Question

A data consists of 20 observations $x_1, x_2, \dots, x_{20}$. 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
• B. 3:1
• C. 3:2
• D. 4:1

Hint:

Expand the squared terms and subtract/add the two given summations to find the sum of observations and the sum of their squares.

Answer 1

The Correct Option is B

To solve this problem, we start by expanding the given summations for the data of 20 observations. We are given two equations:
$$ \sum_{i=1}^{20} (x_i + 5)^2 = 2500 \quad \dots (1) $$
$$ \sum_{i=1}^{20} (x_i - 5)^2 = 100 \quad \dots (2) $$
Expanding the squared terms inside the summation of equation (1):
$$ \sum (x_i^2 + 10x_i + 25) = 2500 $$
$$ \sum x_i^2 + 10 \sum x_i + 20(25) = 2500 $$
$$ \sum x_i^2 + 10 \sum x_i = 2000 \quad \dots (3) $$
Similarly, expanding the summation of equation (2):
$$ \sum (x_i^2 - 10x_i + 25) = 100 $$
$$ \sum x_i^2 - 10 \sum x_i + 500 = 100 $$
$$ \sum x_i^2 - 10 \sum x_i = -400 \quad \dots (4) $$
Subtracting equation (4) from equation (3) helps us isolate the sum of the observations:
$$ 20 \sum x_i = 2400 \implies \sum x_i = 120 $$
The mean ($\bar{x}$) of the data is given by $\bar{x} = \frac{\sum x_i}{n}$ where $n=20$:
$$ \bar{x} = \frac{120}{20} = 6 $$
Adding equations (3) and (4) gives us the sum of squares of the observations:
$$ 2 \sum x_i^2 = 1600 \implies \sum x_i^2 = 800 $$
Standard deviation ($\sigma$) is calculated using the formula $\sigma = \sqrt{\frac{\sum x_i^2}{n} - (\bar{x})^2}$:
$$ \sigma = \sqrt{\frac{800}{20} - 6^2} = \sqrt{40 - 36} = \sqrt{4} = 2 $$
Finally, the ratio of mean to standard deviation is:
$$ \text{Ratio} = \bar{x} : \sigma = 6 : 2 = 3 : 1 $$