A data consists of 20 observations x1, x2, ..., x20. If ∑(xi + 5)^2 = 2500 and ∑(xi - 5)^2 = 100, then the ratio of mean to standard deviation of this data is:

The correct answer is Option 2. Full mathematical solution steps and hints are accessible in the interactive StemCET workspace.

Maths Q9 – JEE-MAIN 2026

A data consists of 20 observations

x_{1}, x_{2}, . . ., x_{20}

. If

\sum_{i = 1}^{20} (x_{i} + 5)^{2} = 2500

and

\sum_{i = 1}^{20} (x_{i} - 5)^{2} = 100

, then the ratio of mean to standard deviation of this data is:

Expand the squared terms in both given equations to express them in terms of $\sum x_{i}^{2}$ and $\sum x_{i}$ .

Step 1: Expand and Simplify Given Sums✦ Active

Given $n = 20$ observations. Expand the two given sum equations:

\sum_{i = 1}^{20} (x_{i} + 5)^{2} = \sum (x_{i}^{2} + 10 x_{i} + 25) = \sum x_{i}^{2} + 10 \sum x_{i} + 20 \times 25 = 2500

\sum x_{i}^{2} + 10 \sum x_{i} + 500 = 2500 ⟹ \sum x_{i}^{2} + 10 \sum x_{i} = 2000 (1)

\sum_{i = 1}^{20} (x_{i} - 5)^{2} = \sum (x_{i}^{2} - 10 x_{i} + 25) = \sum x_{i}^{2} - 10 \sum x_{i} + 20 \times 25 = 100

\sum x_{i}^{2} - 10 \sum x_{i} + 500 = 100 ⟹ \sum x_{i}^{2} - 10 \sum x_{i} = - 400 (2)

Step 2: Solve for Sum of

x_{i}

and Sum of

x_{i}^{2}

○ Expand

Add equations (1) and (2) to find $\sum x_{i}^{2}$ , and subtract (2) from (1) to find $\sum x_{i}$ :

(1) + (2) : (\sum x_{i}^{2} + 10 \sum x_{i}) + (\sum x_{i}^{2} - 10 \sum x_{i}) = 2000 + (- 400)

2 \sum x_{i}^{2} = 1600 ⟹ \sum x_{i}^{2} = 800

(1) - (2) : (\sum x_{i}^{2} + 10 \sum x_{i}) - (\sum x_{i}^{2} - 10 \sum x_{i}) = 2000 - (- 400)

20 \sum x_{i} = 2400 ⟹ \sum x_{i} = 120

💡 Teacher's Secret Hint

Remember that $n = 20$ is the number of observations, which is used when calculating the sum of constants.

Step 3: Calculate Mean, Standard Deviation, and Their Ratio○ Expand

Calculate the mean $\bar{x}$ and variance $σ^{2}$ , then the standard deviation $σ$ , and finally the required ratio:

\bar{x} = \frac{\sum x_{i}}{n} = \frac{120}{20} = 6

σ^{2} = \frac{\sum x_{i}^{2}}{n} - (\bar{x})^{2} = \frac{800}{20} - (6)^{2} = 40 - 36 = 4

σ = \sqrt{4} = 2

Ratio of mean to standard deviation = \frac{\bar{x}}{σ} = \frac{6}{2} = 3

The ratio is $3 : 1$ , which corresponds to option 2.

💡 Teacher's Secret Hint

Ensure you use the correct formula for variance: $σ^{2} = \frac{\sum x_{i}^{2}}{n} - (\bar{x})^{2}$ .

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