Given the mean of a normal distribution, which of the following best describes the $standard deviation$?

• A. It represents the maximum value in the data set.
• B. It is always equal to the $mean$.
• C. It measures the average distance of data points from the $mean$.
• D. It is the sum of all data points divided by the number of points.

Answer: (C)

Verified step by step guidance

1. Understand the concept of standard deviation: it is a measure of how spread out the values in a data set are around the mean.
2. Recall that the mean is the average value of the data set, calculated as \(\bar{x} = \frac{1}{n} \sum_{i=1}^n x_i\), where \(x_i\) are the data points and \(n\) is the number of points.
3. Recognize that the standard deviation quantifies the average distance of each data point from the mean, not the maximum value or the mean itself.
4. The formula for the standard deviation \(\sigma\) of a population is \(\sigma = \sqrt{\frac{1}{n} \sum_{i=1}^n (x_i - \bar{x})^2}\), which calculates the square root of the average squared deviations from the mean.
5. Therefore, the standard deviation provides insight into the variability or dispersion of the data points relative to the mean.