# Variance

## Variance

The variance of a variate is the square of standard deviation and it is denoted by σ² or Var(x).

Var(x) = σ² = (standard deviation)

Coefficient of dispersion = σ/x

Step I: Firstly, write the given data, if it is given

Step II: If either change of origin or change of scale is given, then apply it.

Step III: If other condition is given, the. n apply it.

Step IV: Simplify it. Use the formula of variance.

Find the variance for the following frequency distribution.

 Class 0 – 30 30 – 60 60 – 90 90 – 120 120 – 150 150 – 180 180 – 210 Frequency 2 3 5 10 3 5 2

Step I: Write the given data.

 Class Frequency (fᵢ) Mid – value xᵢ Deviation from mean$${{d}_{i}}=\frac{{{x}_{i}}-A}{h}$$, A = 105 dᵢ² fᵢdᵢ fᵢdᵢ² 0-30 2 15 – 3 9 -6 18 30-60 3 45 – 2 4 -6 12 60-90 5 75 – 1 1 -5 5 90-120 10 105 0 0 0 0 120-150 3 135 1 1 3 3 150-180 5 165 2 4 10 20 180-210 2 195 3 9 6 18 Total 30 2 76

Step II: Use the formula and simplify it

Variance$$=\left[ \frac{\sum{{{f}_{i}}d_{i}^{2}}}{\sum{{{f}_{i}}}}-{{\left( \frac{\sum{{{f}_{i}}{{d}_{i}}}}{\sum{{{f}_{i}}}} \right)}^{2}} \right]\times {{h}^{2}}\left[ \frac{76}{30}-{{\left( \frac{2}{30} \right)}^{2}} \right]\times {{\left( 30 \right)}^{2}}$$,

$$=\left[ \frac{76}{30}-\frac{4}{30\times 30} \right]\times 900$$,

$$=\left( \frac{2280-4}{900} \right)\times 900$$,

$$=\frac{2276}{900}\times 900=2276$$.