The variance is the measure of variability about the mean and is symbolized by "σ^{2}”

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;

Variablexi |
Deviation from the mean di = ( xi – x ) |
di^{2} |

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 |

Σdi^{2} = 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

**σ ^{2} = Σ f**

**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:**

Variablexi |
Frequency fi |
fixi |
di = (xi – x ) |
di^{2} |
fi di^{2} |

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 di^{2} = 754 |

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

Variablexi |
Frequency fi |
di = xi – A |
di^{2} |
fidi^{2} |
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 fidi |
Σfidi = -22 |

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