{ "category": "Statistical functions", "display_name": "KURT", "entry_kind": "built_in_function", "eval_api": { "method": "POST", "url": "https://handbook.dnacalc.com/api/eval" }, "function_id": "FUNC.KURT", "implementation_state": { "answer_bank": { "excel_builds": [ "20430" ], "larger_numeric": 145, "last_bit": 1154, "matched": 3209, "other_mismatch": 0, "refused": 0, "rows": 4508, "unread": 0, "worst_ulp": 32768 }, "detail": "Of 4,514 recorded answers (4,508 in the answer bank and 6 workbook cells), 1,299 do not match: 145 differ by more than 16 ULP, 1,154 differ by at most 16 ULP.", "dnacalc_commit": "9e1884369a678158dd82d3d74188daf986995e87", "in_catalogue": true, "state": "differs", "state_label": "Known larger differences", "summary": "Known larger differences on 1,299 of 4,514 recorded answers", "workbook_corpus": { "cells": 6, "matched": 6 } }, "implementations": [ { "commit": "9e1884369a678158dd82d3d74188daf986995e87", "flavour": "excel-compatibility-candidate", "language": "Rust", "name": "CalcFunc", "own_code": [ "https://github.com/DnaCalc/DnaCalc/blob/9e1884369a678158dd82d3d74188daf986995e87/crates/calc-func/src/catalogue/statistics.rs#L388-L390", "https://github.com/DnaCalc/DnaCalc/blob/9e1884369a678158dd82d3d74188daf986995e87/crates/calc-func/src/ported/moment_stats_family.rs#L7-L39" ], "repository": "https://github.com/DnaCalc/DnaCalc", "shared_code": [ "catalogue/statistics::all", "ported/moment_stats_family::mean" ], "source": "https://github.com/DnaCalc/DnaCalc/blob/9e1884369a678158dd82d3d74188daf986995e87/crates/calc-func/src/catalogue/statistics.rs#L388-L390", "state": "differs", "suite": null, "summary": "Known larger differences on 1,299 of 4,514 recorded answers" } ], "json_url": "https://handbook.dnacalc.com/api/functions/FUNC.KURT.json", "name": "KURT", "notes": [], "schema": "efh.function-reference/v1", "section_order": [ "what-it-does", "try-it", "quirks", "how-to-implement", "version-differences", "alternatives" ], "sections": { "alternatives": "\n### Relationships\n\n- **[SKEW.P](FUNC.SKEW.P.md)** — the population (plug-in) skewness. Note the asymmetry in the\n published surface: Excel offers both a sample and a population *skewness*, but only the\n bias-corrected sample *kurtosis*. There is no `KURT.P`. A reader who wants the plug-in\n `g₂ = μ₄/μ₂² − 3` must compute it from `DEVSQ` and a fourth-power sum by\n hand, or invert the `G₂` correction algebraically.\n- **[STDEV.S](FUNC.STDEV.S.md)** and **`VAR.S`** — supply the `s` in the\n denominator; any disagreement about `s` propagates to the fourth power.\n- **`DEVSQ`** — the second-moment building block, and the natural surface to\n compare against when checking whether the centred sums agree.\n- **Confused with:** \"peakedness\". See above. Also confused with `SKEW`, since both are called\n \"shape statistics\" and both fail on small samples, but they fail at different `n`.\n\n\n", "how-to-implement": "\n### Numerical notes\n\n\n`KURT` is the hardest of the elementary descriptive statistics to compute well, for a reason\nthat is entirely structural: it is a **ratio of a fourth central moment to the square of a\nsecond central moment**, and both are differences of large quantities.\n\n**Cancellation.** The one-pass \"computational\" route accumulates `Σx`, `Σx²`, `Σx³`, `Σx⁴`\nand reconstructs the central moments by the binomial expansion\n\n μ₄ = (Σx⁴ − 4x̄Σx³ + 6x̄²Σx² − 4x̄³Σx + n x̄⁴)/n\n\nFor data with a small coefficient of variation this is catastrophic: the terms are of order\n`n·x̄⁴` while the result is of order `n·σ⁴`, so the relative error is inflated by\n`(x̄/σ)⁴`. A sample with mean 10⁶ and spread 1 loses roughly twenty-four decimal digits — far\nmore than a double has. The estimator is then not merely inaccurate; it is arbitrary, and can\ncome out negative where the true value is positive. The canonical treatment is Chan, Golub and\nLeVeque (1983) for the variance, extended to arbitrary order by Pébay, *Formulas for Robust,\nOne-Pass Parallel Computation of Covariances and Arbitrary-Order Statistical Moments* (Sandia\nreport SAND2008-6212), whose update relations are the standard modern reference for a stable\nstreaming fourth moment.\n\n**What a careful implementation does.** Two passes: compute `x̄` first (with a compensated or\npairwise sum), then accumulate the centred powers `Σ(xᵢ − x̄)²` and `Σ(xᵢ − x̄)⁴` directly.\nOptionally refine `x̄` with a second-pass correction term `Σ(xᵢ − x̄)/n`, which is zero in\nexact arithmetic and is a cheap error-cancelling trick. Standardise before the fourth power —\nthat is, accumulate `Σz⁴` with `z = (xᵢ − x̄)/s` — to keep the summands near unity and avoid\nthe overflow and underflow bands mentioned above. Apply the two exactly-representable rational\nfactors last, and do the final subtraction in the order the formula is written, since the two\nterms are of comparable size for near-normal data and their difference is the answer.\n\n**The subtraction at the end is itself a cancellation.** For a near-normal sample the first\nterm is close to `3(n−1)²/((n−2)(n−3))` and the excess is a small difference of two moderate\nnumbers. Even with a perfect fourth-moment sum, a `KURT` near zero has few correct significant\ndigits in the relative sense — which is fine, because the quantity of interest is the absolute\nexcess, but it means a relative-error residual plate for `KURT` will look alarming near zero\nand should be read as an absolute-error plate instead.\n\n**The published record on Excel's statistical accuracy** is relevant background here and is\nworth naming precisely: McCullough and Wilson's papers on the accuracy of Excel's statistical\nprocedures, and Morten Welinder's work on Gnumeric's statistical functions, both document\none-pass moment accumulation as a recurring defect class in spreadsheet implementations. The\nHandbook does not assert what Excel does internally, and neither of those sources is a\nmeasurement of this surface.\n\n\n", "quirks": "\n### Result and edge cases\n\n\nReturns `Number`.\n\n- **All values identical** → `s = 0`, the standardisation divides by zero, `#DIV/0!`.\n- **Exactly four points** is the boundary case and the one worth testing: the factors are\n `20/6` and `27/2`, both exactly representable, so a correct implementation returns the\n uniform-distribution value for equally spaced points with only the fourth-power sum\n contributing rounding.\n- **Very large or very small values.** Standardising *before* raising to the fourth power is\n what keeps the sum in range; forming `Σ(xᵢ − x̄)⁴` first overflows for `|xᵢ − x̄| >` about\n `1.3·10⁷⁷` and underflows below about `10⁻⁷⁷`. See Numerical notes.\n- **Error values** anywhere in the data propagate.\n- Arrays are consumed by scanning, not lifted elementwise; `KURT` is not a lift kernel.\n\n\n\n### Errors\n\n\n| Error | Condition | Basis |\n|---|---|---|\n| `#DIV/0!` | Fewer than four data points, or the sample standard deviation is zero | Documented by Microsoft on the `KURT` page; also what the defining formula requires |\n| `#VALUE!` | A directly-passed argument that cannot be converted to a number | Shared coercion rule, chapter 02 |\n| propagated | An error value among the data | Shared coercion rule, chapter 02 |\n\n", "sources": "- Joanes and Gill, *Comparing measures of sample skewness and kurtosis*, Journal of the Royal\n Statistical Society Series D 47 (1998) — the `g₂` / `G₂` / `b₂` taxonomy that identifies\n which estimator a package returns.\n- Chan, Golub and LeVeque, *Algorithms for Computing the Sample Variance* (1983) — the\n cancellation analysis that generalises to higher moments.\n- Pébay, *Formulas for Robust, One-Pass Parallel Computation of Covariances and\n Arbitrary-Order Statistical Moments*, Sandia SAND2008-6212 — stable update relations for\n third and fourth moments.\n- McCullough and Wilson, *On the accuracy of statistical procedures in Microsoft Excel*; and\n Morten Welinder's work on Gnumeric's statistical functions — the published record on\n spreadsheet statistical accuracy. Named as literature, not as evidence about this surface.\n- Handbook [Coercion and lifting](../model/02-coercion-and-lifting.md) — the direct-argument\n versus range-scan split.\n- Handbook projections `data/functions/FUNC.KURT.json` (arity 1–255, `xlfKurt`,\n `AggregateDirectAndRangeDualPolicy`) and `data/presence/FUNC.KURT.json` (module\n `moment_stats_family.rs`, shared with `SKEW`, `SKEW.P`, `STEYX`, `TRIMMEAN`; upstream defect\n stream `BUG-FUNC-021` named on the module).\n\n", "version-differences": "The recorded environments below distinguish a reading from a same-input comparison. Counts describe captured inputs in each environment; overlapping captures and comparisons are not totals of unique worksheet calls.\n\n| Excel 16.0 build | Platform and CPU | Captured inputs |\n|---|---|---:|\n| 20430 | Windows, 64-bit, AMD Ryzen 7 4800H with Radeon Graphics | 3,004 |\n\nThe record contains no same-input comparison between these environments. Reading different inputs on different builds does not establish either a version change or cross-build agreement.\n\n[Recorded environments and comparison details](https://github.com/DnaCalc/ExcelFunctionsHandbook/blob/5dc75391aa5125ccc5abdb095da85950980d6fd8/content/evidence/environments/FUNC.KURT.json).", "what-it-does": "\n`KURT(number1, [number2], …)` returns the **bias-corrected sample excess kurtosis** of the\nnumeric values it admits. The estimator Microsoft documents is\n\n KURT = [ n(n+1) / ((n−1)(n−2)(n−3)) ] · Σᵢ ( (xᵢ − x̄) / s )⁴\n − 3(n−1)² / ((n−2)(n−3))\n\nwhere `x̄` is the sample mean and `s` is the **sample** standard deviation with the `n − 1`\ndenominator, the same `s` that [STDEV.S](FUNC.STDEV.S.md) returns.\n\nThree separate conventions are baked into that one line, and every disagreement between\nstatistical packages about \"the kurtosis\" is a disagreement about which of them applies:\n\n1. **Excess, not raw.** The trailing term subtracts the kurtosis of the normal distribution,\n so a Gaussian sample has expectation zero rather than three. The population quantity being\n estimated is `β₂ − 3 = μ₄/μ₂² − 3`, where `μₖ` is the `k`-th central moment.\n2. **Bias-corrected, not the plug-in moment ratio.** The `n(n+1)/((n−1)(n−2)(n−3))` factor is\n the standard `G₂` correction, which makes the estimator unbiased for the excess kurtosis\n when the parent distribution is normal. In the taxonomy of Joanes and Gill (1998) this is\n `G₂`, not the plug-in `g₂` and not the `b₂` variant; SAS and SPSS use `G₂`, several other\n systems do not.\n3. **Sample, not population, scaling of `s`.** Using `n − 1` inside the standardisation and\n then applying the `G₂` factor on top is what makes the algebra come out.\n\n**Domain.** The formula requires `n ≥ 4` — the factor `(n−3)` in both denominators is the\nbinding constraint — and `s > 0`. **Range:** the sample excess kurtosis of `n` points is\nbounded below by `−2·(n−1)/(n−2) + 3(n−1)²/((n−2)(n−3))`-type expressions and above by a\nfinite value determined by `n`; unlike the population quantity it cannot be arbitrarily\nlarge for fixed `n`. The population excess kurtosis is bounded below by `−2` (attained by the\ntwo-point distribution) and unbounded above.\n\nUseful reference values, all exact and all cheap to check: four equally spaced points give\n`−1.2`, which is the excess kurtosis of the continuous uniform distribution; a sample that is\nexactly normal in its order statistics tends to `0`; heavy-tailed samples give large positive\nvalues. Kurtosis is **not** a measure of \"peakedness\" — that reading is a persistent\ntextbook error. It is a standardised fourth moment, and the fourth power means it is driven\nalmost entirely by the observations farthest from the mean. A single outlier moves `KURT`\nmore than the entire rest of the sample.\n\n\n" }, "signature": { "display": "KURT(number1, [number2], ...)", "parameters": [ { "description": null, "name": "number1", "optional": false, "repeats": false }, { "description": null, "name": "number2", "optional": true, "repeats": true } ], "trailing_repeats": true }, "sources": { "metadata": { "admission": { "label": "implemented", "reason": null }, "arity": { "max": 255, "min": 1 }, "category": "Statistical functions", "classification": { "arg_preparation_profile": "ValuesOnlyPreAdapter", "coercion_lift_profile": "AggregateDirectAndRangeDualPolicy", "determinism_class": "Deterministic", "error_collapse_profile": "None", "fec_dependency_profile": "None", "host_interaction_class": "None", "kernel_signature_class": "NumsToNum", "rich_value_usage": "RichBlind", "surface_fec_dependency_profile": "RefOnly", "thread_safety_class": "SafePure", "volatility_class": "NonVolatile" }, "descriptions": { "long": null, "short": "Sample excess kurtosis: how heavy the tails of the data are", "short_source": "dna-calc-own" }, "docs": { "english_description": "Returns the kurtosis of a data set", "microsoft_url": "https://support.microsoft.com/en-us/office/kurt-function-bc3a265c-5da4-4dcb-b7fd-c237789095ab" }, "entry_kind": "built_in_function", "localized_family_names": null, "localized_names": [ { "locale": "ar-SA", "name": "KURT" }, { "locale": "bg-BG", "name": "KURT" }, { "locale": "cs-CZ", "name": "KURT" }, { "locale": "da-DK", "name": "TOPSTEJL" }, { "locale": "de-DE", "name": "KURT" }, { "locale": "el-GR", "name": "KURT" }, { "locale": "en-AU", "name": "KURT" }, { "locale": "en-GB", "name": "KURT" }, { "locale": "en-US", "name": "KURT" }, { "locale": "es-ES", "name": "CURTOSIS" }, { "locale": "et-EE", "name": "KURT" }, { "locale": "fi-FI", "name": "KURT" }, { "locale": "fr-FR", "name": "KURTOSIS*" }, { "locale": "he-IL", "name": "KURT" }, { "locale": "hr-HR", "name": "KURT" }, { "locale": "hu-HU", "name": "CSÚCSOSSÁG" }, { "locale": "id-ID", "name": "KURT" }, { "locale": "it-IT", "name": "KURT" }, { "locale": "ja-JP", "name": "KURT" }, { "locale": "ko-KR", "name": "KURT" }, { "locale": "lt-LT", "name": "KURT" }, { "locale": "lv-LV", "name": "KURT" }, { "locale": "nb-NO", "name": "KURT" }, { "locale": "nl-NL", "name": "KURTOSIS" }, { "locale": "pl-PL", "name": "KURTOZA" }, { "locale": "pt-BR", "name": "KURT" }, { "locale": "pt-PT", "name": "CURT" }, { "locale": "ro-RO", "name": "KURT" }, { "locale": "ru-RU", "name": "ЭКСЦЕСС" }, { "locale": "sk-SK", "name": "KURT" }, { "locale": "sl-SI", "name": "KURT" }, { "locale": "sr-Latn-RS", "name": "KURT" }, { "locale": "sv-SE", "name": "TOPPIGHET" }, { "locale": "th-TH", "name": "KURT" }, { "locale": "tr-TR", "name": "BASIKLIK işlevi" }, { "locale": "uk-UA", "name": "KURT" }, { "locale": "vi-VN", "name": "KURT" }, { "locale": "zh-CN", "name": "KURT" }, { "locale": "zh-HK", "name": "KURT" }, { "locale": "zh-TW", "name": "KURT" } ], "signature": { "display": "KURT(number1, [number2], ...)", "parameters": [ { "description": null, "name": "number1", "optional": false, "repeats": false }, { "description": null, "name": "number2", "optional": true, "repeats": true } ], "trailing_repeats": true }, "surface_name": "KURT", "version_marker": null, "xlcall": { "code": "322", "symbol": "xlfKurt" } }, "microsoft": "https://support.microsoft.com/en-us/office/kurt-function-bc3a265c-5da4-4dcb-b7fd-c237789095ab" }, "try_it": { "formula": "=KURT({1;2;3;4;6})" }, "url": "https://handbook.dnacalc.com/functions/KURT.html", "variants": [], "versions": [ { "behaviour": "3,004 captured inputs are indexed for this environment. Capture dates: 2026-09-24 to 2026-09-25. This is a reading count, not a comparison with another build.", "excel_build": "16.0 build 20430", "platform": "Windows, 64-bit, AMD Ryzen 7 4800H with Radeon Graphics", "sources": [ "https://github.com/DnaCalc/ExcelFunctionsHandbook/blob/5dc75391aa5125ccc5abdb095da85950980d6fd8/content/evidence/environments/FUNC.KURT.json" ] } ] }