Find the variance for the following set of data. $$ 20,7,15,4,22,17,7,21,13 $$ 36.45 39.78 37.56 38.67

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1. Compute the mean $$ \mu $$ of the data set:
   $$
   \mu = \frac{20+7+15+4+22+17+7+21+13}{9} = \frac{126}{9} = 14
   $$

2. Calculate the squared differences from the mean:
   $$
   \begin{aligned}
   (20-14)^2 &= 6^2 = 36,\[5mm]
   (7-14)^2 &= (-7)^2 = 49,\[5mm]
   (15-14)^2 &= 1^2 = 1,\[5mm]
   (4-14)^2 &= (-10)^2 = 100,\[5mm]
   (22-14)^2 &= 8^2 = 64,\[5mm]
   (17-14)^2 &= 3^2 = 9,\[5mm]
   (7-14)^2 &= (-7)^2 = 49,\[5mm]
   (21-14)^2 &= 7^2 = 49,\[5mm]
   (13-14)^2 &= (-1)^2 = 1.\[5mm]
   \end{aligned}
   $$

3. Sum all the squared differences:
   $$
   36 + 49 + 1 + 100 + 64 + 9 + 49 + 49 + 1 = 358
   $$

4. Since we are finding the population variance, divide the total by the number of data points $$ n = 9 $$:
   $$
   \sigma^2 = \frac{358}{9} \approx 39.78
   $$

Thus, the variance of the data is approximately $$ 39.78 $$.
The variance of the data is approximately 39.78.

Find the variance for the following set of data. \( 20,7,15,4,22,17,7,21,13 \) 36.45 39.78 37.56 38.67

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