Calculate Standard Deviation from Grouped Frequencies

Decide whether the groups are exact values or intervals

A table of exact values and frequencies preserves every value needed for a variance calculation. A table of intervals preserves only which class contains each observation. For interval data, replacing each observation with its class midpoint is an approximation. The estimated spread may differ from the original data's spread because the locations within classes are unknown. Clearly identify which kind of table you have before describing the result as exact or estimated.

Build a midpoint table

Consider classes [0,10), [10,20) and [20,30), with frequencies 2, 3 and 1. Their midpoints are 5, 15 and 25, and total frequency is six. Use a consistent boundary convention so an observation on a shared endpoint is counted only once. The midpoint-weighted total is 2×5+3×15+1×25=80, giving an estimated mean of 80/6=13.333333. Do not take the unweighted mean of the three midpoints: their frequencies are not equal.

Calculate the weighted squared deviations

Subtract the estimated mean from each midpoint, square the differences and multiply each square by its frequency. The weighted total is 2(5−40/3)²+3(15−40/3)²+(25−40/3)²=850/3≈283.333333. The frequency multiplier accounts for all observations represented by a midpoint. If you forget it, the calculation describes three equally weighted midpoint values rather than six midpoint-coded observations and produces the wrong centre as well as the wrong spread.

Choose the population or sample denominator

Dividing 850/3 by the total count six gives the midpoint population-variance estimate 425/9≈47.222222, so the corresponding SD is √425/3≈6.871843. Dividing instead by five gives sample variance 170/3≈56.666667 and sample SD≈7.527727. Use the sample convention when estimating population variance from a suitable sample, and the population convention when describing the grouped finite population. The interval approximation remains present whichever denominator you select.

Verify the supported step in the local form

Enter 5, 5, 15, 15, 15, 25 as the raw list. The calculator should show the two standard deviations above and a count of six. It treats the repeated midpoints as observations, which is exactly the numerical approximation being checked. Do not paste interval labels such as 0–10 or alternate midpoint and frequency columns into the same field. Those are not the form's input format. For large counts, a frequency-aware calculation avoids building an impractically long expanded list.

Check a different table

For classes [0,10), [10,20), [20,30), [30,40) with frequencies 1, 4, 3, 2, the midpoint-coded list is 5, 15, 15, 15, 15, 25, 25, 25, 35, 35. Its mean is 21 and weighted squared-deviation sum is 840. Population variance is 84 and SD≈9.165151; sample variance is 840/9≈93.333333 and SD≈9.660918. This example checks both unequal frequencies and a different total count, rather than simply rescaling the first result.

Understand information that grouping removes

Two original data sets can place the same numbers of observations in every interval but have different means and standard deviations. Values clustered near one boundary and values spread across a class share the same midpoint code. Therefore, the grouping error cannot generally be determined from frequencies alone. Smaller class widths can reduce information loss, but do not make arbitrary midpoint assumptions exact. Open-ended classes also lack a defined midpoint unless an additional, justified convention or model supplies one.

Keep frequency weights distinct from other weights

Repeating a value by an integer frequency is different from applying survey, reliability or inverse-variance weights. Weighted sample-variance formulas can depend on the meaning of the weights and sampling design; n−1 after summing arbitrary weights is not a universal rule. For this guide, frequencies are counts of observations. Report the table, total count, midpoint convention and chosen denominator with the final estimate, and keep the original ungrouped observations when they are available for an exact calculation.

Related guides

Mean, Median and Mode for Grouped Data

Distinguish exact frequency tables from interval groups, estimate the mean with midpoints, and interpret grouped median and modal-class calculations.

Find Standard Deviation with a Checked Worked Example

Calculate squared deviations, choose the sample or population denominator, take the square root, and interpret standard deviation in original units.

Calculate Variance: Population and Sample Examples

Follow two variance calculations from mean to squared deviations, verify n versus n−1, and check the result using an independent identity.