Question

find the population variance and standard deviation
9, 18, 30, 36, 42

the population variance is (type an integer or a decimal.)
the standard deviation is (type an integer or a decimal.)

Explanation:

⚡ Using what you learned: Variance_Definition · Standard Deviation

Step 1: Calculate the population mean

The given data set is: \( 9, 18, 30, 36, 42 \)

The number of data points is \( N = 5 \).

Calculate the population mean (\( \mu \)):

$$ \mu = \frac{9 + 18 + 30 + 36 + 42}{5} = \frac{135}{5} = 27 $$

Step 2: Calculate the squared deviations from the mean

Subtract the mean from each data value and square the result:

$$ (9 - 27)^2 = (-18)^2 = 324 $$

$$ (18 - 27)^2 = (-9)^2 = 81 $$

$$ (30 - 27)^2 = (3)^2 = 9 $$

$$ (36 - 27)^2 = (9)^2 = 81 $$

$$ (42 - 27)^2 = (15)^2 = 225 $$

Step 3: Calculate the population variance

Sum the squared deviations and divide by the population size \( N = 5 \):

$$ \sigma^2 = \frac{324 + 81 + 9 + 81 + 225}{5} = \frac{720}{5} = 144 $$

Step 4: Calculate the population standard deviation

Take the square root of the population variance:

$$ \sigma = \sqrt{144} = 12 $$

Answer:

The population variance is 144
The standard deviation is 12