Ratio of Mean to Standard Deviation for Given Observations

To find the ratio of the mean to the standard deviation of the given $20$ observations $x_1, x_2, \ldots, x_{20}$, let us define: $S_1 = \sum_{i=1}^{20} x_i \quad \text{and} \quad S_2 = \sum_{i=1}^{20} x_i^2$

We are given two equations:

1. $\sum_{i=1}^{20} (x_i + 5)^2 = 2500$
2. $\sum_{i=1}^{20} (x_i - 5)^2 = 100$

Expanding the first equation: $\sum_{i=1}^{20} (x_i^2 + 10 x_i + 25) = 2500$ $S_2 + 10 S_1 + 25 \times 20 = 2500$ $S_2 + 10 S_1 + 500 = 2500$ $S_2 + 10 S_1 = 2000 \quad \text{--- (1)}$

Expanding the second equation: $\sum_{i=1}^{20} (x_i^2 - 10 x_i + 25) = 100$ $S_2 - 10 S_1 + 25 \times 20 = 100$ $S_2 - 10 S_1 + 500 = 100$ $S_2 - 10 S_1 = -400 \quad \text{--- (2)}$

Subtracting equation (2) from equation (1): $(S_2 + 10 S_1) - (S_2 - 10 S_1) = 2000 - (-400)$ $20 S_1 = 2400 \implies S_1 = 120$

Adding equation (1) and equation (2): $2 S_2 = 1600 \implies S_2 = 800$

Now, we calculate the mean ($\bar{x}$) of the data: $\bar{x} = \frac{S_1}{n} = \frac{120}{20} = 6$

Next, we calculate the variance ($\sigma^2$): $\sigma^2 = \frac{S_2}{n} - (\bar{x})^2 = \frac{800}{20} - 6^2 = 40 - 36 = 4$

Thus, the standard deviation ($\sigma$) is: $\sigma = \sqrt{4} = 2$

Finally, the ratio of the mean to the standard deviation is: $\text{Ratio} = \frac{\bar{x}}{\sigma} = \frac{6}{2} = \frac{3}{1} = 3:1$

Hence, the correct option is B.