What is the relationship between the $\mathrm{variance}$ and the $\mathrm{standard\; deviation}$ for a sample data set?

• A. The $\mathrm{variance}$ is always less than the $\mathrm{standard\; deviation}$.
• B. The $\mathrm{variance}$ is the $\sqrt{\mathrm{standard\; deviation}}$.
• C. The $\mathrm{standard\; deviation}$ is the $\sqrt{\mathrm{variance}}$.
• D. The $\mathrm{standard\; deviation}$ is equal to the $\mathrm{variance}$.

Answer: (C)

Verified step by step guidance

1. Understand the definitions: Variance measures the average squared deviation of each data point from the mean, while standard deviation measures the average deviation in the original units of the data.
2. Recall the formula for variance of a sample: \(S^2 = \frac{1}{n-1} \sum_{i=1}^n (X_i - \bar{X})^2\), where \(S^2\) is the sample variance, \(X_i\) are the data points, \(\bar{X}\) is the sample mean, and \(n\) is the sample size.
3. Recall the formula for standard deviation of a sample: \(S = \sqrt{S^2}\), which means the standard deviation is the positive square root of the variance.
4. Interpret the relationship: Since variance is in squared units, taking the square root (to get the standard deviation) brings the measure back to the original units of the data, making it easier to interpret.
5. Conclude that the correct relationship is: The standard deviation is the square root of the variance.