{
  "category": "Statistical functions",
  "display_name": "STANDARDIZE",
  "entry_kind": "built_in_function",
  "eval_api": {
    "method": "POST",
    "url": "https://handbook.dnacalc.com/api/eval"
  },
  "function_id": "FUNC.STANDARDIZE",
  "implementation_state": {
    "answer_bank": {
      "excel_builds": [
        "20430"
      ],
      "larger_numeric": 0,
      "last_bit": 0,
      "matched": 10302,
      "other_mismatch": 0,
      "refused": 8,
      "rows": 10310,
      "unread": 0,
      "worst_ulp": 0
    },
    "detail": "Of 10,315 recorded answers (10,310 in the answer bank and 5 workbook cells), 8 do not match: 8 are refused.",
    "dnacalc_commit": "9e1884369a678158dd82d3d74188daf986995e87",
    "in_catalogue": true,
    "state": "differs",
    "state_label": "Known larger differences",
    "summary": "Refuses 8 of 10,315 recorded calls; the rest match",
    "workbook_corpus": {
      "cells": 5,
      "matched": 5
    }
  },
  "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#L402-L405",
        "https://github.com/DnaCalc/DnaCalc/blob/9e1884369a678158dd82d3d74188daf986995e87/crates/calc-func/src/ported/standardize_fn.rs#L3-L13"
      ],
      "repository": "https://github.com/DnaCalc/DnaCalc",
      "shared_code": [],
      "source": "https://github.com/DnaCalc/DnaCalc/blob/9e1884369a678158dd82d3d74188daf986995e87/crates/calc-func/src/catalogue/statistics.rs#L402-L405",
      "state": "differs",
      "suite": null,
      "summary": "Refuses 8 of 10,315 recorded calls; the rest match"
    }
  ],
  "json_url": "https://handbook.dnacalc.com/api/functions/FUNC.STANDARDIZE.json",
  "name": "STANDARDIZE",
  "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\n- **[NORM.S.DIST](FUNC.NORM.S.DIST.md) and [NORM.DIST](FUNC.NORM.DIST.md)** are the functions\n  `STANDARDIZE` feeds. The composition identity above is exact mathematically, and\n  [NORM.DIST](FUNC.NORM.DIST.md) exists as a separate function precisely so that callers do not\n  have to route through it.\n- **[NORM.S.INV](FUNC.NORM.S.INV.md) and [NORM.INV](FUNC.NORM.INV.md)** run the inverse map:\n  `NORM.INV(p, mu, sigma) = mu + sigma * NORM.S.INV(p)`, which is `STANDARDIZE` read backwards.\n- **[AVERAGE](FUNC.AVERAGE.md), [STDEV.S](FUNC.STDEV.S.md) and [STDEV.P](FUNC.STDEV.P.md)** are\n  the usual sources of the two parameters. Choosing between the sample and population standard\n  deviation is a statistical decision that `STANDARDIZE` does not make and cannot see.\n- **[Z.TEST](FUNC.Z.TEST.md)** is the hypothesis test built on the same standardisation, with the\n  standard error `sigma/sqrt(n)` in place of `sigma`.\n- **`PERCENTRANK.INC`** is the non-parametric alternative for \"where\n  does this value sit\": it uses the empirical distribution instead of a location-scale model, and\n  needs no parameters.\n- Readers confuse `STANDARDIZE` with normalisation to `[0, 1]`, which is `(x - min)/(max - min)`\n  — a different affine map with different invariances. `STANDARDIZE` centres on the mean and\n  scales by spread; min-max scaling pins the extremes. Neither is \"the\" normalisation.\n\n\n",
    "how-to-implement": "\n### Numerical notes\n\n\nThe function is one subtraction and one division, both correctly rounded, so the *implementation*\ncontributes at most one ulp of relative error to each. All of the interesting behaviour is in the\nconditioning of the map itself.\n\n**Catastrophic cancellation at the centre.** `x - mu` is a subtraction of two nearby quantities\nwhenever `x` is close to `mu`, and that is exactly the region where users spend most of their\ntime. If `x` and `mu` are each around `10^6` and differ by around `10^{-6}`, the subtraction is\nexact in binary64 (Sterbenz's lemma covers the case where the two operands are within a factor of\ntwo of each other), but the *inputs* generally are not: `mu` came from an\n[AVERAGE](FUNC.AVERAGE.md) that carried its own rounding, and that error is transmitted to the\nresult multiplied by `1/sigma`. The relative error of the z-score is unbounded near `x = mu` even\nthough its absolute error is fine. This matters for anything downstream that cares about the\nrelative accuracy of a small z-score, and it does not matter at all for a tail probability, where\nthe absolute accuracy is what governs.\n\nThe general statement: the condition number of `z` with respect to `mu` is `|mu|/|x - mu|`, which\nblows up at the centre; the condition number with respect to `sigma` is `1`, uniformly. Errors in\nthe mean are dangerous; errors in the standard deviation are not.\n\n**Why the composition identity is not a computational recipe.** `NORM.DIST` is mathematically\n`NORM.S.DIST(STANDARDIZE(...))`, but computing it that way costs accuracy twice. First, the\nstandardisation rounds `z`, and then the distribution function amplifies that rounding by its own\nderivative — which in the far tail is the density, so the *relative* error of the tail probability\ngrows like `|z|` times the relative error in `z`. Second, the composition throws away any\nopportunity for the distribution routine to use a better-conditioned form of its argument. This\nis the standard reason that libraries expose a general-parameter entry point rather than telling\ncallers to standardise first, and it is the reason [NORM.DIST](FUNC.NORM.DIST.md) is a separate\nfunction rather than a wrapper. Do not use `STANDARDIZE` as a preprocessing step for a tail\nprobability if accuracy in the tail matters.\n\n**The guard is exact, not tolerant.** `sigma <= 0` is an exact comparison. A subnormal `sigma`\npasses it and produces an enormous quotient, possibly an infinity. There is no tolerance and no\nscaling, which is a defensible choice — any threshold would be arbitrary — but it means the\nfunction has no \"nearly degenerate\" diagnostic, only a degenerate one.\n\n**Overflow and underflow.** `(x - mu)/sigma` can overflow when the numerator is large and the\nscale is subnormal, and can underflow to zero when the reverse holds. Neither is diagnosed. A\nscale-aware implementation would compute the quotient by rescaling both operands, as `hypot`\ndoes; this one does not need to for ordinary data and would need to for adversarial data.\n\n",
    "quirks": "\n### Result and edge cases\n\n\nReturns `Number`, dimensionless — the units of `x` and `mu` cancel against those of `sigma`.\n\n- **`x = mean`.** Exactly `0`, and exactly `+0` for `mu` finite, since `x - mu` is exactly zero\n  and `0/sigma` is `+0` for positive `sigma`.\n- **`standard_dev` negative.** `#NUM!`. A negative scale would give a well-defined\n  order-reversing affine map, and it is refused rather than accepted — the function insists on\n  the orientation convention.\n- **Very small `standard_dev`.** Subnormal scales are accepted by the `<= 0` test and produce\n  correspondingly enormous or infinite results.\n- **Errors in any argument** propagate as themselves; under the lifted path an error in one\n  element of an array argument stays in that element.\n\n\n\n### Errors\n\n\n| Error | Condition | Source |\n|---|---|---|\n| `#NUM!` | `standard_dev <= 0` | documented and implemented |\n| `#VALUE!` | Any argument is text that does not parse as a number | shared coercion rule |\n| propagated | An error value in any argument surfaces as that error | shared coercion rule |\n\nThere is no `#DIV/0!` reachable from this function: the only division is by a value the guard has\nalready established to be strictly positive.\n\n\n",
    "sources": "- Microsoft, *STANDARDIZE function* —\n  <https://support.microsoft.com/en-us/office/standardize-function-81d66554-2d54-40ec-ba83-6437108ee775>\n  (signature, the stated equation, and the `#NUM!` condition when `standard_dev` is not positive).\n  Retrieval was blocked by the upstream host for this page; the documented behaviour above is\n  stated as documented behaviour and should be re-checked against the page.\n- M. Abramowitz and I. A. Stegun, *Handbook of Mathematical Functions*, chapter 26 section 26.2 —\n  the standardising substitution that reduces a general normal to the standard normal, and the\n  Jacobian factor in the density identity.\n- N. J. Higham, *Accuracy and Stability of Numerical Algorithms*, 2nd ed., chapter 2 (Sterbenz's\n  lemma and the conditioning of subtraction) — the basis of the cancellation discussion.\n- Handbook [coercion and lifting](../model/02-coercion-and-lifting.md) — the elementwise lift and\n  the element-local error rule.\n- Handbook `data/functions/FUNC.STANDARDIZE.json` (the declared real-result policy allowing\n  non-finite results), `data/presence/FUNC.STANDARDIZE.json`.\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, CPU not recorded | 10,310 |\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.STANDARDIZE.json).",
    "what-it-does": "\n`STANDARDIZE(x, mean, standard_dev)` returns the **z-score** of `x` relative to a distribution\nwith the stated centre and scale:\n\n    STANDARDIZE(x, mu, sigma)  =  ( x - mu ) / sigma        for sigma > 0\n\nIt is the affine map `T(x) = (x - mu)/sigma`, and everything about it follows from that being an\naffine map with positive slope:\n\n- It is a **bijection of the real line onto itself**, strictly increasing, with inverse\n  `x = mu + sigma*z`. Nothing is lost and nothing is folded.\n- It is the **unique** affine map sending `mu` to `0` and `mu + sigma` to `1`. That is what fixes\n  the convention: `STANDARDIZE(mu, mu, sigma) = 0` and a value one standard deviation above the\n  mean maps to exactly `1`.\n- Applied to a whole data set with its own mean and standard deviation, the image has mean `0`\n  and standard deviation `1`. Applied to a random variable `X` with `E[X] = mu` and\n  `SD(X) = sigma`, the image `Z` has `E[Z] = 0` and `SD(Z) = 1` — for *any* distribution, not\n  only the normal one. Standardisation is a change of units, not a normality assumption.\n\nIts role in the statistical category is as the **argument reduction** for the normal family. The\nsubstitution `z = (x - mu)/sigma` is what reduces a general normal density and distribution\nfunction to the standard ones — it is the substitution behind Abramowitz & Stegun's treatment in\nchapter 26, where every general-normal quantity is tabulated through its standard form. In Excel\nterms the mathematical identity is\n\n    NORM.DIST(x, mu, sigma, TRUE)  =  NORM.S.DIST( STANDARDIZE(x, mu, sigma), TRUE )\n\nand the corresponding one for the density carries an extra `1/sigma` from the Jacobian:\n\n    NORM.DIST(x, mu, sigma, FALSE)  =  NORM.S.DIST( STANDARDIZE(x, mu, sigma), FALSE ) / sigma\n\nBoth are exact in real arithmetic. Neither is a good way to compute the left-hand side in\nfloating point — see Numerical notes.\n\nThe name is slightly misleading in one respect worth stating: `STANDARDIZE` does not compute\n`mean` or `standard_dev` from data. It is a three-argument scalar transform, and the caller\nsupplies the parameters. Standardising a column against its own statistics means writing\n`STANDARDIZE(A1, AVERAGE($A$1:$A$100), STDEV.S($A$1:$A$100))`, which is where the population\nversus sample choice actually enters.\n\n\n\n### Arguments\n\n\n| Argument | Meaning | Admissible |\n|---|---|---|\n| `x` | The value to standardise. Required. | any finite number |\n| `mean` | The centre of the distribution. Required. | any finite number |\n| `standard_dev` | The scale. Required. | strictly positive |\n\n"
  },
  "signature": {
    "display": "STANDARDIZE(x, mean, standard_dev)",
    "parameters": [
      {
        "description": null,
        "name": "x",
        "optional": false,
        "repeats": false
      },
      {
        "description": null,
        "name": "mean",
        "optional": false,
        "repeats": false
      },
      {
        "description": null,
        "name": "standard_dev",
        "optional": false,
        "repeats": false
      }
    ],
    "trailing_repeats": false
  },
  "sources": {
    "metadata": {
      "admission": {
        "label": "implemented",
        "reason": null
      },
      "arity": {
        "max": 3,
        "min": 3
      },
      "category": "Statistical functions",
      "classification": {
        "arg_preparation_profile": "ValuesOnlyPreAdapter",
        "coercion_lift_profile": "Custom",
        "determinism_class": "Deterministic",
        "error_collapse_profile": "None",
        "fec_dependency_profile": "None",
        "host_interaction_class": "None",
        "kernel_signature_class": "Custom",
        "rich_value_usage": "RichBlind",
        "surface_fec_dependency_profile": "RefOnly",
        "thread_safety_class": "SafePure",
        "volatility_class": "NonVolatile"
      },
      "descriptions": {
        "long": null,
        "short": "z-score: distance from the mean in standard deviations",
        "short_source": "dna-calc-own"
      },
      "docs": {
        "english_description": "Returns a normalized value",
        "microsoft_url": "https://support.microsoft.com/en-us/office/standardize-function-81d66554-2d54-40ec-ba83-6437108ee775"
      },
      "entry_kind": "built_in_function",
      "localized_family_names": null,
      "localized_names": [
        {
          "locale": "ar-SA",
          "name": "STANDARDIZE"
        },
        {
          "locale": "bg-BG",
          "name": "STANDARDIZE"
        },
        {
          "locale": "cs-CZ",
          "name": "STANDARDIZE"
        },
        {
          "locale": "da-DK",
          "name": "STANDARDISER"
        },
        {
          "locale": "de-DE",
          "name": "STANDARDIZE"
        },
        {
          "locale": "el-GR",
          "name": "STANDARDIZE"
        },
        {
          "locale": "en-AU",
          "name": "STANDARDIZE"
        },
        {
          "locale": "en-GB",
          "name": "STANDARDIZE"
        },
        {
          "locale": "en-US",
          "name": "STANDARDIZE"
        },
        {
          "locale": "es-ES",
          "name": "NORMALIZACION"
        },
        {
          "locale": "et-EE",
          "name": "STANDARDIZE"
        },
        {
          "locale": "fi-FI",
          "name": "NORMITA"
        },
        {
          "locale": "fr-FR",
          "name": "CENTREE.REDUITE*"
        },
        {
          "locale": "he-IL",
          "name": "STANDARDIZE"
        },
        {
          "locale": "hr-HR",
          "name": "STANDARDIZE"
        },
        {
          "locale": "hu-HU",
          "name": "NORMALIZÁLÁS"
        },
        {
          "locale": "id-ID",
          "name": "STANDARDIZE"
        },
        {
          "locale": "it-IT",
          "name": "STANDARDIZE"
        },
        {
          "locale": "ja-JP",
          "name": "STANDARDIZE"
        },
        {
          "locale": "ko-KR",
          "name": "STANDARDIZE"
        },
        {
          "locale": "lt-LT",
          "name": "STANDARDIZE"
        },
        {
          "locale": "lv-LV",
          "name": "STANDARDIZE"
        },
        {
          "locale": "nb-NO",
          "name": "STANDARDIZE"
        },
        {
          "locale": "nl-NL",
          "name": "NORMALISEREN"
        },
        {
          "locale": "pl-PL",
          "name": "NORMALIZUJ"
        },
        {
          "locale": "pt-BR",
          "name": "STANDARDIZE"
        },
        {
          "locale": "pt-PT",
          "name": "NORMALIZAR"
        },
        {
          "locale": "ro-RO",
          "name": "STANDARDIZE"
        },
        {
          "locale": "ru-RU",
          "name": "НОРМАЛИЗАЦИЯ"
        },
        {
          "locale": "sk-SK",
          "name": "STANDARDIZE"
        },
        {
          "locale": "sl-SI",
          "name": "STANDARDIZE"
        },
        {
          "locale": "sr-Latn-RS",
          "name": "STANDARDIZE"
        },
        {
          "locale": "sv-SE",
          "name": "STANDARDISERA"
        },
        {
          "locale": "th-TH",
          "name": "STANDARDIZE"
        },
        {
          "locale": "tr-TR",
          "name": "STANDARTLAŞTIRMA"
        },
        {
          "locale": "uk-UA",
          "name": "STANDARDIZE"
        },
        {
          "locale": "vi-VN",
          "name": "STANDARDIZE"
        },
        {
          "locale": "zh-CN",
          "name": "STANDARDIZE"
        },
        {
          "locale": "zh-HK",
          "name": "STANDARDIZE"
        },
        {
          "locale": "zh-TW",
          "name": "STANDARDIZE"
        }
      ],
      "signature": {
        "display": "STANDARDIZE(x, mean, standard_dev)",
        "parameters": [
          {
            "description": null,
            "name": "x",
            "optional": false,
            "repeats": false
          },
          {
            "description": null,
            "name": "mean",
            "optional": false,
            "repeats": false
          },
          {
            "description": null,
            "name": "standard_dev",
            "optional": false,
            "repeats": false
          }
        ],
        "trailing_repeats": false
      },
      "surface_name": "STANDARDIZE",
      "version_marker": null,
      "xlcall": {
        "code": "297",
        "symbol": "xlfStandardize"
      }
    },
    "microsoft": "https://support.microsoft.com/en-us/office/standardize-function-81d66554-2d54-40ec-ba83-6437108ee775"
  },
  "try_it": {
    "formula": "=STANDARDIZE(1000000.0000001, 1000000, 0.0000001)"
  },
  "url": "https://handbook.dnacalc.com/functions/STANDARDIZE.html",
  "variants": [],
  "versions": [
    {
      "behaviour": "10,310 captured inputs are indexed for this environment. Capture dates: 2026-09-29 to 2026-09-29. This is a reading count, not a comparison with another build.",
      "excel_build": "16.0 build 20430",
      "platform": "Windows, 64-bit, CPU not recorded",
      "sources": [
        "https://github.com/DnaCalc/ExcelFunctionsHandbook/blob/5dc75391aa5125ccc5abdb095da85950980d6fd8/content/evidence/environments/FUNC.STANDARDIZE.json"
      ]
    }
  ]
}