,100

Standard Deviation Tk. 600 Tk. 540

Exercise For two firms A and B belonging to same industry, the following

details are available:

1. Which firm pays out larger amount as wages?

2. Which firm shows greater variability in the distribution of wages?

3. Find average monthly wages and the standard deviation of the wages of

all employees in both the firms.

Firm A Firm B

No. of Employees 100 200

Average Monthly Wage Tk. 2,400 Tk. 1,800

Standard Deviation Tk. 60 Tk. 80

Measurements of Variation Standard Score: The standard deviation is also useful in

describing how far individual items in a distribution departfrom the mean of the distribution. A measure called thestandard score gives us the number of standard deviationsof a particular observation lies below or above the mean.

where,

x = Observation from the population

μ = Population mean

σ = Population Standard Deviation

σ

-xz Score, Standard Population

Chebyshev’s Theorem