First slide

Introduction to statistics

Question

The variance of the numbers 2, 3,11 and x is $\frac{49}{4}$ .find the value x .

Moderate

A

$6, \frac{14}{3}$
B

$6, \frac{14}{5}$
C

$6, \frac{16}{3}$
D

None of these

Solution

From given data, we make the following table
x x2
2 4
3 9
11 121
x x²
$Σx = 16 + x$ ${Σx}^{2} = 134 + x^{2}$
But we know that, variance $= \frac{{Σx}^{2}}{n} - {(\frac{Σx}{n})}^{2}$
$\begin{array}{lr} \Rightarrow \frac{134 + x^{2}}{4} - {(\frac{16 + x}{4})}^{2} = \frac{49}{4} \\ \Rightarrow \frac{134 + x^{2}}{4} - \frac{(256 + x^{2} + 32 x)}{16} = \frac{49}{4} \\ \Rightarrow \frac{3 x^{2} - 32 x + 280}{16} = \frac{49}{4} \\ \Rightarrow 280 + 3 x^{2} - 32 x = \frac{49}{4} \times 16 \\ \Rightarrow 280 + 3 x^{2} - 32 x = 196 \\ \Rightarrow 3 x^{2} - 32 x + 84 = 0 \\ \Rightarrow (x - 6) (3 x - 14) = 0 \\ \Rightarrow x = 6, x = \frac{14}{3} \end{array}$
Therefore, the values of x are 6 and $\frac{14}{3}$ .

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