Find the mean and variance of frequency distribution given below:

x_i: 1 ≤ x < 3 3 ≤ x < 5 5 ≤ x < 7 7 ≤ x < 10

f_i: 6 4 5 1

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Solution

x_i Mid-Values(y_i) y_i² f_i f_i y_i f_iy_i²

1–3 2 4 6 12 24

3–5 4 16 4 16 64

5–7 6 36 5 30 180

7–10 8.5 72.25 1 8.5 72.25

N = $\sum_{} f_{i}$ = 16 $\sum_{} f_{i} y_{i} = 66.5$ $\sum_{} f_{i} {y_{i}}^{2} = 340.25$

Therefore,

Mean = $\frac{\sum_{} f_{i} y_{i}}{\sum_{} f_{i}} = \frac{66.5}{16} = 4.16$

Variance = $(\frac{1}{N} \sum_{} f_{i} y_{i}^{2}) - {(\frac{1}{N} \sum_{} f_{i} y_{i})}^{2} = \frac{1}{16} \times 340.25 - {(\frac{1}{16} \times 66.5)}^{2} = 21.26 - 17.22 = 4.04$

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Find the mean and variance of frequency distribution given below :

$\begin{matrix} x_{i} : & 1 \leq x < 3 & 3 \leq x < 5 & 5 \leq x < 7 & 7 \leq x < 10 f_{i} : & 6 & 4 & 5 & 1 \end{matrix}$

f (x) = {\begin{matrix} 3 e^{- x} & 0 < x < \infty 0 & elsewhere \end{matrix}

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