In descriptive statistics, what is the relationship between variance and standard deviation for a dataset with variance $s^2$ and standard deviation $s$?

• A. $s^2=\sqrt{s}$, so variance is the square root of the standard deviation
• B. $s=s^2$, so standard deviation equals variance
• C. $s=\frac{1}{s^2}$, so standard deviation is the reciprocal of the variance
• D. $s=\sqrt{s^2}$, so standard deviation is the square root of the variance

Answer: (D)

Verified step by step guidance

1. Recall the definitions: Variance, denoted as \(s^{2}\), measures the average squared deviation of each data point from the mean, while standard deviation, denoted as \(s\), measures the average deviation in the original units of the data.
2. Understand that variance is calculated by squaring the standard deviation, which means variance is the square of the standard deviation: \(s^{2} = (s)^{2}\).
3. To find the standard deviation from the variance, you take the square root of the variance, which reverses the squaring process: \(s = \sqrt{s^{2}}\).
4. This relationship shows that the standard deviation is always the positive square root of the variance, ensuring it is expressed in the same units as the original data.
5. Therefore, the correct relationship is: \(s = \sqrt{s^{2}}\), meaning the standard deviation is the square root of the variance.