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\(=\left[ \frac{\sum{{{f}_{i}}d_{i}^{2}}}{\sum{{{f}_{i}}}}-{{\left( \frac{\sum{{{f}_{i}}{{d}_{i}}}}{\sum{{{f}_{i}}}} \right)}^{2}} \right]\times {{h}^{2}}\left[ \frac{76}{30}-{{\left( \frac{2}{30} \right)}^{2}} \right]\times {{\left( 30 \right)}^{2}}\),
\(=\left[ \frac{76}{30}-\frac{4}{30\times 30} \right]\times 900\),
\(=\left( \frac{2280-4}{900} \right)\times 900\),
\(=\frac{2276}{900}\times 900=2276\).