Q34 / 75 · JEE Main 2026 (06 April Shift 1) · Mathematics
A data consists of $20$ observations $x_1, x_2, \ldots, x_{20}$. 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:
- A$2:1$
- Correct answerB$3:1$
- C$3:2$
- D$4:1$
Solution
Given $\sum_{i=1}^{20}(x_i + 5)^2 = 2500$ and $\sum_{i=1}^{20}(x_i - 5)^2 = 100$.
Expanding the first equation:
$\sum_{i=1}^{20} (x_i^2 + 10x_i + 25) = 2500$
$\sum_{i=1}^{20} x_i^2 + 10 \sum_{i=1}^{20} x_i + 500 = 2500$
$\sum_{i=1}^{20} x_i^2 + 10 \sum_{i=1}^{20} x_i = 2000$ --- (1)
Expanding the second equation:
$\sum_{i=1}^{20} (x_i^2 - 10x_i + 25) = 100$
$\sum_{i=1}^{20} x_i^2 - 10 \sum_{i=1}^{20} x_i + 500 = 100$
$\sum_{i=1}^{20} x_i^2 - 10 \sum_{i=1}^{20} x_i = -400$ --- (2)
Subtracting (2) from (1):
$20 \sum_{i=1}^{20} x_i = 2400 \Rightarrow \sum_{i=1}^{20} x_i = 120$
Adding (1) and (2):
$2 \sum_{i=1}^{20} x_i^2 = 1600 \Rightarrow \sum_{i=1}^{20} x_i^2 = 800$
Mean ($\mu$) is given by:
$\mu = \dfrac{\sum_{i=1}^{20} x_i}{20} = \dfrac{120}{20} = 6$
Variance ($\sigma^2$) is given by:
$\sigma^2 = \dfrac{\sum_{i=1}^{20} x_i^2}{20} - \mu^2 = \dfrac{800}{20} - 6^2 = 40 - 36 = 4$
Standard deviation ($\sigma$) is $\sqrt{4} = 2$.
The ratio of mean to standard deviation is:
$\dfrac{\mu}{\sigma} = \dfrac{6}{2} = \dfrac{3}{1}$
Answer: $3:1$
Expanding the first equation:
$\sum_{i=1}^{20} (x_i^2 + 10x_i + 25) = 2500$
$\sum_{i=1}^{20} x_i^2 + 10 \sum_{i=1}^{20} x_i + 500 = 2500$
$\sum_{i=1}^{20} x_i^2 + 10 \sum_{i=1}^{20} x_i = 2000$ --- (1)
Expanding the second equation:
$\sum_{i=1}^{20} (x_i^2 - 10x_i + 25) = 100$
$\sum_{i=1}^{20} x_i^2 - 10 \sum_{i=1}^{20} x_i + 500 = 100$
$\sum_{i=1}^{20} x_i^2 - 10 \sum_{i=1}^{20} x_i = -400$ --- (2)
Subtracting (2) from (1):
$20 \sum_{i=1}^{20} x_i = 2400 \Rightarrow \sum_{i=1}^{20} x_i = 120$
Adding (1) and (2):
$2 \sum_{i=1}^{20} x_i^2 = 1600 \Rightarrow \sum_{i=1}^{20} x_i^2 = 800$
Mean ($\mu$) is given by:
$\mu = \dfrac{\sum_{i=1}^{20} x_i}{20} = \dfrac{120}{20} = 6$
Variance ($\sigma^2$) is given by:
$\sigma^2 = \dfrac{\sum_{i=1}^{20} x_i^2}{20} - \mu^2 = \dfrac{800}{20} - 6^2 = 40 - 36 = 4$
Standard deviation ($\sigma$) is $\sqrt{4} = 2$.
The ratio of mean to standard deviation is:
$\dfrac{\mu}{\sigma} = \dfrac{6}{2} = \dfrac{3}{1}$
Answer: $3:1$
Hence, the correct answer is option: 'B'.