Disparsion Analysis

1. Variance: The variance is the measure of
how much spread or variability is present in
the sample.
 If the all the numbers in the sample are very
close to each other, the variance is close to
zero.
If the numbers well dispersed or scatter the
variance will tend to be large.
The variance of a set of data is defined as the
sum of squares of the deviations of
observations from their mean divided by their
number of observations.
It is denoted by σ2 (sigma square).

2. Mathematically, For Ungrouped data
variance can be written as

3. That is, the mean of squared
deviations of observations
from their mean is known as
the variance.
Then find variance of the given
data 5,9,11, 4,3, 20

4. Variance of Group data can be
mathematically written as

5. Standard Deviation: The
positive square root of
variance is known as Standard
deviation.
Standard deviation is denoted
by σ (sigma) or s

6. Standard Deviation is written as

7. The standard deviation measures the
absolute variation of the distribution.
The greater the amount of variation,
the greater the standard deviation
(SD).
 A small standard deviation means a
high degree of uniformity and
homogeneity of the observations. A
large standard deviation means the
opposite.

8. 1. Find the standard deviation from
the weekly wages of the workers
working in a factory
Worker Wages Worker Wages
A 320 F 340
B 310 G 345
C 315 H 330
D 325 I 333
E 350 J 342

9. An analysis of production rejects result in the
following table. Find standard deviation
No. of rejected No. Of operator
Product
20-25 5
25-30 15
30-35 28
35-40 42
40-45 15
45-50 12
50-55 3

10. An analysis of production rejects result in the
following table. Find standard deviation
class x f fx fx^2
20-25 22.5 5 113 2531.3
25-30 27.5 15 413 11344
30-35 32.5 28 910 29575
35-40 37.5 42 1575 59063
40-45 42.5 15 638 27094
45-50 47.5 12 570 27075
50-55 52.5 3 158 8268.8
TOTAL 120 4375 164950
mean 36.5 1374.6

11. Mathematical Properties of Sd:
1. The Standard Deviation(SD) of first n
natural numbers can be obtain as
Hence find the standard deviation of
the data
1,2,3,4………………………………..500

12. 2. Standard deviation is independent of
change of origin but dependent of scale.
3. For symmetrical distribution
Mean±1σ covers 68.27% data
Mean±2σ covers 95.45% data
Mean±3σ covers 99.73% data
4. Relation between measures of variations
Q.D=2/3 SD
A.D=4/5 SD

13. Merits of SD:
1. SD is the best measures of variation
2. SD is used in comparing two or
more distribution.
3. It is possible to calculate the
combined SD
4. SD is most prominently used for
further statistical work.

14. Limitation of SD:
1.It is difficult to compute.
2.It gives more weight to
extreme values and less
weight to near mean.

15. Example: The performance of run
of two player is given. Select who
is better one of the two?
Tamim Mushfik
No. of Matches 100 100
Average of Run 120 80
SD of Run 45 20

16. Co-efficient of Variation(CV): When
we do not get an accurate picture
just by comparing two sets of data by
standard deviation then Co-efficient
of variation solve this difficulty.
Co-efficient of variation(CV) is
computed as a ratio of the standard
deviation to the mean of the same
distribution.

17. Mathematically, Co-efficient of Variation(CV)

18. # For which sample Co-efficient of
variation is greater is said to be less
consistent, less uniform, less stable
or less homogeneous and more
variability.
# For which Co-efficient of variation
is less is said to be more consistent,
more uniform, more stable or more
homogeneous and less variability.

19. The price of tea company shares in Dhaka
and Chittagong share markets during the
last Four months are recorded as below:
Month Dhaka Chittagong
January 106 108
February 120 112
March 100 110
April 130 114
In Which Market Share prices are more
Stable?

20. Home Work: A purchasing agent obtained samples of
60 watt bulbs from two companies. He had the
samples tested in his own laboratory for length of
life with the following results:
Length of life Company A Company B
(in Hours)
1700-1900 10 3
1900-2100 16 40
2100-2300 20 12
2300-2500 8 3
2500-2700 6 2
i) Which company’s bulb do you think are better
in terms average life?
ii) If prices of bulbs are the same, which
company’s bulbs would you buy and why?

21.  For Ungrouped data
variance can
The Standard Deviation(SD) of
first n natural numbers
 For Ungrouped data variance
Standard Deviation is written as
Co-efficient of Variation(CV)