Grouped Data Standard Deviation Calculator
Estimate the mean, sample variance, and sample standard deviation of a frequency distribution using each class interval's midpoint.
Grouped frequency data
Choose 2 – 10 class intervals. Changing this preserves existing rows where possible.
Results
Sample standard deviation estimated from class midpoints.
Standard deviation 36.05; mean 172.50 across 30 observations.
Enter complete ranges and at least two total observations to calculate results.
Frequency by range
Compare how many observations fall in each entered interval. Bar height encodes frequency from a zero baseline.
The 160 – 189 interval has the highest frequency in the demonstration data.
| Range | Frequency |
|---|
Frequency chart unavailable until at least two ranges contain positive frequencies.
Calculation breakdown
| Range | Min | Max | Midpoint | Frequency | Midpoint × frequency | Frequency × midpoint² |
|---|---|---|---|---|---|---|
| Range 1 | 100.00 | 129.00 | 114.50 | 5 | 572.50 | 65,551.25 |
| Range 2 | 130.00 | 159.00 | 144.50 | 4 | 578.00 | 83,521.00 |
| Range 3 | 160.00 | 189.00 | 174.50 | 12 | 2,094.00 | 365,403.00 |
| Range 4 | 190.00 | 219.00 | 204.50 | 6 | 1,227.00 | 250,921.50 |
| Range 5 | 220.00 | 249.00 | 234.50 | 3 | 703.50 | 164,970.75 |
| Totals | – | – | – | 30 | 5,175.00 | 930,367.50 |
Each interval is represented by its midpoint. Grouping therefore produces an estimate: the exact within-interval observations are not known.
No calculation rows are available for the current state.
How to use this grouped data standard deviation calculator
What this calculator does
This calculator estimates the center and spread of data that have already been summarized into class intervals. Instead of requiring every original observation, it represents each interval by its midpoint, weights that midpoint by the interval's frequency, and returns the grouped-data Mean, Variance, and Standard deviation. It also reports the Number of samples and Ranges used. The variance and standard deviation follow the sample convention with an n – 1 denominator. Because the original values inside each class are unknown, these outputs are midpoint-based estimates, not a reconstruction of the ungrouped dataset. OpenStax's definitions of a frequency table, midpoint, variance, and standard deviation provide useful background for interpreting the calculation.
When to use it
Use the calculator when you have a frequency distribution rather than raw observations – for example, customer ages grouped into bands, test scores summarized into score intervals, measured sizes binned into ranges, or operational observations reported only as class counts. It is especially useful for checking hand calculations, preparing a compact descriptive-statistics summary, or comparing how changes in grouped frequencies affect the estimated spread.
How to calculate
- The calculator opens with a complete five-range demonstration, so the results, chart, breakdown table, and Excel workbook are usable immediately. Select a different Number of ranges only if your frequency table has a different number of classes.
- For every class, replace Min value, Max value, and Frequency with your data. Results update live; no separate Calculate button is needed.
- Read Standard deviation as the estimated typical spread around the grouped mean, then use Mean, Variance, and Number of samples to verify the scale and sample size. The Frequency by range chart shows the same counts visually, while the Calculation breakdown shows the midpoint-weighted terms.
- Choose Download Excel to export the current validated inputs and results as a real XLSX workbook. Reset clears the demonstration values, results, chart, table, and workbook state; after Reset, Excel export stays disabled until you enter a complete valid dataset again.
Input guide
Number of ranges is a required whole-number selector from 2 through 10. A realistic example is 5. Increasing it adds blank class rows; reducing it removes trailing rows. Match this setting to the classes in your source frequency table rather than splitting data only to obtain a particular standard deviation.
Min value is the required lower endpoint of each interval. Enter an ordinary decimal number using a period as the decimal separator; properly grouped thousands such as 1,250.5 are accepted. Negative endpoints are allowed, and the supported magnitude is up to 1 trillion in absolute value. Example: 100. The common mistake is entering a lower endpoint that is not strictly below the same row's maximum.
Max value is the required upper endpoint in the same units as Min value, with the same decimal format and magnitude limit. Example: 129. A wider interval changes its midpoint, which can move both the mean and spread. Keep interval conventions consistent with your source data, and avoid overlapping classes unless overlap is genuinely part of the data definition.
Frequency is the required nonnegative whole-number count for a range, from 0 to 1,000,000,000. Example: 5. Higher frequency gives that interval's midpoint more weight in the mean and variance. Zero is allowed for a class, but the total across all ranges must be at least two observations. Do not enter percentages or decimal frequencies.
Output guide
Standard deviation is the primary output, shown to two decimals in the same unit as the interval endpoints. A value near zero means the represented midpoints are tightly clustered; a larger value means more spread. Variance is the corresponding squared-unit measure and equals the standard deviation squared. Mean is the frequency-weighted midpoint average. Number of samples is the exact sum of frequencies, and Ranges used is the selected class count.
The Frequency by range chart is a zero-baseline comparison of interval counts; every bar is driven directly by the current Frequency values. The Calculation breakdown columns show Range, Min, Max, Midpoint, Frequency, Midpoint × frequency, and Frequency × midpoint². These last terms let you audit the weighted mean and the standard grouped-data variance shortcut. The table values and workbook use the same canonical calculation model as the displayed results.
Worked example
The opening example uses five intervals: 100 – 129 with frequency 5, 130 – 159 with 4, 160 – 189 with 12, 190 – 219 with 6, and 220 – 249 with 3. Their midpoints are 114.5, 144.5, 174.5, 204.5, and 234.5. The frequencies sum to 30 observations, and the weighted midpoint sum is 5,175, giving a mean of 5,175 ÷ 30 = 172.50. Applying the sample variance calculation to the midpoint-weighted deviations gives 1,299.31; its square root is the displayed 36.05 standard deviation. This is the same numerical state exported by the initial workbook.
How the grouped-data formula works
For each interval, the midpoint M is (minimum + maximum) ÷ 2. The grouped mean is the sum of M × frequency divided by the total frequency n. The sample variance can then be computed as the sum of frequency × (M – mean)² divided by n – 1, and the sample standard deviation is the square root of that variance. The calculator uses this deviation form internally because it is numerically stable; it is algebraically equivalent to the common shortcut based on Σ(fM²) and n × mean².
mean = Σ(fM) / n · sample variance = Σ[f(M – mean)²] / (n – 1) · sample standard deviation = √variance
The NIST Engineering Statistics Handbook discussion of measures of scale explains why variance and standard deviation summarize spread and why standard deviation returns the spread to the original measurement units.
Interpreting grouped results responsibly
Grouping compresses information. Two datasets can produce the same frequency table while containing different exact observations inside each interval, so their true ungrouped means or standard deviations can differ from the midpoint estimates. Narrow, sensible classes usually preserve more detail than very wide classes. If you still have the raw observations and need exact descriptive statistics, calculate directly from the raw data instead of discarding detail through grouping.
The frequency chart helps reveal where observations concentrate, but it is a category bar view of the entered intervals rather than a probability-density estimate. For a formal histogram – especially when class widths differ – bar area and normalization choices matter. The NIST histogram guidance describes how frequency and normalized histograms represent distributions. Use the table for exact entered counts and the chart for a quick comparison of concentration across ranges.