Question:

The mean of 100 observations is 50 and their standard deviation is 5. Then the sum of squares of all observations is

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To calculate the sum of squares of observations, use the formula $ \sum{x^2} = n \times \sigma^2 + n \times \bar{x}^2 $, where $ \sigma^2 $ is the variance and $ \bar{x} $ is the mean. This allows you to find the sum of squares directly from the given statistics like mean and standard deviation.

Updated On: Mar 29, 2025

250000
50000
255000
252500

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The Correct Option is D

Approach Solution - 1

The correct answer is (D) : 252500.

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Approach Solution -2

The correct answer is: (D) 252500.

We are given the following information about the 100 observations:

The mean ($\bar{x}$) of the observations is 50.
The standard deviation ($\sigma$) of the observations is 5.
The number of observations (n) is 100.

We need to find the sum of squares of all observations. To do this, we can use the following relationship between the sum of squares, variance, and mean:

The formula for the variance ($\sigma^2$) is:

$ \sigma^2 = \frac{\sum{x^2}}{n} - \bar{x}^2 $

Rearranging to solve for the sum of squares ($\sum{x^2}$):

$ \sum{x^2} = n \times \sigma^2 + n \times \bar{x}^2 $

Substituting the given values:

n = 100
$\sigma = 5$, so $\sigma^2 = 25$
$\bar{x} = 50$, so $\bar{x}^2 = 2500$

Substituting into the formula:

$ \sum{x^2} = 100 \times 25 + 100 \times 2500 = 2500 + 250000 = 252500 $

Therefore, the sum of squares of all observations is (D) 252500.