Grouped Data Standard Deviation Calculator

To calculate the standard deviation of grouped data, the calculator treats each frequency interval as if all its data points are concentrated at the class midpoint. First, the midpoint (x) for each range is found by averaging the lower and upper bounds. Next, the calculator multiplies each midpoint by its corresponding frequency (f) to find the sum of all values, allowing it to determine the grouped mean. Finally, it applies the variance formula: s squared equals the sum of f times the quantity of midpoint minus mean squared, divided by the total number of data points minus one for a sample. The standard deviation is simply the square root of this resulting variance.

Imagine you have a grouped frequency distribution representing test scores divided into three intervals: 50 to 60 (frequency of 2), 60 to 70 (frequency of 5), and 70 to 80 (frequency of 3). First, find the midpoints (x) for each interval: 55, 65, and 75. Next, multiply each frequency by its midpoint: (2 * 55 = 110), (5 * 65 = 325), and (3 * 75 = 225), giving a sum of 660. With a total frequency of 10, the mean is 660 / 10 = 66. Then, calculate squared deviations from the mean for each group, multiply by their frequencies, sum them up to find the variance, and take the square root to yield the final standard deviation.

When setting up your frequency distribution, ensure that your class intervals are mutually exclusive and of equal width whenever possible to minimize grouping error. Keep in mind that calculations using grouped data are approximations of the true dataset because individual data point identities are lost inside the intervals. Always double-check your frequency sums and interval bounds before entering them into the calculator to prevent skewed variance results.