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Statistics question

2004 · Shift 0 · Q74

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2004 · Shift 0 · Q74

Statistics question

2004 · Shift 0 · Q74

JEE MainMathematicsStatisticsMCQ+4 / −1

In a series of 2n observations, half of them equal

a

and remaining half equal

–a

. If the standard deviation of the observations is 2, then

|a|

equals

A
2
B
$\sqrt 2$
C
${1 \over n}$
D
${{\sqrt 2 } \over n}$

View written solutionFree

Correct answer: A

Given data

There are $2n$ observations:

$n$ observations are equal to $a$
$n$ observations are equal to $-a$

We are given that the standard deviation is $2$ .

Find the mean

The mean is

\bar{x} = \frac{na + n(-a)}{2n} = \frac{na-na}{2n} = 0.

So, the mean is $0$ .

Find the variance

Variance is

\sigma^2 = \frac{1}{2n}\sum (x_i-\bar{x})^2.

Since $\bar{x}=0$ , each observation contributes:

for $a$ : $(a-0)^2=a^2$
for $-a$ : $(-a-0)^2=a^2$

Thus all $2n$ observations have squared deviation $a^2$ .

So,

\sigma^2 = \frac{1}{2n}(2n\cdot a^2)=a^2.

Hence,

\sigma = |a|.

Use the given standard deviation

Given $\sigma=2$ , therefore

|a|=2.

Check options

A: $2$ ✅
B: $\sqrt{2}$ ❌
C: $\dfrac{1}{n}$ ❌
D: $\dfrac{\sqrt{2}}{n}$ ❌

Therefore, the correct option is A.

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