In statistics, how are variance and standard deviation related?

• A. The standard deviation is the variance divided by the sample size.
• B. The variance is the square root of the standard deviation.
• C. Variance and standard deviation are always equal.
• D. The standard deviation is the square root of the variance.

Answer: (D)

Verified step by step guidance

1. Understand that both variance and standard deviation are measures of spread or dispersion in a data set, indicating how much the data points deviate from the mean.
2. Recall the definition of variance: it is the average of the squared differences between each data point and the mean, mathematically expressed as \(\text{Variance} = \sigma^2 = \frac{1}{n} \sum_{i=1}^n (x_i - \mu)^2\) for a population, or \(s^2 = \frac{1}{n-1} \sum_{i=1}^n (x_i - \bar{x})^2\) for a sample.
3. Recognize that the standard deviation is the positive square root of the variance, which brings the units back to the original scale of the data, expressed as \(\text{Standard Deviation} = \sigma = \sqrt{\text{Variance}}\) for a population, or \(s = \sqrt{s^2}\) for a sample.
4. Note that the statement 'The standard deviation is the variance divided by the sample size' is incorrect because variance is not divided by the sample size to get standard deviation; rather, variance itself involves division by the sample size or (n-1) in the sample case.
5. Conclude that the correct relationship is: the standard deviation is the square root of the variance, which means standard deviation and variance are related but not equal, and standard deviation is always in the original units of the data.