Find the population variance and population standard deviation of the data set: 2, 4, 6, 8.

Step 1: Find the population mean.

Mean = (2 + 4 + 6 + 8) / 4
Mean = 20 / 4
Mean = 5

Step 2: Find each deviation from the mean.

2 - 5 = -3
4 - 5 = -1
6 - 5 = 1
8 - 5 = 3

Step 3: Square each deviation.

(-3)^2 = 9
(-1)^2 = 1
1^2 = 1
3^2 = 9

Step 4: Add the squared deviations.

9 + 1 + 1 + 9 = 20

Step 5: Find the population variance.

Because the data represent the entire population, divide by N = 4.

Population variance = 20 / 4
Population variance = 5

Step 6: Find the population standard deviation.

The standard deviation is the square root of the variance:

σ = √5
σ ≈ 2.24

Therefore, the population variance is 5 and the population standard deviation is approximately 2.24.