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13.Statistics
medium
If the variance of the frequency distribution is $3$ then $\alpha$ is ......
| $X_i$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ |
| Frequency $f_i$ | $3$ | $6$ | $16$ | $\alpha$ | $9$ | $5$ | $6$ |
A
$5$
B
$4$
C
$3$
D
$2$
(JEE MAIN-2023)
Solution
| $x_i$ | $f_i$ | $d_i=x_i-5$ | $f _{ i } d _{ i }^2$ | $f _{ i } d _{ i }$ |
| $2$ | $3$ | $-3$ | $27$ | $-9$ |
| $3$ | $6$ | $-2$ | $24$ | $-12$ |
| $4$ | $16$ | $-1$ | $16$ | $-16$ |
| $5$ | $\alpha$ | $0$ | $0$ | $0$ |
| $6$ | $9$ | $1$ | $9$ | $9$ |
| $7$ | $5$ | $2$ | $20$ | $10$ |
| $8$ | $6$ | $3$ | $54$ | $18$ |
$\sigma_{ x }^2=\sigma_{ d }^2=\frac{\sum f _{ i } d _{ i }^2}{\sum f _{ i }}-\left(\frac{\sum f _{ i } d _{ i }}{\sum f _{ i }}\right)^2$
$=\frac{150}{45+\alpha}-0=3$
$\Rightarrow 150=135+3 \alpha$
$\Rightarrow 3 \alpha=15 \Rightarrow \alpha=5$
Standard 11
Mathematics
