In statistics, does the value of the $\mathrm{standard\; deviation}$ of a data set depend on the value of the $\mathrm{mean}$ of that data set?

• A. No, the $\mathrm{standard\; deviation}$ measures the spread of the data and does not depend on the value of the $\mathrm{mean}$.
• B. Yes, the $\mathrm{standard\; deviation}$ increases as the $\mathrm{mean}$ increases.
• C. Yes, the $\mathrm{standard\; deviation}$ is always equal to the $\mathrm{mean}$.
• D. No, the $\mathrm{standard\; deviation}$ is always $0$ regardless of the $\mathrm{mean}$.

Answer: (A)

Verified step by step guidance

1. Understand the definition of the mean: it 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.
2. Understand the definition of the standard deviation: it measures the spread or dispersion of the data points around the mean, calculated as \(s = \sqrt{\frac{1}{n-1} \sum_{i=1}^n (x_i - \bar{x})^2}\) for a sample.
3. Notice that the standard deviation depends on the differences between each data point and the mean, not on the mean itself. It quantifies how much the data varies, regardless of the mean's value.
4. Recognize that changing the mean by adding a constant to all data points shifts the data but does not change the spread, so the standard deviation remains the same.
5. Conclude that the standard deviation does not depend on the value of the mean; it only depends on how spread out the data points are around the mean.