Grouped Standard Deviation Calculator NZ

This calculator finds the mean and standard deviation of grouped or frequency data, where instead of a list of individual values you have categories or class intervals and how often each occurs. Data often arrives summarised this way: a frequency table of exam marks grouped into bands, ages grouped into ranges, or any survey where responses are tallied by category. To find the spread of such data you cannot simply use the raw values, because you only have the groups and their counts. The standard approach uses the midpoint of each class to represent its values, weights each midpoint by its frequency, and works the mean and standard deviation from those weighted figures. This tool does exactly that. You enter the class midpoints in one box and the matching frequencies in another, in the same order, and the calculator returns the sample standard deviation, the mean, the population standard deviation, and the total number of observations. The results update as you type. Use it for statistics homework involving frequency tables, for analysing survey or grouped data, or whenever your data comes as categories with counts rather than a raw list. The mean is the sum of each midpoint times its frequency, divided by the total frequency. The standard deviation measures how spread out the data is around that mean, using the squared deviations of the midpoints weighted by frequency. The calculator gives both the sample standard deviation, which divides by one less than the total count and is used when your data is a sample, and the population version, which divides by the full count. Because midpoints stand in for the actual values within each class, grouped statistics are an approximation, very good when classes are reasonably narrow and the data is evenly spread within them.

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Class midpoints

Frequencies (same order)

11.19

sample standard deviation

Mean24

Population SD10.91

Total count (n)20

Uses class midpoints weighted by frequency. Mean = sum(f x) / sum(f). Sample SD divides by n - 1; population SD by n. An approximation since midpoints stand in for values.

How it works

Each class midpoint is multiplied by its frequency and summed, then divided by the total frequency to get the mean. The standard deviation comes from the squared difference of each midpoint from the mean, weighted by frequency and summed; dividing by one less than the total count and taking the square root gives the sample standard deviation, while dividing by the total count gives the population version.

Worked example

For midpoints 5, 15, 25, 35, 45 with frequencies 2, 5, 8, 3, 2, the total count is 20 and the weighted sum is 480, so the mean is 24. The weighted squared deviations sum to 2,380, so the population standard deviation is the square root of 2,380 over 20, about 10.91, and the sample standard deviation, dividing by 19, is about 11.19.

Related calculators

Standard Deviation: raw data SD.
Mean Median Mode: averages.
Coefficient of Variation: relative spread.
Weighted Average: weighted mean.

Frequently asked questions

What is grouped data?

Grouped data is summarised into classes or categories with a frequency for each, rather than listed as individual values. A frequency table of test scores in bands is a common example. To find statistics from it, the midpoint of each class represents its values.

How do you find the standard deviation of grouped data?

Use each class midpoint, weighted by its frequency. Find the mean as the sum of midpoint times frequency over the total frequency, then take the frequency-weighted squared deviations from the mean, divide by one less than the count for a sample, and take the square root.

Why is it an approximation?

Because the actual values within each class are unknown, the midpoint stands in for all of them. This is a good approximation when classes are narrow and values are spread evenly within them, but it can differ slightly from the standard deviation you would get from the raw data.