Enter responses
Example data table
These sample responses demonstrate how a few extreme values can shift kurtosis.
| # | Response | Comment |
|---|---|---|
| 1 | 12 | Typical observation |
| 2 | 14 | Typical observation |
| 3 | 15 | Typical observation |
| 4 | 13 | Typical observation |
| 5 | 14 | Typical observation |
| 6 | 16 | Typical observation |
| 7 | 11 | Typical observation |
| 8 | 10 | Typical observation |
| 9 | 22 | High-end value affects tail weight |
| 10 | 2 | Low-end value affects tail weight |
Formula used
Let x₁…xₙ be the cleaned response values and x̄ their mean.
- m₂ = (1/n) Σ(xᵢ − x̄)² and m₄ = (1/n) Σ(xᵢ − x̄)⁴
- β₂ = m₄ / m₂² (Pearson kurtosis)
- Excess kurtosis = β₂ − 3
Bias-corrected sample excess kurtosis (requires n ≥ 4):
s² = Σ(xᵢ−x̄)² / (n−1)
This page also reports a normal-theory SE estimate for excess kurtosis: SE ≈ √(24/n).
How to use this calculator
- Paste your numeric responses into the input box.
- Select an estimator and choose excess or Pearson output.
- Optionally apply outlier handling, then press Calculate.
- Review the result, notes, and quick statistics above the form.
- Use the download buttons to export results or cleaned data.
Why response kurtosis matters
Response kurtosis describes how tightly responses cluster near the middle while still allowing occasional extreme values. Excess kurtosis near zero suggests tails that resemble a normal pattern, while values above one imply heavier tails where a small minority can drive volatility. In customer satisfaction, incident severity, or quality audits, a few very low scores may signal pockets of failure even when the average looks steady and reassuring over time.
Choosing an estimator and output
This calculator can show Pearson kurtosis, often called beta two, or excess kurtosis, which is beta two minus three. Pearson equals three for normal shaped data, while excess equals zero, making comparisons simpler across surveys. The bias corrected sample estimator reduces small sample distortion and matches many spreadsheet conventions. Moment estimators are straightforward, but they can be biased when the number of responses is limited.
Effect of sample size and uncertainty
Kurtosis is noisy because it depends on fourth power deviations from the mean. The page reports an approximate standard error for excess kurtosis: the square root of twenty four divided by n. With twenty five responses the error is about 0.98, and with one hundred responses it drops near 0.49. Use the displayed ninety five percent range to judge whether a difference is meaningful or likely sampling fluctuation for your audience.
Outlier handling and data cleaning
Because tails dominate kurtosis, one data entry error can inflate results dramatically. Z score filtering removes values that are extreme relative to the mean and sample standard deviation. The IQR fence uses quartiles to flag unusually distant points. Winsorization clips extremes to chosen percentiles and preserves the full sample size. Compare raw and adjusted outputs to understand sensitivity, and document thresholds used for transparency.
Interpreting results for decisions
Positive excess kurtosis suggests more extreme responses than expected, which can motivate segmentation, robust modeling, or targeted process fixes. Negative excess kurtosis indicates flatter distributions where extremes are less frequent, often after capping, rounding, or strong standardization. Combine kurtosis with skewness, median, and min and max to communicate distribution shape clearly. When tails are heavy, percentiles and median based targets often describe experience better than averages alone.
FAQs
What is the difference between Pearson and excess kurtosis?
Pearson kurtosis (β₂) equals 3 for normal-shaped data. Excess kurtosis subtracts 3, so normal data is near 0. Excess is often easier to interpret because positive values imply heavier tails and negative values imply lighter tails.
Which estimator should I choose for typical survey analysis?
Use the bias-corrected sample estimator when you have limited samples or want results comparable to many spreadsheets. Use moment estimators for quick exploratory work or when you need a simpler definition across software.
How many responses do I need for a reliable kurtosis estimate?
Kurtosis can be unstable with small samples because it relies on fourth powers. As a rule of thumb, treat results below about 50 observations as highly uncertain and use the shown standard error and range to judge stability.
Can I compute kurtosis on ordinal scales like 1–5 ratings?
Yes, but interpret cautiously. Discrete scales cap extremes, which can reduce observed tail weight. Compare groups using the same scale, and consider supplementing kurtosis with percentiles or category frequencies for clearer communication.
Should I remove outliers before reporting kurtosis?
Outliers can be real signals or data errors. Use the outlier tools to test sensitivity, then report both raw and cleaned results if conclusions change. Always document the rule, thresholds, and remaining sample size.
Why does kurtosis change a lot when I add one extreme value?
The fourth-power term heavily amplifies extreme deviations from the mean. One very large or very small observation can dominate Σ(x−x̄)⁴, raising kurtosis sharply. That sensitivity is exactly why kurtosis is useful for tail-risk checks.