Statistics • Numerical Descriptive Measures
Enter class midpoints and their frequencies. This tool uses the shortcut formulas for grouped data to compute the approximate mean, variance, and standard deviation for both a population and a sample, following the procedure in the textbook tables.
Enter the class midpoint m and the class frequency f for each class interval, using one row per class. If you only have class limits, compute the midpoint first and then enter it with the frequency.
Grouped-data formulas treat every value in a class as if it equals the class midpoint. This midpoint assumption loses within-class detail, so the computed mean, variance, and standard deviation are approximations.
Let N = sum f_i be the total frequency. The grouped-data mean is x-bar = (sum(m_i f_i)) / N, where each midpoint is weighted by its frequency.
Population variance divides by N, while sample variance divides by n - 1 to estimate a population variance from a sample. In a grouped frequency table, n equals the total frequency N = sum f_i.
It uses sums of m_i f_i and m_i^2 f_i. For example, population variance can be computed as sigma^2 = (sum(m_i^2 f_i) - (sum(m_i f_i)^2)/N) / N, and standard deviation is sqrt(sigma^2).