What Is Standard Deviation for Grouped Data?
When data are grouped into class intervals, you no longer have the individual values. The calculation uses the class midpoint as the representative value for each class, so the mean and standard deviation are estimates rather than exact figures.
The calculator finds each midpoint, the mean of the grouped data, the sum f(x − x̄)², the sample variance and standard deviation, the coefficient of variation and an estimated median. The working table shows each class so you can follow or check a textbook solution.
Below you will find the formulas, a variables table, a worked example and answers to common questions such as why grouped results are approximate and whether to divide by N or N − 1.
How to Use the Standard Deviation for Grouped Data Calculator
- Type one class per line as “lower, upper, frequency”.
- Choose sample or population.
- Check the working table; it matches the layout most textbooks use.
- Remember the answer is an approximation because the exact values inside each class are unknown.
Standard Deviation for Grouped Data Formula
Class midpoint
Mean
Sample variance and SD
Median of grouped data
Standard Deviation for Grouped Data Variables Explained
| Symbol | Meaning | Notes |
|---|---|---|
| x | Class midpoint | (lower + upper) ÷ 2 |
| f | Class frequency | |
| N | Total frequency | Σf |
| x̄ | Mean of grouped data | Σfx ÷ N |
| s | Standard deviation | √(Σf(x − x̄)² ÷ (N − 1)) |
Standard Deviation for Grouped Data Worked Example
With the example values loaded in the calculator above, the steps are:
- Find the midpoint x of every class: (lower + upper) ÷ 2.
- Mean x̄ = Σ f·x ÷ N = 1165 ÷ 45 = 25.8889.
- Σ f(x − x̄)² = 5564.44.
- Variance = 5564.44 ÷ (45 − 1) = 126.465.
- Standard deviation = √126.465 = 11.2457.
Result: Sample standard deviation (s) = 11.2457; Mean of grouped data = 25.8889.
How to Read a Grouped Standard Deviation
With grouped data each value is replaced by its class midpoint, so the mean and standard deviation are estimates. The calculator weights each squared distance from the grouped mean by the class frequency, adds them and divides by N − 1 for a sample.
The estimate improves with narrower classes and gets worse with wide or open-ended ones. The table shows each class midpoint and its contribution so you can follow the working.
How accurate is a grouped estimate?
Grouping throws away information, so the result is an estimate. The estimate is best when classes are narrow, the data within each class are roughly evenly spread, and there are many observations. Wide classes inflate the variance slightly because every value is placed at the midpoint; a correction called Sheppard’s correction subtracts w²/12 from the variance for equal-width classes, but most courses do not require it.
Open-ended and unequal classes
Classes such as “50 and over” have no midpoint and cannot be used directly; you must assume an upper limit. Unequal class widths are fine for this calculation, but a histogram of unequal widths should plot frequency density, not raw frequency. This tool draws the bar chart by class midpoint when the classes are not contiguous.
Discrete frequency tables
If your table lists exact values (for example, 1 child: 8 families), enter each value as a class from value to value plus a small width, or use the mean and standard deviation calculator after expanding the list. For a discrete table the midpoint method is not required.
Where Standard Deviation for Grouped Data Is Used
Frequency tables
Get the SD from classes and frequencies.
Published data
Estimate spread when only a grouped table is available.
Exam questions
Check midpoints, f·x and f·x² columns.
Median estimate
See the estimated median for the same table.
Common Standard Deviation for Grouped Data Mistakes to Avoid
- Class midpoints stand in for the data, so the result is an estimate.
- Use N − 1 for a sample and N for a population.
- Open-ended classes need a chosen upper or lower limit.
- Zero-frequency classes contribute nothing but can stay in the table.
Standard Deviation for Grouped Data Terms and Definitions
- Class midpoint
- The centre of a class interval; used to represent all values in that class.
- Frequency
- The number of observations in a class.
Standard Deviation for Grouped Data FAQs
How do you find the standard deviation of grouped data?
Compute class midpoints, find the mean Σfx ÷ N, then take Σf(x − mean)², divide by N − 1 (sample) or N (population) and take the square root.
Why are the results only estimates?
The individual values inside each class are unknown, so each class is represented by its midpoint.
Should I divide by N or N − 1?
Divide by N − 1 for a sample and N for a full population. Many school textbooks use N for grouped data, so check your course.
Can I use unequal class widths?
Yes. The calculation uses midpoints and frequencies, so widths can differ.
What if a class has frequency 0?
Include it with frequency 0 or omit it. It does not change the result.
How do I find the mean of grouped data?
Multiply each midpoint by its frequency, add the products and divide by the total frequency.