For 10 observations x1, x2, ..., x10. If sum from i=1 to 10 of (xi + 2)^2 = 180 and sum from i=1 to 10 of (xi - 1)^2 = 90, then their standard deviation is:

The correct answer is Option 4. Full mathematical solution steps and hints are accessible in the interactive StemCET workspace.

Maths Q8 – JEE-MAIN 2026

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Home Maths StatisticsQuestion #427

For 10 observations

x_{1}, x_{2}, . . ., x_{10}

. If

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

and

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

, then their standard deviation is:

Clue 1 of 3: Understanding Summation Properties

Recall how summation interacts with algebraic expressions, specifically how $\sum (a + b)^{2}$ expands to $\sum a^{2} + 2 \sum a b + \sum b^{2}$ .

🧠 Full Solution Path

Step 1: Expand and Simplify Given Equations✦ Active

Expand the two given summation equations to express them in terms of $\sum x_{i}^{2}$ and $\sum x_{i}$ . Let $N = 10$ be the number of observations.

\sum_{i = 1}^{10} (x_{i} + 2)^{2} = \sum (x_{i}^{2} + 4 x_{i} + 4) = \sum x_{i}^{2} + 4 \sum x_{i} + 4 N = 180 ⟹ \sum x_{i}^{2} + 4 \sum x_{i} + 4 (10) = 180 ⟹ \sum x_{i}^{2} + 4 \sum x_{i} = 140 (1)

\sum_{i = 1}^{10} (x_{i} - 1)^{2} = \sum (x_{i}^{2} - 2 x_{i} + 1) = \sum x_{i}^{2} - 2 \sum x_{i} + N = 90 ⟹ \sum x_{i}^{2} - 2 \sum x_{i} + 10 = 90 ⟹ \sum x_{i}^{2} - 2 \sum x_{i} = 80 (2)

Step 2: Solve for

\sum x_{i}

and

\sum x_{i}^{2}

○ Expand

Solve the system of linear equations (1) and (2) to find the values of $\sum x_{i}$ and $\sum x_{i}^{2}$ .

(1) - (2) : (\sum x_{i}^{2} + 4 \sum x_{i}) - (\sum x_{i}^{2} - 2 \sum x_{i}) = 140 - 80 ⟹ 6 \sum x_{i} = 60 ⟹ \sum x_{i} = 10

Substitute $\sum x_{i} = 10$ into equation (2):

\sum x_{i}^{2} - 2 (10) = 80 ⟹ \sum x_{i}^{2} - 20 = 80 ⟹ \sum x_{i}^{2} = 100

Step 3: Calculate Standard Deviation○ Expand

Use the formula for standard deviation, $σ = \sqrt{\frac{\sum x_{i}^{2}}{N} - {(\frac{\sum x_{i}}{N})}^{2}}$ , with $N = 10$ , $\sum x_{i} = 10$ , and $\sum x_{i}^{2} = 100$ .

σ = \sqrt{\frac{100}{10} - {(\frac{10}{10})}^{2}} σ = \sqrt{10 - (1)^{2}} σ = \sqrt{10 - 1} σ = \sqrt{9} σ = 3

The standard deviation is 3, which corresponds to option 4.

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Maths Question 8 – JEE-MAIN 2026

Clue 1 of 3: Understanding Summation Properties

🧠 Full Solution Path