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FISHERINV Function in Apple Numbers: Inverse Fisher z

Statistical Advanced Apple Numbers

Reverse Fisher z-transformed correlations in Apple Numbers with FISHERINV, confidence intervals, round trips, and error guidance.

Verification status: Official documentation referenced; example outcomes independently reasoned or arithmetically checked where stated. Not executed in Apple Numbers. How formula examples are checked

Quick answer: =FISHERINV(0.5) has the mathematically expected result 0.46211715726001. The FISHERINV function reverses a Fisher-transformed correlation and returns the original correlation scale. Apple’s FISHERINV help confirms the exact one-argument syntax and its inverse relationship to FISHER. It is not an inverse matrix function and it does not calculate a correlation from raw paired observations. [A1, C1]

Exact syntax, types and defaults

=FISHERINV(y-value)
Position Argument Type Required Default Meaning
1 y-value Number value Yes None Fisher-transformed z value to convert back into a correlation.

The sole parameter is required. Apple’s page does not add a range, base, degrees-of-freedom or confidence-level parameter. It imposes no explicit finite numeric-domain bound on y-value in its argument description; this does not imply that every extreme magnitude or nonnumeric input has been tested in Numbers. Use one scalar per formula. [A1, A2]

The independent mathematical formula is:

r = (EXP(2z) − 1)/(EXP(2z) + 1) = tanh(z).

For positive finite z, the correlation is positive but below +1; for negative z the result is negative but above −1. FISHERINV(0)=0. For very large |z|, computed floating-point results may round to ±1; no specific Numbers precision or overflow threshold has been established here. [C1]

Reproducible worksheet: back-transform six Fisher values

In a Numbers table named Transformed, set B1 to Fisher z and C1 to Correlation r. Enter the numeric inputs below in B2:B7, put =FISHERINV(B2) in C2, and fill down. The output is a correlation coefficient, not a probability or an odds ratio. [A1, A3]

Row B: Fisher z C: Formula Expected result in C
2 0 =FISHERINV(B2) 0
3 0.5 =FISHERINV(B3) 0.46211715726
4 1 =FISHERINV(B4) 0.761594155956
5 -1 =FISHERINV(B5) -0.761594155956
6 1.5 =FISHERINV(B6) 0.905148253645
7 -0.5 =FISHERINV(B7) -0.46211715726

The sign symmetry is visible in z=1 versus z=−1. For a named cross-table reference, =FISHERINV(Transformed::B3) refers to the B3 cell of the Transformed table; Apple’s reference conventions use double colons. The expected result for that example is 0.46211715726. [A3, C1]

Apple’s published round-trip examples

The official page demonstrates =FISHERINV(FISHER(0.75)) → 0.75, =FISHERINV(0) → 0, and =FISHERINV(FISHER(0.5)) → 0.5. The two nested formulas test inverse composition, not merely a lookup of fixed constants. Independent calculation confirms these identities to normal floating-point tolerance. [A1, C1]

When comparing workbooks, be aware that =FISHERINV(FISHER(0.5)) is expected to give approximately 0.5 mathematically; floating-point arithmetic need not guarantee bit-for-bit equality to a separately typed decimal.

Advanced worksheet: 95% interval endpoints on the correlation scale

Suppose a data analysis reports r=0.6 for 25 independent pairs. If you apply the conventional approximate Fisher-z confidence method, the endpoints must be back-transformed. Put the observed r into B2 and n into C2 of the same Numbers table. [C1]

=FISHERINV(FISHER(B2)-1.96/SQRT(C2-3))
=FISHERINV(FISHER(B2)+1.96/SQRT(C2-3))

The expected endpoints are 0.268525236321 and 0.804422855475. These correspond to lower/upper correlation limits, not to z values. The formula first transforms r with FISHER, moves 1.96 estimated standard errors in z space, and then uses FISHERINV to return to the correlation scale. [C1]

This approximate normal-theory interval generally assumes independent pairs, a sufficiently large sample, and suitable distributional conditions (classically bivariate normal). n must exceed three for the displayed square-root calculation. The function itself has no special knowledge of sample size or of these statistical assumptions; they are properties of the worksheet formula, not hidden FISHERINV defaults. [C1]

Pooling transformed results: illustration, not a meta-analysis template

Imagine three equally weighted hypothetical transformed estimates for r values 0.2, 0.5, and 0.8. Create B2:B4 with those correlations and compute C2=FISHER(B2) and fill through C4. Then in an empty result cell use:

=FISHERINV(AVERAGE(C2:C4))

Independent computation yields 0.54895452789, versus the untransformed arithmetic mean of r, which is 0.5. This demonstrates that averaging transformed values differs from averaging raw coefficients. In practice, appropriate Fisher-z pooling often uses weights based on each study’s precision or sample size; an equal-weight example is not automatically an appropriate meta-analysis. [C1]

Troubleshooting and limits

Situation Interpretation What to do
An output is between −1 and +1 even when input is greater than 1 The input is on Fisher-z scale; the output is a correlation. Keep z and r in distinctly labelled columns to avoid unit/scale confusion.
=FISHERINV(0.5) differs from 0.5 The function is not an identity for arbitrary z. Check against the expected value 0.462117157, or use =FISHERINV(FISHER(0.5)) for a round trip.
A range B2:B7 is supplied directly Apple specifies one number value, not a collection. Use one formula per output row instead of assuming array spill behaviour.
Text, blank, or Boolean input The function reference does not enumerate every coercion/error behaviour. Verify the source is numeric and test the actual sheet; do not invent error-code details.
Numeric overflow/precision concern at extreme z Mathematical tanh approaches ±1, but the app’s numeric precision limit has not been verified. Test the relevant extreme values in the installed Numbers version before relying on them.
Function missing in older Numbers Only the 15.4 function reference and current release context were checked. Confirm the app version; do not assume the oldest compatible version.

General diagnostic context: Apple formula errors. Nothing in this review implies a native app error was observed. [A4]

Use FISHER for the forward transform, and CORREL or PEARSON to calculate a correlation from matched source columns. Apple’s FISHERINV article links to FISHER and CORREL; those official URLs were inspected. [A1]

Although inverse Fisher transforms are also used in other statistical software, no other spreadsheet was run or tested for this guide. Mathematical equivalence is not proof of identical error handling, coercion, or version support.

Platform, version, locale and validation boundaries

This guide uses the English Apple Formulas and Functions Help 15.4; the reference explains formula arguments for Numbers and the wider iWork functions browser, not a live test in any particular installation. Apple’s Numbers 15.4 Mac and iPhone user guides describe the current release and OS requirements (Mac: macOS 26 or later; iPhone: iOS 26 or later). Apple’s release notes say new statistical functions were added in 15.4, but they do not individually certify this formula on older releases. Check Numbers > About Numbers on Mac or the iPhone app version and test on your own device if backwards compatibility matters. [A5, A6, A7, A8]

The formulas here use the English-language identifiers and decimal points; English examples separate multiple arguments with commas. Formula names, numeric punctuation and argument separators are locale-dependent. For example, Apple’s Brazilian Portuguese help translates EXPONDIST as DISTEXPON and shows semicolons as separators with decimal commas; do not create a second English-language function page or silently replace a translated name with an English name in that environment. This regional demonstration establishes a specific documented locale example, not a blanket rule for every locale. [A9]

All displayed results are mathematically derived expected values, with checks in an independent Python calculation harness. The documentation and links were inspected on 2026-10-13 (UTC). No Apple Numbers workbook was executed and no screenshot, production link, Excel import/export behaviour, coercion for every cell type, or installed-version compatibility was tested. Differences in rounding or presentation can occur if the cell’s displayed number format has fewer decimals. [C1]

Apple references and verification methods