Which of the following statements is true about $standarddeviation$?

• A. $Standarddeviation$ is unaffected by outliers in the data.
• B. A $standarddeviation$ of $0$ means the data are widely spread out.
• C. $Standarddeviation$ measures the average distance of data points from the $mean$.
• D. $Standarddeviation$ is always negative.

Answer: (C)

Verified step by step guidance

1. Understand the definition of standard deviation: it is a measure of the amount of variation or dispersion in a set of data values.
2. Recall that standard deviation is calculated as the square root of the variance, where variance is the average of the squared differences from the mean. The formula for standard deviation (population) is: \(\\sigma = \\sqrt{\\frac{1}{N} \\sum_{i=1}^N (x_i - \\mu)^2}\), where \(x_i\) are data points, \(\\mu\) is the mean, and \(N\) is the number of data points.
3. Recognize that standard deviation is always non-negative because it is a square root of a squared quantity, so it cannot be negative.
4. Note that a standard deviation of zero means all data points are exactly equal to the mean, so there is no spread, not that data are widely spread out.
5. Understand that outliers affect the standard deviation because they increase the average distance of data points from the mean, thus increasing the standard deviation.