Which three quantities are required to calculate the variance of a data set?

• A. The $\mathrm{median}$, the $\mathrm{mode}$, and the $\mathrm{range}$
• B. The $\mathrm{minimum}$ value, the $\mathrm{maximum}$ value, and the $\mathrm{mean}$
• C. The $\sum \mathrm{data}$, the $\mathrm{median}$, and the $\mathrm{number}\mathrm{of}\mathrm{data}\mathrm{values}$
• D. The $\mathrm{mean}$, each $\mathrm{data}$ value, and the $\mathrm{number}\mathrm{of}\mathrm{data}\mathrm{values}$

Answer: (D)

Verified step by step guidance

1. Recall the formula for the variance of a data set, which is given by: \[\text{Variance} = \frac{1}{n} \sum_{i=1}^{n} (x_i - \bar{x})^2\] where \(n\) is the number of data values, \(x_i\) represents each individual data value, and \(\bar{x}\) is the mean of the data set.
2. Identify the quantities needed in the formula: - The mean (\(\bar{x}\)) is required to find the difference between each data value and the average. - Each data value (\(x_i\)) is needed to calculate these differences and then square them. - The number of data values (\(n\)) is necessary to divide the sum of squared differences to get the average squared deviation.
3. Understand why other options are incorrect: - The median and mode are measures of central tendency but are not used in the variance formula. - The minimum and maximum values help describe the range but do not directly contribute to variance calculation. - The sum of the data can be used to find the mean, but the median is not involved in variance calculation.
4. Summarize that to calculate variance, you must know the mean, each individual data value, and the total number of data points.
5. Therefore, the correct quantities required to calculate variance are: the mean, each data value, and the number of data values.