Statistics & Linear ProgrammingmediumPYQ · JEE Main · 4 Apr 2026 · Shift 2 (Afternoon)Free

Standard Deviation of 10 Observations = 3 | JEE Main 2026

JEE Maths question with a full step-by-step solution.

Question

For 10 observations

x_1,x_2,\ldots,x_{10}

, if

\displaystyle\sum_{i=1}^{10}(x_i+2)^2=180

and

\displaystyle\sum_{i=1}^{10}(x_i-1)^2=90

, then their standard deviation is

2

\sqrt3

2\sqrt2

3

correct

Solution

Step 1: Expand the first sum:

\sum x_i^2+4\sum x_i+40=180\ \Rightarrow\ \sum x_i^2+4\sum x_i=140.\quad(1)

Step 2: Expand the second sum:

\sum x_i^2-2\sum x_i+10=90\ \Rightarrow\ \sum x_i^2-2\sum x_i=80.\quad(2)

Step 3: Subtract (2) from (1):

6\sum x_i=60\Rightarrow\sum x_i=10

. Then from (2),

\sum x_i^2=80+2(10)=100

. Step 4: Variance:

\sigma^2=\frac{\sum x_i^2}{N}-\left(\frac{\sum x_i}{N}\right)^2=\frac{100}{10}-\left(\frac{10}{10}\right)^2=10-1=9\ \Rightarrow\ \sigma=3.

Correct answer: (4)

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