Fisher's Z-Transformation Table (Complete Lookup Chart & PDF)

Unweighted Average

Convert all k correlations to Fisher Z values. Compute the mean Z̄ = (Z₁ + Z₂ + ... + Zₖ) / k. Convert Z̄ back to r using r = tanh(Z̄). This gives the unweighted average correlation across studies.

Weighted Average (Preferred)

Weight each Zᵢ by wᵢ = nᵢ − 3, where nᵢ is the sample size of study i. Compute Z̄_w = Σ(wᵢ × Zᵢ) / Σwᵢ. Convert back to r. Studies with larger samples receive more weight, which gives a more precise estimate of the population correlation.

Mistake 1 — Using n − 2 instead of n − 3 in the SE formula

The standard error of Fisher Z is 1/√(n − 3), not 1/√(n − 2). Confusing this with the degrees of freedom used in the Pearson r significance test (df = n − 2) is common. With n = 28, the correct SE is 1/√25 = 0.200. Using n − 2 gives 1/√26 = 0.196 — a small error, but it compounds in confidence interval calculations.

Mistake 2 — Averaging r values directly instead of averaging Z values

The mean of several r values is not an unbiased estimate of the population mean correlation. Fisher Z values must be averaged first, then converted back. For example, averaging r₁ = 0.30 and r₂ = 0.90 gives r̄ = 0.60, but tanh[(arctanh(0.30) + arctanh(0.90))/2] = tanh[(0.3095 + 1.4722)/2] = tanh(0.8909) = 0.713, a substantially different result.

Mistake 3 — Applying the transformation to r = ±1

Fisher Z is undefined at r = ±1 because ln(0) is undefined. In practice, a correlation of exactly ±1 cannot arise from real data unless the variables are perfectly linearly dependent (which would make the correlation trivially obvious). Software will return ±Inf or an error at r = ±1.

Mistake 4 — Reporting the Fisher Z value as if it were a correlation

Fisher Z is not a correlation coefficient. It is unbounded (it can exceed 1 in absolute value, unlike r). Reporting Z = 0.87 as a "correlation of 0.87" is wrong — Z = 0.87 corresponds to r = tanh(0.87) = 0.70. All final results should be reported in the original r metric after back-transformation.

R — Complete CI Procedure

Python (NumPy / SciPy) — Complete CI Procedure

Excel — Cell Formulas

SPSS

In SPSS, use COMPUTE FisherZ = ATANH(r) in the Syntax editor to transform a correlation variable. CI bounds are then computed manually: COMPUTE CI_lower = TANH(FisherZ - 1.96 * (1 / SQRT(n - 3))). SPSS does not have a dedicated Fisher Z confidence interval dialog in standard editions.

Problem 1 — Healthcare (CI Construction)

A health researcher finds r = 0.42 between blood pressure and sodium intake in n = 78 patients. Construct a 95% confidence interval for ρ.

Problem 2 — Psychology (Comparing Two Correlations)

Study A (n = 60, r = 0.45) and Study B (n = 80, r = 0.28) both examine the correlation between extraversion and social media use. Do these correlations differ significantly at α = 0.05?

Problem 3 — Education (Meta-Analysis Average)

Three studies report correlations between feedback quality and student learning: r₁ = 0.35 (n = 45), r₂ = 0.52 (n = 62), r₃ = 0.41 (n = 30). Calculate the weighted average correlation using Fisher Z.

Can Fisher's Z be applied to Spearman r or Kendall's tau?

The standard Fisher Z procedure assumes that r is a Pearson product-moment correlation from bivariate normal data. Applying it to Spearman or Kendall correlations is technically incorrect because their sampling distributions differ. Some researchers do apply it as an approximation to Spearman r, but this should be noted as a limitation. For rigorous inference on Spearman r, bootstrapping is preferred.

Is Fisher's Z accurate for small samples?

The normal approximation underlying the Fisher Z procedure improves as n increases. With very small samples (n < 10), the approximation is rough and confidence intervals may be inaccurate. A minimum of n = 20–25 is commonly recommended, with n ≥ 50 producing reliable intervals for most values of ρ. For small samples, the bias-corrected bootstrap CI is a more reliable alternative.

What is the minimum n needed for Fisher's Z?

The formula requires n ≥ 4 because SE = 1/√(n − 3) is undefined at n = 3 and produces a very large standard error at n = 4. For practical purposes, the standard recommendation is n ≥ 20 before the normal approximation becomes reliable. APA guidelines suggest reporting confidence intervals alongside r regardless of sample size.

Does Fisher's Z assume bivariate normality?

Yes. The derivation of SE = 1/√(n − 3) assumes that (X, Y) follow a bivariate normal distribution. When this assumption is violated — for example, with ordinal data or heavily skewed variables — the procedure is not strictly valid, though it is often still used as an approximation. Checking for bivariate normality through scatter plots and tests is good practice before applying the Fisher Z method.

How do I test whether r differs from a specific non-zero value?

Convert both the sample r and the hypothesized value ρ₀ to Fisher Z. Then compute z = (Z − Z₀) / SE, where Z₀ = arctanh(ρ₀) and SE = 1/√(n − 3). Compare |z| to 1.96 (α = 0.05) or 2.576 (α = 0.01). This is more accurate than testing r directly against ρ₀ because the Z scale is symmetric and approximately normal regardless of ρ₀.

Fisher, R. A. (1915). Frequency distribution of the values of the correlation coefficient in samples from an indefinitely large population. Biometrika, 10(4), 507–521. doi:10.2307/2331838 — The original paper in which Fisher derived the transformation and its sampling distribution.

Fisher, R. A. (1921). On the "probable error" of a coefficient of correlation deduced from a small sample. Metron, 1, 3–32. — Fisher's follow-up paper establishing the z' notation and providing practical tables for confidence intervals.

Cohen, J., Cohen, P., West, S. G., & Aiken, L. S. (2003). Applied Multiple Regression/Correlation Analysis for the Behavioral Sciences (3rd ed.). Lawrence Erlbaum. doi:10.4324/9780203774441 — The standard behavioral science reference for correlation inference including Fisher Z procedures.

NIST/SEMATECH e-Handbook of Statistical Methods (2013). Section 1.3.5.11: Fisher Z Transformation. National Institute of Standards and Technology. itl.nist.gov — Government reference covering the formula, assumptions, and applications of the Fisher Z transformation.

Hedges, L. V., & Olkin, I. (1985). Statistical Methods for Meta-Analysis. Academic Press. — The foundational reference for using Fisher Z in combining and comparing correlations across studies in meta-analysis.

Penn State STAT 505: Applied Multivariate Statistical Analysis. Lesson 7.3 — Testing for Differences Between Two Correlations. Pennsylvania State University. online.stat.psu.edu/stat505/ — Open-access course material covering the Fisher Z comparison procedure with worked examples.