Measures of Dispersion

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 distributions

o Distribution A

200 200 200 200 200oDistribution B

200 205 202 203 190oDistribution C

1 989 2 3 5

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

Objectives of Measuring Variation

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 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 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

Example

Compute Quartile deviation

Marks 10 20 30 40 50 60

Students 4 7 15 8 7 2

Example

• 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 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

37

143

-13

-11

-09

-08

-03

-02

+01

+05

+20

+23

169

121

81

64

9

4

1

25

400

529

1403

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

Example

Calculate standard deviation-

Size of item Frequency Size of item Frequency

3.5 3 7.5 85

4.5 7 8.5 32

5.5 22 9.5 8

6.5 60

ExampleCalculate standard deviation –

Marks No. of students0-10 5

10-20 1220-30 3030-40 4540-50 5050-60 3760-70 21

Uses of Standard Deviation

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%

Varianceo = 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

Example

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

Tools that can be used to describe the shape of a distribution of data.

Two measures of shape-SkewnessKurtosis

SKEWNESSA distribution of data in which the right half is a mirror

image of the left half is symmetrical(no skewness)

Distribution lacks symmetry i.e. asymmetrical i.e skewness

The skewed portion is long, thin part of the curve. Skewed left or negatively skewed Skewed right or positively skewed.

Data sparse at one end and piled up at the other . - patients suffering from diabetes

SYMMETRIC

MODE = MEAN = MEDIAN

MEAN MODE MEDIAN

SKEWED LEFT(NEGATIVELY)

MEDIAN MEAN MODE

SKEWED RIGHT(POSITIVELY)

Kurtosis

Describes the amount of peakedness of a distribution.

Lepto-kurtic: more peaked than normal curve Platy-kurtic: more flat-tapped than normal curve Meso-kurtic: a normal shape in a frequency distribution

Case: Coca Cola goes small in Russia

Coca-cola is No.1 seller of soft drinks in world.

An average of more than 1 billion servings of coca- cola and its products around the world everyday.

World’s largest production and distribution system for soft drinks.

Sells more than twice soft drinks than its nearest competitor.

Its products are sold in more than 200 companies around the globe.

For several reasons, The company believes it will continue to grow internationally.

Disposable income is rising.World is getting younger.Reaching world market is easier as

political barriers fall and transportation difficulties are overcome.

Sharing of ideas, cultures, news creates market opportunities.

Company’s Mission- To maintain the world’s most powerful

trademark and effectively utilize the world’s most effective and pervasive distribution system.

In June 99,Coca cola introduced a 200 ml Coke bottle in Volgograd, Russia in order to market coke to its poorest customers.

This strategy is successful in many countries like India.

The bottles sold for 12 cents, making it affordable to everyone.

In 2001, Coca cola enjoyed a 25% volume growth in Russia including an 18% increase in unit case sales of Coca- Cola.

Because of the variability of bottling machinery, it is likely that every 200 ml bottle of Coke does not contain exactly 200 ml of fluid.

Some bottles may contain more fluid or other less.

A production engineer wants to test some of the bottles from the first production runs to determine how close they are to be 200 ml specification.

The following data are the fill measurements from a random sample of 50 bottles.

Use techniques Measures of Central tendency, Variability, Skewness.

Based on this analysis, how is the bottling process working?

Show Data

Case 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