How to calculate population variance for a finite dataset treated as the entire population?

Conditions

• The dataset represents the entire population, not a sample.
• The number of data points $n$ is finite and greater than zero.

Reasoning, step by step

1. Calculate the arithmetic mean $\mu$ by summing all data values and dividing by $n$.
2. Subtract the mean $\mu$ from each individual data point $x_i$ to find the deviations.
3. Square each deviation to ensure all values are non-negative: $(x_i - \mu)^2$.
4. Sum all the squared deviations together.
5. Divide the total sum of squared deviations by the population size $n$ to obtain the variance $\sigma^2$.

Example

For the weights 67, 79, 55, 61, and 88 kg, the mean is 70. The squared deviations are $(-3)^2, 9^2, (-15)^2, (-9)^2,$ and $18^2$, which sum to 720. Dividing by $n=5$ gives a population variance of 144.

Common misconceptions

• Dividing by $n-1$ instead of $n$, which is the formula for sample variance, not population variance.
• Forgetting to square the deviations before summing them, which would result in a sum of zero.
• Calculating the mean after finding the deviations instead of before.

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Connected concepts

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• Why is the denominator $n$ instead of $n-1$ for population variance?
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