What Sheppard's Correction Fixes
Sheppard's correction is an adjustment for variance and standard deviation estimates calculated from grouped continuous data. It is used when values have been binned into equal-width class intervals and each class is represented by its midpoint.
The midpoint method is practical, but it treats every observation in a class as if it fell exactly at the center. For a smooth continuous distribution, that grouping process tends to add about h² / 12 of artificial variance, where h is the class width. Sheppard's correction subtracts that amount before taking the square root.
This is a correction to variance, not directly to SD
If you need the basic midpoint workflow first, start with Standard Deviation from a Frequency Table. If your problem is about simplifying by-hand grouped calculations, read Assumed Mean Method for Standard Deviation.
| Term | Meaning | Why it matters |
|---|---|---|
| h | Common class width | The correction is h² / 12 |
| Grouped variance | Variance computed from class midpoints | This is the value adjusted by the correction |
| Corrected variance | Grouped variance minus h² / 12 | Take the square root of this value for corrected SD |
| Raw-data variance | Variance computed from original observations | No Sheppard correction is needed |
The Formula
For equal-width grouped data, first compute the standard grouped-data variance using class midpoints mᵢ and frequencies fᵢ.
Population grouped variance
Sample grouped variance
Then subtract Sheppard's correction term:
Sheppard-corrected variance
Sheppard-corrected standard deviation
Do not confuse corrections
For the sample-versus-population denominator choice, see Sample vs Population. For the ungrouped formula behind these calculations, see Standard Deviation Formula Explained.
Worked Example
Suppose delivery times are only available as grouped class intervals. The class width is h = 10 minutes.
| Class interval | Frequency f | Midpoint m | f × m | f × m² |
|---|---|---|---|---|
| 10-19 | 2 | 14.5 | 29.0 | 420.5 |
| 20-29 | 5 | 24.5 | 122.5 | 3001.25 |
| 30-39 | 7 | 34.5 | 241.5 | 8331.75 |
| 40-49 | 4 | 44.5 | 178.0 | 7921.0 |
| 50-59 | 2 | 54.5 | 109.0 | 5940.5 |
| Total | 20 | - | 680.0 | 25615.0 |
Compute the grouped mean
Compute the midpoint-based population variance
Compute Sheppard's correction term
Subtract before taking the square root
Convert variance to standard deviation
Sample version of the same example
You can compare the uncorrected midpoint calculation with the site's variance calculator, sample standard deviation calculator, or population standard deviation calculator by expanding the midpoint values according to their frequencies.
When to Use It
Sheppard's correction is most defensible when grouping is an artificial reporting step applied to an underlying continuous variable. For example, measurements might be rounded into equal 10-minute, 5-point, or 0.1-unit bins even though the original variable is continuous.
Good fit
Weak fit
The correction is usually small when class widths are narrow compared with the standard deviation. It becomes more important when bins are wide enough that the grouping itself contributes visible spread.
When Not to Use It
- Raw data are available:Use the original observations instead. Sheppard's correction is only an approximation for information lost through grouping.
- Class widths differ:The simple **h² / 12** correction assumes one common class width.
- Classes are open-ended:A class such as **60 and above** has no natural midpoint or width for this correction.
- The corrected variance is negative:That means the assumptions are not compatible with the grouped variance. Do not report an imaginary standard deviation.
- The data are inherently discrete:Counts, Likert ratings, and small integer scales may not behave like continuous values uniformly spread inside each bin.
Report the method
Grouped-Data Correction Checklist
- Confirm that the table uses equal-width class intervals.
- Compute the standard grouped-data variance from class midpoints first.
- Use the correct denominator: N for a population or n - 1 for a sample.
- Calculate h² / 12 from the common class width.
- Subtract the correction from variance, not from standard deviation.
- Take the square root only after the variance has been corrected.
- Avoid the correction for open-ended, unequal-width, highly skewed, or very small grouped tables.
- Use raw observations with the descriptive statistics calculator whenever the original data can be recovered.
Sheppard's correction is not a magic way to reconstruct raw data. It is a specific adjustment for a specific source of bias: extra variance introduced by replacing continuous observations with equal-width class midpoints.
Further Reading
Sources
References and further authoritative reading used in preparing this article.
- Sheppard's Correction — Wolfram MathWorld
- Sheppard's correction
- Grouped data variance methods — R Project