Q.The mean of 100 observations is 50 and their standard deviation is 5. The sum of all squares of all the observations is
(A) 50000
(B) 250000
(C) 252500
(D) 255000

The core concept here is the relationship between the mean, variance (or standard deviation), and the sum of squares of observations.

The variance $\sigma^2$ of $n$ observations $x_1, x_2, \dots, x_n$ with mean $\bar{x}$ is given by the formula:

We are given:

• Number of observations, $n = 100$
• Mean, $\bar{x} = 50$
• Standard deviation, $\sigma = 5$

1. First, calculate the variance: $\sigma^2 = 5^2 = 25$.
2. Substitute the given values into the variance formula: $25 = \frac{\sum x_i^2}{100} - (50)^2$ …

When we talk about a set of data, the mean tells us its central tendency, essentially the "average" value. The standard deviation, on the other hand, tells us about the spread or dispersion of the data points around that mean. A small standard deviation means data points are clustered close to the mean, while a large one means they are spread out.

The key insight here is that both the mean and the standard deviation are defined using sums of observations. Specifically, the mean involves the sum of the observations ($\sum x_i$), and the standard deviation (or its square, the variance) involves the sum of the squares of the observations ($\sum x_i^2$). This direct mathematical link allows us to find one quantity if the others are known.

The standard deviation formula that is most useful for this problem is:

This formula directly connects the standard deviation ($\sigma$), the number of observations ($n$), the mean ($\bar{x}$), and the sum of squares of observations ($\sum x_i^2$). We are given $n$, $\bar{x}$, and $\sigma$, and we need to find $\sum x_i^2$.

Let's work through the problem step-by-step.

1. Identify the given information.

   We are given:

   • Number of observations, $n = 100$
   • Mean of observations, $\bar{x} = 50$
   • Standard deviation of observations, $\sigma = 5$

2. Recall the formula for standard deviation involving the sum of squares.

   The standard deviation $\sigma$ is given by the formula:

Our goal is to find $\sum x_i^2$. To do this, we need to rearrange this formula.

3. Rearrange the formula to isolate the sum of squares.

First, square both sides of the equation to remove the square root:

Now, add $\bar{x}^2$ to both sides to isolate the term containing $\sum x_i^2$: …