measure of dispersion

1. Measures of Dispersion

2. 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

3. Compare the following
distributions
o Distribution A
200 200 200 200 200
o Distribution B
200 205 202 203 190
o Distribution C
1 989 2 3 5
Arithmetic mean is same for all the
series but distribution differ widely from
one another.

4. 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.

5. The Range
o Difference between largest value and
smallest value in a set of data
o Indicates how spread out the data are
o 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

6. Uses of Range
o Quality Control
o Weather Forecasting
o Fluctuations in Share/Gold prices

7. Quartile Deviation/Inter-Quartile
Range
The quartiles divide the data into four parts. There
are a total of three quartiles which are usually denoted
by Q1, Q2 and Q3.

8. 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

9. Example
 Compute Quartile deviation
Marks 10 20 30 40 50 60
Students 4 7 15 8 7 2

10. 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

11. 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.

12. For Ungrouped Data
For a set of ungrouped data x1, x2, …, xn,
x x x x x x
( ) ( ) ( )
f x x
n
n
n
i
i
n




        

1
2
1
2 2
2
2
1
( )
Standard deviation 
where x is the mean and n is the total number of data.
•It is the average of the distances of the
observed values from the mean value for a set
of data

13. SD =
Sum of squares of individual deviations from arithmetic mean
Number of items
Example
:
Scores
Deviations
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
No. of scores = 10
M = 143/10 = 14
SD =
1403
10
= 11.8

14. For Grouped Data
f x x f x x f x x
(  )  (  )      ( 
)
f x x






     

n
i
i
n
i
i
n
n n
f
f f f
1
1
2
1
1 2
2 2
2 2
2
1 1
( )
Standard deviation 
where f is the frequency of the i th group of data, x is the mean and
i
n
is the total number of data.

15. 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

16. Example
 Calculate 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

17. 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.

18. NORMAL DISTRIBUTION CURVE
1 Standard Deviation
-1 +1
2.2
9.6
14
11
25.8
12.4
68%

19. NORMAL DISTRIBUTION CURVE
2 Standard Deviations
-2 +2
01
8.2
14
11
37
13.8
95%

20. NORMAL DISTRIBUTION CURVE
3 Standard Deviations
-3 +3
01
6.8
14
11
37
15.2
99.7%

21. Variance
o = Variance= (standard deviation)2
2 
o For comparing the variability of two or
more distributions,
o Coefficient of variation = s.d. X 100
o mean

C.V. X100
x

More C.V., More variability

22. Example
Organization A Organization B
Number of employees 100 200
Average wage per
employee
5000 8000
Variance of wages per
employee
6000 10000
 Which organization is more uniform wages?

23. Measure of Shape
 Tools that can be used to describe
the shape of a distribution of data.
 Two measures of shape-
 Skewness
 Kurtosis

24. SKEWNESS
 A 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

25. MODE = MEAN = MEDIAN
SYMMETRIC
MEAN MODE
MEDIAN
SKEWED LEFT
(NEGATIVELY)
MEDIAN
MEAN
MODE
SKEWED RIGHT
(POSITIVELY)

26. 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

28. Case: Coca Cola goes small in
Russia

29.  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.

30.  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.

31.  Company’s Mission-
To maintain the world’s most powerful
trademark and effectively utilize the
world’s most effective and pervasive
distribution system.

32.  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.

33.  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.

34.  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

35. 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.

36. 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