If the total number of observations is 20. `Sigmax_(i)= 1000` and `Sigmax_(i)^(2)=84000` then what is the variance of the distribution?

Options

A. 1500
B. 1600
C. 1700
D. 1800

Answer

To find the variance of the distribution given the total number of observations \( n = 20 \), the sum of observations \( \Sigma x_i = 1000 \), and the sum of squares of observations \( \Sigma x_i^2 = 84000 \), we can follow these steps:

### Step 1: Write the formula for variance

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

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

### Step 2: Substitute the known values into the formula

We know:
- \( \Sigma x_i^2 = 84000 \)
- \( n = 20 \)
- \( \Sigma x_i = 1000 \)

Substituting these values into the variance formula:

\[ \sigma^2 = \frac{84000}{20} - \left(\frac{1000}{20}\right)^2 \]

### Step 3: Calculate \( \frac{\Sigma x_i^2}{n} \)

Now, calculate \( \frac{84000}{20} \):

\[ \frac{84000}{20} = 4200 \]

### Step 4: Calculate \( \frac{\Sigma x_i}{n} \)

Next, calculate \( \frac{1000}{20} \):

\[ \frac{1000}{20} = 50 \]

### Step 5: Square the result from Step 4

Now, square the result from Step 4:

\[ \left(\frac{1000}{20}\right)^2 = 50^2 = 2500 \]

### Step 6: Substitute back into the variance formula

Now substitute back into the variance formula:

\[ \sigma^2 = 4200 - 2500 \]

### Step 7: Calculate the variance

Finally, calculate \( 4200 - 2500 \):

\[ \sigma^2 = 1700 \]

Thus, the variance of the distribution is:

\[ \boxed{1700} \]