How to calculate the standard deviation of a set of population data?

Conditions

• The dataset represents a full population, not a sample.
• The number of data points $n$ is finite and non-empty.
• The data values are real numbers.

Reasoning, step by step

1. Calculate the mean of the data: $\bar{x} = \frac{\sum x_i}{n}$.
2. Subtract the mean from each data point and square the result: $(x_i - \bar{x})^2$.
3. Sum all the squared deviations: $\sum (x_i - \bar{x})^2$.
4. Divide the sum of squared deviations by the number of data points $n$ to find the population variance: $\sigma^2 = \frac{\sum (x_i - \bar{x})^2}{n}$.
5. Take the non-negative square root of the variance to find the standard deviation: $\sigma = \sqrt{\sigma^2}$.

Example

For the population data 6, 7, 8, 9, 10, the mean is 8. The population variance is $\sigma_1^2 = \frac{(6-8)^2+(7-8)^2+(8-8)^2+(9-8)^2+(10-8)^2}{5} = \frac{10}{5} = 2$. The standard deviation is $\sigma_1 = \sqrt{2} \approx 1.41$.

Common misconceptions

• Using $n-1$ in the denominator instead of $n$; this applies to sample variance, not population variance.
• Forgetting to take the square root of the variance, leaving the answer as the variance itself.
• Calculating the absolute differences instead of the squared differences.

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Connected concepts

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