Which of the following best describes the relationship between $\mathrm{variance}$ and $\mathrm{standard\; deviation}$ for a data set?

• A. The $\mathrm{variance}$ is the square root of the $\mathrm{standard\; deviation}$.
• B. The $\mathrm{variance}$ and $\mathrm{standard\; deviation}$ are always equal.
• C. The $\mathrm{standard\; deviation}$ is always twice the $\mathrm{variance}$.
• D. The $\mathrm{standard\; deviation}$ is the square root of the $\mathrm{variance}$.

Answer: (D)

Verified step by step guidance

1. Recall the definitions of variance and standard deviation for a data set. Variance measures the average squared deviation from the mean, while standard deviation measures the average deviation from the mean in the original units.
2. Understand that variance is calculated as the average of the squared differences between each data point and the mean, expressed as \(\sigma^2 = \frac{1}{n} \sum_{i=1}^n (x_i - \mu)^2\) for a population, where \(\sigma^2\) is the variance.
3. Recognize that the standard deviation is the positive square root of the variance, given by \(\sigma = \sqrt{\sigma^2}\), which brings the units back to the original scale of the data.
4. Analyze the given options by comparing them to the relationship \(\sigma = \sqrt{\sigma^2}\): the variance is not the square root of the standard deviation, they are not equal, and the standard deviation is not twice the variance.
5. Conclude that the correct relationship is that the standard deviation is the square root of the variance, which aligns with the mathematical definitions and preserves the units of measurement.