Math Assignment Help With Variance and Standard Deviation

10.3.2 Variance and Standard Deviation:

Math Assignment Help Order Now The variance is the measure of variability about the mean and is symbolized by "σ2

The square root of variance is called the standard deviation. It is represented by σ.

10.3.2.1 Methods of finding variance and standard deviation:

Case I : when each term has frequency 1

Let x1 x2……xn are the n given observations and let x be their mean. Then the variance is given by;

methods of finding variance and standard deviation
Variable
xi
Deviation from the mean
di = (xi – x )
di2
10
11
8
15
16
10 – 12 = -2
11 – 12 = -1
8 – 12 = -4
15 – 12 = 3
16 – 12 = 4
4
1
16
9
16
Σdi2 = 46

Therefore,

Variance = σ2 = Σdi2 = 46/5 = 9.2

n

and standard deviation = σ = √9.2 = 3.303

Case II: When the frequencies of the variable are given. The variance is given by

σ2 = Σ fi di2 & S.D = σ = Σ fi di2

Σf i Σf i

Example: find the variance and standard deviation from the given distribution table.

Variable (xi) 2 4 6 10 12
Frequency (fi) 4 4 5 5 6

Solution:

Variable
xi
Frequency
fi
fixi di = (xi – x ) di2 fi di2
2
4
6
8
10
12
14
16
4
4
5
15
8
5
4
5
8
16
30
120
80
60
56
80
-7
-5
-3
-1
1
3
5
7
49
25
9
1
1
9
25
49

196

100
45
15
8
45
100
245
Σ fi = 50 Σfixi = 450 Σfi di2 = 754

Statistics

Example: The scores of 10 students in an examination, in which maximum marks were 50 is given. Find the variance?

Variable
xi
Frequency
fi
di = xi – A di2 fidi2 fidi
19
22
27
28
34
35
36
41
48
1
1
1
2
1
1
1
1
1
-15
-12
-7
-6
0
1
2
7
14
225
144

49

36

0

1

4

49

196

225

144

49

72

0

1

4

49

196

-15

-12

-7

-12

0

1

2

7

14

Σfi = 10

Σ fidi2 = 740

Σfidi = -22

variance and standard deviation

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