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;
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 10045 15 8 45 100 245 |
Σ fi = 50 | Σfixi = 450 | Σfi di2 = 754 |
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 |
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