measure of dispersion

1. MMeeaassuurreess ooff DDiissppeerrssiioonn

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. CCoommppaarree tthhee ffoolllloowwiinngg
ddiissttrriibbuuttiioonnss
o Distribution A
200 200 200 200 200
o Distribution B
300 205 202 203 190
o Distribution C
1 989 2 3 5
Arithmetic mean is same for all the
series but distribution differ widely from
one another.

4. OObbjjeeccttiivveess ooff MMeeaassuurriinngg
VVaarriiaattiioonn
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. TThhee RRaannggee
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. UUsseess ooff RRaannggee
o Quality Control
o Weather Forecasting
o Fluctuations in Share/Gold prices

7. Quartile Deviation/Inter-Quartile
Range
1
QThe quartiles divide the data into four parts. There
are a total of three quartiles which are usually denoted
by , Qand Q.
2 3

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. EExxaammppllee
• 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

10. SSttaannddaarrdd DDeevviiaattiioonn
• 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.

11. For Ungrouped Data
For a set of ungrouped data x1
, x2, …, xn,
x x x x x x
= - + - + ××× + -
Standard deviation s ( ) ( ) ( )
f x x
n
n
n
i
i
n
å=
-
=
1
2
1
2 2
2
2
1
( )
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

12. 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

13. For Grouped Data
f x x f x x f x x
= - + - + ××× + -
Standard deviation s ( ) ( ) ( )
f x x
å
å
=
=
-
=
+ + ××× +
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
( )
where fi is the frequency of the i th group of data, x
is the mean and
n
is the total number of data.

14. EExxaammppllee
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

15. UUsseess ooff SSttaannddaarrdd DDeevviiaattiioonn
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.

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

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

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

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

20. EExxaammppllee
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?

21. MMEEAASSUURREE OOFF SSHHAAPPEE--
SSKKEEWWNNEESSSS
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.

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

23. CCaassee LLeett ffoorr pprraaccttiiccee
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.

24. 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