Visual SolutionHard

Question

Let the observations

x_i

(

1 \leq i \leq 10

) satisfy

\sum_{i=1}^{10}(x_i - 5) = 10

and

\sum_{i=1}^{10}(x_i - 5)^2 = 40

. Then the variance of

x_1, x_2, \ldots, x_{10}

is:

(A)

3

(B)

4

(C)

5

(D)

6

Solution Path

Substitute

d_i = x_i - 5

. Compute Var

(d) = 40/10 - 1 = 3

. Since adding 5 does not change variance, Var

(x) = 3

01Question Setup

1/4

Given

\sum(x_i - 5) = 10

and

\sum(x_i - 5)^2 = 40

for 10 observations. Find Var

(x)

Find variance of

x_1, \ldots, x_{10}

02Change of Origin

2/4

Let

d_i = x_i - 5

. Then

\bar{d} = 10/10 = 1

and

\sum d_i^2 = 40

. Since

x_i = d_i + 5

, Var

(x)

= Var

(d)

Adding a constant does not change variance

03Compute Var(d)KEY INSIGHT

3/4

Var

(d) = \frac{\sum d_i^2}{n} - \bar{d}^2 = \frac{40}{10} - 1^2 = 4 - 1 = 3

\text{Var}(d) = 3

04Final Answer

4/4

Var

(x)

= Var

(d) = 3

. Answer: (A) 3.

\boxed{\text{Var}(x) = 3}

- Answer (A)

Concepts from this question2 concepts unlocked

★

Variance Shortcut Formula

EASY

Variance = E(X²) - [E(X)]². Compute the mean of squares minus the square of the mean. Avoids computing individual deviations.

\sigma^2 = \frac{\sum f_i x_i^2}{N} - \left(\frac{\sum f_i x_i}{N}\right)^2

Nearly every JEE variance question is faster with the shortcut. The direct formula with (xᵢ - mean)² takes longer and has more room for arithmetic errors.

Variance computationSD from raw dataGrouped data variance

Practice (10 Qs) →

★

Effect of Linear Transformation on Mean and Variance

STANDARD

If yᵢ = axᵢ + b, then mean(y) = a * mean(x) + b, and Var(y) = a² * Var(x). Adding a constant does not change variance. Multiplying by a constant scales variance by a².

\bar{y} = a\bar{x} + b, \quad \sigma_y^2 = a^2 \sigma_x^2

JEE frequently asks 'if 5 is added to each observation' or 'if each value is doubled'. This single rule answers all such questions instantly.

Change of origin/scaleEffect of adding constantEffect of multiplying constant

Practice (9 Qs) →

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