JAMIA 2023 — Mathematics PYQ

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Question

JAMIA 2023 — Mathematics PYQ

JAMIA | Mathematics | 2023

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

Choose the correct answer:

Explanation

Step 1: Given Data

Number of observations ( $n$ ) = $100$
Mean ( $\bar{x}$ ) = $50$
Standard Deviation ( $\sigma$ ) = $5$

Step 2: Formula for Variance

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

\sigma^2 = \frac{\sum x_i^2}{n} - (\bar{x})^2

Where $\sum x_i^2$ is the sum of squares of all observations.

Step 3: Calculation

First, find the variance ( $\sigma^2$ ):

\sigma^2 = 5^2 = 25

Now, substitute the values into the formula:

25 = \frac{\sum x_i^2}{100} - (50)^2

25 = \frac{\sum x_i^2}{100} - 2500

Add $2500$ to both sides:

25 + 2500 = \frac{\sum x_i^2}{100}

2525 = \frac{\sum x_i^2}{100}

Now, multiply by $100$ :

\sum x_i^2 = 2525 \times 100

\sum x_i^2 = 252500

Final Answer:

The sum of squares of all observations is:

252500