Kurtosis Calculator
Tail weight, honestly measured: excess kurtosis g₂ against the normal’s 0 reference, with the adjusted G₂ beside — and the peakiness myth corrected on the page, because kurtosis is about TAILS, not pointy tops.
Kurtosis Calculator
Results recalculate instantly on every keystroke. Nothing you type is transmitted.
What this result does not account for
- Moment-based only — no tail-index estimation
- Small samples: the raw and adjusted forms can disagree in sign at n near 4
In short: Data 2, 4, 4, 4, 5, 5, 7, 9: excess kurtosis g₂ = −0.218750, adjusted G₂ = 0.940625 — the raw moment says slightly lighter tails than the normal, the small-sample-corrected form says heavier, and the gap between those sentences is exactly why both are printed at n = 8. The reference is the normal curve at 0: negative values name THINNER tails than the bell (fewer surprises), positive name fatter (more shocks than the normal bargains for — the number risk models exist to check). Kurtosis measures TAILS: the popular “peakedness” reading is the myth the page corrects.
Formula
g₂ = m₄/m₂² − 3 · G₂ = [(n+1)g₂ + 6](n−1)/[(n−2)(n−3)]
The − 3 pins the normal curve at exactly 0, so the number reads directly as heavier (+) or lighter (−) tails than the bell. The adjusted form is the Excel-KURT convention.
Worked Example
- Paste 4 to 500 values.
- Read the adjusted G₂ against the normal’s 0 reference — positive is fat tails.
- The raw g₂ sits beside it with the correction gap priced.
- Read the tails card before any risk decision: that is what the number actually measures.
Defaults: g₂ = −0.218750, G₂ = 0.940625 — the two sentences disagree at n = 8, which is the lesson. Constant lists and n < 4 refused (the correction divides by (n−2)(n−3)).
Strengths & Limits Of This Model
Where this engine is strong
- Normal-referenced reading (excess form) with the correction gap priced on your list
- The peakiness myth corrected where the number is interpreted
Where it stops
- No significance test for kurtosis
- Says nothing about WHERE in the tails the mass sits
Practical Use Cases
Risk
fat-tail check before trusting normal bounds
QC
does the process throw outliers more than normal?
Teaching
moments beyond skewness
Methodology & Editorial Standards
Computation runs in IEEE-754 double precision at full internal precision; rounding to two decimal places occurs strictly at the display layer, so no cumulative drift enters the result. All monetary outputs use accounting presentation — grouped thousands, two decimals, negatives in parentheses — so figures can be transcribed directly into a model or working paper. Division-by-zero and out-of-domain inputs return an em-dash rather than a misleading number.
This engine was reconciled against an independent reference implementation and hand-verified for the worked example above before release. Our full five-stage review process is published on the About Us page.
Disclaimer. This calculator is provided for informational and modelling purposes only and does not constitute financial, tax, legal, medical, or engineering advice. Verify all figures with a qualified professional before acting on them.
Kurtosis Calculator — 8 Expert FAQs
8 analyst-written answers to the questions practitioners actually ask — optimised for voice and answer-engine retrieval.
Why “excess” kurtosis?
Because the raw fourth-moment ratio m₄/m₂² equals 3 for the normal curve — subtracting 3 pins the bell at exactly 0 so the number reads directly: positive means fatter tails than normal, negative thinner. On the defaults the raw ratio sits just below the line at g₂ = −0.218750.
Why do my two numbers disagree?
Small samples again: G₂ = [(n+1)g₂ + 6](n−1)/[(n−2)(n−3)] is the bias-corrected form (Excel’s KURT), and at n = 8 it says 0.940625 where the raw moment says −0.218750 — different sentences about the same list. The page prints both because the gap IS the uncertainty, not noise to hide.
Is kurtosis about the peak of the curve?
Mostly myth. The arithmetic weights SQUARED deviations four powers deep — what moves kurtosis is mass in the TAILS, not the height of the top: a uniform distribution is flat-topped AND thin-tailed; a Laplace is pointy-topped AND fat-tailed. Read it as tail weight, and the risk use case makes sense.
Why does kurtosis matter for risk?
Because normal-based intervals underprice shocks exactly when tails are fat: a G₂ well above 0 says extreme values arrive more often than the bell promises, so “3-sigma events” are not rare at all. The normal page’s tail probabilities are exact FOR the curve — this number is the check that your data wear it.
Why the n ≥ 4 floor?
The adjusted form divides by (n−2)(n−3): below 4 observations the correction does not exist. And like skewness, kurtosis estimates stabilize slowly — expect jumpiness far past n = 20; the correction gap card prices it on your list.
My constant list was refused — why?
m₄/m₂² needs spread: a constant list has m₂ = 0 and the ratio is 0/0, not a flat-tailed distribution. Same refusal as the skewness page — shape statistics die without variance.
Can kurtosis be negative?
Yes — “platykurtic” in the old vocabulary: lighter tails than the bell, like the uniform’s −1.2. The raw default here is mildly negative (−0.218750); the adjusted form disagrees, and at n = 8 that disagreement is the honest report.
How do skewness and kurtosis work together?
As the two shape coordinates around the two-parameter normal: skewness prices asymmetry, kurtosis prices tail weight, and both feed the same judgment — how far your data sit from the curve the normal page prices probabilities on. Jointly they are a normality CHECK, not a normality test; no p-value is printed here.