“STANDARD DEVIATION”
Standard Deviation:
• Std deviation is the best and scientific method of dispersion.
• It is widely used method used in statistical analysis.
• Std deviation is that method of dispersion where deviations are taken from
mean and while taking deviations algebraic signs are kept in mind.
• Also known as root mean square deviation.
• The greater the standard deviation, the greater will be the
magnitude of the values from their mean.
• A small standard deviation means a high degree of uniformity
of the observations as well as homogeneity of the series.
• The standard deviation is useful in judging the effectiveness
of the mean.
Standard Deviation Individual Series:
=
x2
N
Also
s.d =
X2 -
Where
x=X-X
If Assumed mean is taken
2
2

d

d
 =
N
N
Where
d = X - A (Assumed mean)
N
X
N
2
where X= variables
Example:
Blood Serum cholesterol levels of 10 persons
are as under:
240
260
290
245
255
288
272
263
277
251
Calculate the standard deviation with the help
Of assumed mean.
Standard Deviation (Discrete series)/ Continuous Series
• Actual mean method
• Assumed mean method
• Step Deviation method
Actual mean method:
2
 fx
=
N

Where x = X – (mean)
Assumed Mean Method:
=

fd
2
f d
N
2
N
Step Deviation Method:
=

fd
2
N
Here d = X – A / i
f d
N
2
X i
Example:
The annual salaries of a group of employees are given
in the following table:
Salaries(in 000)
45
50
55
60
65
70
75
80
Number of persons
03
05
08
07
09
07
04
07
Calculate the standard deviation of the salaries.
Example:
Calculate mean and standard deviation of the following
Frequency distribution of marks:
Marks
0-10
10-20
20-30
30-40
40-50
50-60
60-70
No. of Students
05
12
30
45
50
37
21
Coefficient of Variations:
C.V =  / X x 100
Example: The following table shows that monthly expenditures
of 80 students of a university on morning breakfast :
Expenditure
No. of students
78-82
02
73-77
06
68-72
07
63-67
12
58-62
18
53-57
13
48-52
09
43-47
07
38-42
04
33-37
02
Calculate standard deviation and coefficient of variation of above data
Example:
From the prices of shares of X and Y below find out
which is more stable in value.
X
35
54
52
53
56
58
52
50
51
49
Y
108
107
105
105
106
107
104
103
104
101
Variance: It is the square of the standard deviation
i.e..Variance = 2
Example:
The number of employees, wages per employee and the variance
of the wages per employee for two factories is given below
Factory A
Factory B
No. of employees
100
150
Average wage
3200
2800
Variance of wage
625
729
(a) In which factory is there a greater variation in the distribution
of wages per employee.
(b) Suppose in factory B, the wages of an employee were
wrongly noted as Rs. 3050 instead of Rs. 3650, what
would be the correct variance for factory B.
Example:
The mean of 5 observations is 4.4 and the variance is 8.24.
If the three of the five observations are 1, 2 and 6, find the other two.
Example:
The following table gives the marks obtained by a group of
80 students in an examination. Calculate the variance.
Marks obtained
No. of Students
10-14
02
14-18
04
18-22
04
22-26
08
26-30
12
30-34
16
36-38
10
38-42
08
42-46
04
46-50
06
50-54
02
54-58
04
Skew ness:
“It refers to the asymmetry or lack of symmetry
in the shape of a frequency distribution.”
As far the study of central tendency the statistical average is
calculated and for scatter of values, dispersion is measured. In the
same way to study the symmetrical or asymmetrical nature of series,
skew ness is calculated
Types of frequency distribution:
1. Normal frequency distribution
2. Asymmetrical Distribution
1. Normal frequency distribution
One main feature of normal distribution is that mean, median
and mode are found equal .in such a distribution the frequencies
gradually increase, they are maximum in the center and then
decrease. When this distribution is plotted on a graph it will be a
bell-shaped graph. It is also called normal curve.
Mean
Median
Mode
Zero skew ness
Mean =Median =Mode
2. Asymmetrical Distribution:- In a asymmetrical distribution the
rate of increase or decrease of frequencies is not same. Mean,
median and mode are not equal such a distribution is called
asymmetrical distribution. It is of two types:
(i) Positive Skew ness: When in a series mean is more than
median and median is more than mode then skew ness is positive
i.e curve is seen more towards left.
(ii) Negative Skew ness:- When in a series mean is less than
median and median is less than mode then skew ness is negative
i.e curve is seen more towards right.
Positively skewed: Mean and median are to the right of the mode
Mean>Median>Mode
Mode
Mean
Median
Negatively Skewed: Mean and Median are to the left of the Mode.
Mean<Median<Mode
Mean
Mode
Median
Karl Pearson’s Coefficient of Skewness:
Sp = Mean – Mode/ Standard Deviation
Bowley’s Coefficient of Skewness
SB = Q3 + Q1 – 2 Median/Q3 – Q1