A data consists of 20 observations x_1, x_2, , x_ 20 . If ∑_ i=1 ^...

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: $3:1$