Grouped Data Standard Deviation Calculator
Enter class intervals with frequencies to compute the mean and standard deviation for grouped datasets.
Mean
31.7647
Population SD
9.9175
Sample SD
10.0666
| Lower | Upper | Midpoint | Frequency | f · x̄ | f · x̄² |
|---|---|---|---|---|---|
| 10 | 20 | 15.0000 | 5 | 75.0000 | 1125.0000 |
| 20 | 30 | 25.0000 | 9 | 225.0000 | 5625.0000 |
| 30 | 40 | 35.0000 | 12 | 420.0000 | 14700.0000 |
| 40 | 50 | 45.0000 | 8 | 360.0000 | 16200.0000 |
How to Use This Calculator
- Enter each class interval with its frequency (lower bound, upper bound, frequency).
- Confirm intervals are non-overlapping and frequencies are non-negative.
- Review the calculated mean and standard deviations.
- Use the table to verify midpoint computations and f · x terms.
Formula
x̄ = Σ(fi · mi) / Σfi
Population variance = Σ(fi · mi²) / Σfi − x̄²
Population SD = √(population variance)
Sample SD = √[Σ fi(mi − x̄)² / (Σfi − 1)]
Here mi denotes the midpoint of each interval and fi is the frequency associated with that interval.
Full Description
Grouped data standard deviation extends variability analysis to frequency tables where individual observations are not known, only counts per interval. The midpoint approximation yields a useful estimate of overall spread and average.
Accurate results depend on reasonable interval widths and representative midpoints. Consider collecting raw data when feasible for higher precision.
Frequently Asked Questions
Why use midpoints?
Midpoints approximate each class's values. Without raw data, it's standard practice for grouped summaries.
Can I include open-ended intervals?
Yes, but you must choose a representative midpoint or estimate a suitable upper/lower bound manually.
What's the difference between population and sample SD here?
Population SD divides by Σf, while sample SD divides by Σf − 1, reflecting the degrees of freedom adjustment.
Do frequencies need to sum to 1?
No. Frequencies can be counts or weights; the formulas scale accordingly.