Which of the following is the correct formula for the population variance?

• A. ${\sigma }_2=\frac{\sum _{i=1}{(x_i-\mu )}^2}{N}$
• B. ${\sigma }_2=\frac{\sum _{i=1}{(x_i-\mu )}^2}{N+1}$
• C. ${\sigma }_2=\frac{\sum _{i=1}{x}_i}{N}$
• D. $s_2=\frac{\sum _{i=1}{(x_i-\mu )}^2}{N-1}$

Answer: (A)

Verified step by step guidance

1. Understand that the population variance, denoted by \(\sigma^{2}\), measures the average squared deviation of each data point from the population mean \(\mu\).
2. Recall the formula for population variance is the sum of squared differences between each data point \(x_{i}\) and the population mean \(\mu\), divided by the total number of data points \(N\).
3. Write the formula as: \(\sigma^{2} = \frac{\sum_{i=1}^{N} (x_{i} - \mu)^{2}}{N}\)
4. Note that dividing by \(N\) (the population size) distinguishes population variance from sample variance, which divides by \(N-1\).
5. Verify that the correct formula matches the one where the numerator is the sum of squared deviations from the mean and the denominator is \(N\), confirming it represents population variance.