Register or Login to View the Solution or Ask a Question
Question : Find variance and standard deviation of the following frequency distribution.
| x | Frequency (f) |
| 1.5 | 20 |
| 2.0 | 25 |
| 3.5 | 80 |
| 5.0 | 50 |
| 5.5 | 25 |
| 7.2 | 10 |
| 5.3 | 5 |
Solution :
Sum of the frequencies
\[N=\sum_{i=1}^{N}f_i=215\]
(i)
Arithmetic average (mean)
\[\bar{x}=\frac{\sum_{i=1}^{N}{f_i\ast x_i}}{N}=\frac{30+50+280+250+137.5+72+26.5}{215}=\frac{846}{215}=3.93\]
frequency distribution table
| x | Frequency (f) | (x-x ̅) | (x-x ̅)^2 | f*(x-x ̅)^2 |
| 1.5 | 20 | -2.43 | 5.92 | 118.57 |
| 2.0 | 25 | -1.93 | 3.74 | 93.59 |
| 3.5 | 80 | -0.43 | 0.18 | 15.12 |
| 5.0 | 50 | 1.06 | 1.13 | 56.72 |
| 5.5 | 25 | 1.56 | 2.44 | 61.23 |
| 7.2 | 10 | 3.26 | 10.66 | 106.60 |
| 5.3 | 5 | 1.36 | 1.86 | 9.31 |
(ii) Variance
\[V=\frac{\sum_{i=1}^{N}{f_i\ast\left(x_i-\bar{x}\right)^2\ }}{N-1}=\frac{118.57+93.59+15.12+\ 56.72+\ 61.23+106.60+9.31}{215-1}\]
\[V=\frac{461.19}{214}=2.1\] Variance = 2.15
(iii)
Standard deviation
σ=√variance=√2.15=1.46
Register or Login to View the Solution or Ask a Question