Measures of Measures of DispersionDispersion

Measures of Dispersion

• It is the measure of extent to which an individual items vary

• Synonym for variability• Often called “spread” or “scatter”• Indicator of consistency among a

data set• Indicates how close data are

clustered about a measure of central tendency

Compare the following Compare the following distributionsdistributions

o Distribution A

200 200 200 200 200

o Distribution B300 205 202 203 190

o Distribution C1 989 2 3 5

Arithmetic mean is same for all the series but distribution differ widely from one another.

Objectives of Measuring Objectives of Measuring VariationVariation

o To gauge the reliability of an average i.e. dispersion is small, means more reliable.

o To serve as a basis for the control of variability.

o To compare two or more series with regard to their variability.

The RangeThe Rangeo Difference between largest value and

smallest value in a set of datao Indicates how spread out the data areo Dependent on two extreme values.o Simple, easy and less time consuming.

o Calculation: Find largest and smallest number in

data set Range = Largest - Smallest

Uses of RangeUses of Range

o Quality Control

o Weather Forecasting

o Fluctuations in Share/Gold prices

The quartiles divide the data into four parts. There are a total of three quartiles which are usually denoted by Q1, Q2 and Q3.

Quartile Deviation/Inter-Quartile Range

The inter-quartile range is defined as the difference between the upper quartile and the lower quartile of a set of data.

Inter-quartile range = Q3 – Q1

Quartile Deviation/Semi Inter Quartile Range = Q3-Q1 / 2

ExampleExample

• Calculate Quartile Deviation-

Wages in Rs. per week

No. of wages earners

Less than 35 14

35-37 62

38-40 99

41-43 18

Over 43 7

Standard DeviationStandard Deviation• It is most important and widely used

measure of studying variation.

• It is a measure of how much ‘spread’ or ‘variability’ is present in sample.

• If numbers are less dispersed or very close to each other then standard deviation tends to zero and if the numbers are well dispersed then standard deviation tends to be very large.

For a set of ungrouped data x1, x2, …, xn,

n

xxf

n

xxxxxx

n

ii

n

1

21

222

21

)(

)()()( deviation Standard

data. of number total the is and mean the iswhere nx

For Ungrouped Data

•It is the average of the distances of the observed values from the mean value for a set of data

SD =Sum of squares of individual deviations from arithmetic mean

Number of items

Example:

ScoresDeviations From Mean

Squares of Deviations

01

03

05

06

11

12

15

19

34

37143

-13

-11

-09

-08

-03

-02

+01

+05

+20

+23

169

121

81

64

9

4

1

25

400

5291403

M = 143/10 = 14

No. of scores = 10

SD =1403

10= 11.8

n

ii

n

ii

n

nn

f

xxf

fff

xxfxxfxxf

1

1

21

21

2222

211

)(

)()()( deviation Standard

data. of number total the is and mean the isdata, of group th the offrequency the is where

nxifi

For Grouped Data

ExampleExampleCalculate standard deviation –

Marks No. of students

0-10 5

10-20 12

20-30 30

30-40 45

40-50 50

50-60 37

60-70 21

Uses of Standard Uses of Standard DeviationDeviation

o The standard deviation enables us to determine with a great deal of accuracy , where the values of frequency distribution are located in relation to the mean.

o It is also useful in describing how far individual items in a distribution depart from the mean of the distribution.

-1 +12.2 9.6

1411

25.812.4

68%

NORMAL DISTRIBUTION CURVE1 Standard Deviation

-2 +2018.2

1411

3713.8

95%

NORMAL DISTRIBUTION CURVE2 Standard Deviations

NORMAL DISTRIBUTION CURVE3 Standard Deviations

-3 +3016.8

1411

3715.2

99.7%

VarianceVarianceo = Variance= (standard deviation)2

o For comparing the variability of two or more distributions,

o Coefficient of variation = s.d. X 100o mean

2

. . 100CV Xx

More C.V., More variability

ExampleExample

Which organization is more uniform wages?

Organization A Organization B

Number of employees 100 200

Average wage per employee

5000 8000

Variance of wages per employee

6000 10000

MEASURE OF SHAPE- MEASURE OF SHAPE- SKEWNESSSKEWNESS

Distribution lacks symmetry

Data sparse at one end and piled up at the other .

- patients suffering from diabetes

Median is the best measure for skewed data , as it is not highly influenced by the frequency nor is pulled by extreme values.

SYMMETRIC

MODE = MEAN = MEDIAN

MEAN MODE MEDIAN

SKEWED LEFT(NEGATIVELY)

MEDIAN MEAN

MODE

SKEWED RIGHT(POSITIVELY)

Case Let for practiceCase Let for practice At one of the management institutes,

there are mainly six specialists available namely: Marketing, Finance, HR, Operations CRM and International Business.( See Table 1)

Calculate the average salaries for all the specializations, as also for the entire batch. Which specialization has the maximum variation in the salaries offered?

(Ans- Avg. salary for all specializations- 5.27, for mktg-5.24, finance-5.36, HR-4.88,Operations-5.4, CRM-5.5, IB-5.0. Max. Variation was in IB.

3-4 Lacs

4-5 Lacs

5-6 Lacs

6-7 Lacs

7-8 Lacs

Total

Mktg. 23 24 33 23 9 112

Finance 11 17 35 15 6 84

HR 1 7 4 1 0 13

Operations 1 2 5 1 1 10

CRM 1 2 5 2 1 11

IB 2 4 2 1 1 10

Total 39 56 84 43 18 240