Find the population standard deviation of 2, 4, 4, 4, 5, 5, 7, 9.

How we read your problem

Treat the data as the entire population (divide by n).

Step-by-step solution

1. 1

   Find the mean

   $$\bar{x} = \frac{40}{8} = 5$$
2. 2

   Square each deviation from the mean

   $$(2 - 5)^2 = 9 \\ (4 - 5)^2 = 1 \\ (4 - 5)^2 = 1 \\ (4 - 5)^2 = 1 \\ (5 - 5)^2 = 0 \\ (5 - 5)^2 = 0 \\ (7 - 5)^2 = 4 \\ (9 - 5)^2 = 16$$
3. 3

   Add the squared deviations

   $$\sum (x_i - \bar{x})^2 = 32$$
4. 4

   Divide by n to get the population variance

   $$\sigma^2 = \frac{32}{8} = 4$$
5. 5

   Take the square root

   $$\sigma = \sqrt{4} = 2$$

Formulas used

• Sample standard deviation

  $$s = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n - 1}}$$

• Population standard deviation

  $$\sigma = \sqrt{\frac{\sum (x_i - \mu)^2}{n}}$$

Standard deviation measures how spread out the values are around the mean. A small value means the data are clustered near the mean.

Verification

• Variance = 4, and 2² = 4
• Sample SD for comparison: 2.138089935