If for a distribution `sumx_(i)^2=2400` and `sumx_(i)=250` and the total number of observations is 50, then variance is

Options

A. (a) 20
B. (b) 21
C. (c) 22
D. (d) 23

Answer

To find the variance of the given distribution, we will use the formula for variance:

\[ \sigma^2 = \frac{\sum x_i^2}{N} - \left(\frac{\sum x_i}{N}\right)^2 \]

Where:
- \(\sigma^2\) is the variance,
- \(\sum x_i^2\) is the sum of the squares of the observations,
- \(\sum x_i\) is the sum of the observations,
- \(N\) is the total number of observations.

Given:
- \(\sum x_i^2 = 2400\)
- \(\sum x_i = 250\)
- \(N = 50\)

Now, let's calculate the variance step by step:

### Step 1: Calculate \(\frac{\sum x_i^2}{N}\)

\[ \frac{\sum x_i^2}{N} = \frac{2400}{50} \]

Calculating this gives:

\[ \frac{2400}{50} = 48 \]

### Step 2: Calculate \(\frac{\sum x_i}{N}\)

\[ \frac{\sum x_i}{N} = \frac{250}{50} \]

Calculating this gives:

\[ \frac{250}{50} = 5 \]

### Step 3: Calculate \(\left(\frac{\sum x_i}{N}\right)^2\)

Now, we square the result from Step 2:

\[ \left(\frac{\sum x_i}{N}\right)^2 = 5^2 = 25 \]

### Step 4: Substitute values into the variance formula

Now we substitute the values we calculated into the variance formula:

\[ \sigma^2 = 48 - 25 \]

Calculating this gives:

\[ \sigma^2 = 23 \]

### Conclusion

Thus, the variance of the distribution is:

\[ \text{Variance} = 23 \]

### Final Answer

The correct answer is option D: 23.

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