Which of the following correctly represents the formula for the population standard deviation?

• A. $\sqrt{\frac{\sum _{i=1}{x{i}_{}-\mu }^2}{n}}$
• B. $\frac{\sum _{i=1}{x{i}_{}-\mu }^2}{N}$
• C. $\sqrt{\frac{\sum _{i=1}{x{i}_{}-\mu }^2}{N}}$
• D. $\sqrt{\frac{\sum _{i=1}{x{i}_{}-\mu }^2}{N-1}}$

Answer: (C)

Verified step by step guidance

1. Understand that the population standard deviation measures the spread of all data points in an entire population around the population mean (\( \mu \)).
2. Recall the formula for population variance, which is the average of the squared differences between each data point \( x_i \) and the population mean \( \mu \): \[ \\text{Population Variance} = \\sigma^2 = \\frac{\\sum_{i=1}^N (x_i - \\mu)^2}{N} \]
3. Recognize that the population standard deviation is the square root of the population variance, so you take the square root of the above expression: \[ \\text{Population Standard Deviation} = \\sigma = \\sqrt{ \\frac{\\sum_{i=1}^N (x_i - \\mu)^2}{N} } \]
4. Note that \( N \) is the size of the entire population, and the denominator is \( N \) (not \( N-1 \)) because we are dealing with the population, not a sample.
5. Compare the given options and identify the formula that matches the above expression exactly, which includes the square root of the sum of squared deviations divided by \( N \).