Which of the following is not a property of the $s \left(\mathrm{standard\; deviation}\right)$?

• A. The $\mathrm{standard\; deviation}$ is zero if all data values are identical.
• B. The $\mathrm{standard\; deviation}$ is affected by outliers.
• C. The $\mathrm{standard\; deviation}$ is always non-negative.
• D. The $\mathrm{standard\; deviation}$ is equal to the $\mathrm{mean}$ of the data.

Answer: (D)

Verified step by step guidance

1. Step 1: Understand the definition of standard deviation. The standard deviation measures the amount of variation or dispersion in a set of data values. It is calculated as the square root of the variance.
2. Step 2: Review the properties of standard deviation: (a) It is zero if all data values are identical because there is no variation. (b) It is affected by outliers since extreme values increase the spread of data. (c) It is always non-negative because it is a square root of a variance, which cannot be negative.
3. Step 3: Analyze the statement 'The standard deviation is equal to the mean of the data.' This is not a property of standard deviation because the mean and standard deviation measure different aspects of the data: the mean measures central tendency, while the standard deviation measures spread.
4. Step 4: Conclude that the statement 'The standard deviation is equal to the mean of the data' is not a property of the standard deviation, making it the correct answer to the question.
5. Step 5: Summarize that the other statements are true properties of standard deviation, reinforcing the understanding of how standard deviation behaves in relation to data.