Deriving Population Standard Deviation from Population Variance

Conditions

• The population variance $\sigma^2$ has already been calculated.
• The square root taken is the principal (non-negative) root.

Reasoning, step by step

1. Calculate the population variance $\sigma^2$ using the formula $\frac{1}{n} \sum (x_i - \mu)^2$.
2. Take the square root of the variance value: $\sigma = \sqrt{\sigma^2}$.
3. Ensure the result is non-negative, as standard deviation represents a magnitude of spread.
4. Assign the original unit of the data to the standard deviation.

Example

For the cheerleader weights, the population variance was calculated as 144. The population standard deviation is $\sqrt{144} = 12$ kg.

Common misconceptions

• Taking the square root of the mean instead of the variance.
• Forgetting that standard deviation must be non-negative.
• Confusing variance and standard deviation, reporting the variance as the final answer for standard deviation.

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Connected concepts

Explore next

• Why do we take the square root of the variance?
• What is the unit of the population standard deviation?
• How does the standard deviation relate to the spread of the data?