If the mean of the frequency distribution, is 28, then its variance is ______. [2023]

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Solution

Q.
If the mean of the frequency distribution

Class : 0–10 10–20 20–30 30–40 40–50

Frequency : 2 3 $x$ 5 4

is 28, then its variance is _____ . [2023]

Ans.
(151)

$x_{i}$ $f_{i}$ $x_{i} \cdot f_{i}$

5 2 10

15 3 45

25 $x$ $25 x$

35 5 175

45 4 180

$n = 14 + x$ $\sum x_{i} f_{i} = 410 + 25 x$

$Mean, \bar{x} = \frac{410 + 25 x}{14 + x}$ $\Rightarrow 28 = \frac{410 + 25 x}{14 + x}$

$\Rightarrow 392 + 28 x = 410 + 25 x \Rightarrow 3 x = 18$

$\Rightarrow x = 6$

$∴$ $n = 20$

$Variance (σ^{2}) = \frac{\sum f_{i} x_{i}^{2}}{n} - {(28)}^{2}$

$= \frac{2 \times 5^{2} + 3 \times 15^{2} + 6 \times 25^{2} + 5 \times 35^{2} + 4 \times 45^{2}}{20} - {(28)}^{2}$

$= \frac{(2 \times 25) + (3 \times 225) + (6 \times 625) + (5 \times 1225) + (4 \times 2025)}{20} - 784$

$= 935 - 784 = 151$

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