How to Use the FISHER and FISHERINV Functions in Microsoft Excel for Correlation Analysis

In quantitative data analysis, measuring the linear relationship between two continuous variables—such as advertising expenditure and quarterly revenue, or employee training hours and productivity—is typically performed using the Pearson correlation coefficient ($r$). While calculating $r$ using Excel’s CORREL function is straightforward, testing hypotheses about correlation or constructing confidence intervals presents severe mathematical challenges because sample correlation coefficients are not normally distributed, especially as $r$ approaches $+1.0$ or $-1.0$.

To overcome this limitation, statistician Ronald Fisher developed the Fisher $z$-transformation. The transformation maps skewed correlation coefficients onto an approximately normal distribution with constant variance. In Microsoft Excel, the FISHER function performs this forward transformation, while the FISHERINV function reverses the process to convert $z$-scores back into correlation values. This comprehensive tutorial breaks down the mathematical mechanics of Fisher transformations, explains the syntax for FISHER and FISHERINV, and walks through a practical workflow to construct statistically valid confidence intervals for correlation coefficients.

Understanding Why Fisher Transformation Is Essential

When sampling data from a population, the sampling distribution of the Pearson correlation coefficient ($r$) becomes increasingly asymmetric and truncated as the correlation diverges from zero:

• Bounded Boundaries ($-1 \le r \le 1$): Because correlation cannot exceed $+1$ or drop below $-1$, the sampling distribution bunches up against the boundaries, violating the bell-shaped symmetry required for standard statistical tests.
• The Fisher Solution: The transformation applies the inverse hyperbolic tangent ($\text{artanh}$) function to convert $r$ into an unbounded variable $z$:

  z = 0.5 * ln( (1 + r) / (1 - r) )

• Constant Standard Error: Once transformed into $z$, the distribution becomes approximately normal with a predictable standard error dependent solely on sample size ($n$):

  SE(z) = 1 / SQRT(n - 3)

• Hypothesis Testing: Analysts can easily calculate $z$-test critical values, compare two independent correlation coefficients, and determine whether a correlation difference is statistically significant.

Syntax and Constraints for FISHER and FISHERINV

Both functions feature a simple single-argument syntax in Microsoft Excel:

The FISHER Function

=FISHER(y)

• y (Required): A numeric value representing the correlation coefficient you want to transform.
• Constraint: The value of $y$ must be strictly between $-1$ and $+1$ exclusive ($-1 < y < 1$). If $y \le -1$ or $y \ge 1$, Excel will return a #NUM! error because the natural logarithm of zero or negative numbers is undefined.

The FISHERINV Function

=FISHERINV(y)

• y (Required): The transformed Fisher $z$-score you want to convert back into a standard correlation coefficient ($r$).
• Constraint: Accepts any real number from $-\infty$ to $+\infty$, always returning a valid correlation between $-1$ and $+1$.

Step-by-Step Practical Example: Calculating a 95% Confidence Interval for Correlation

Suppose you surveyed a sample of $n = 50$ customers and found a sample correlation coefficient of $r = 0.65$ between customer support response times and customer satisfaction scores. You want to compute the true 95% confidence interval for this correlation in Excel.

Step 1: Transform the Correlation into a Z-Score

1. Enter the correlation in cell B1: 0.65.
2. Enter the sample size in cell B2: 50.
3. In cell B3, transform the correlation using the FISHER function:

   =FISHER(B1)

   (Excel returns approximately 0.7753).

Step 2: Calculate the Standard Error and Margin of Error

1. In cell B4, compute the standard error of the Fisher $z$:

   =1 / SQRT(B2 - 3)

   (For $n = 50$, this evaluates to 1 / SQRT(47) ≈ 0.1459).
2. In cell B5, calculate the 95% margin of error using the standard normal distribution multiplier ($Z = 1.96$):

   =1.96 * B4

   (Returns approximately 0.2859).

Step 3: Establish Lower and Upper Z-Bounds

1. In cell B6, calculate the lower $z$-bound:

   =B3 - B5

   (Returns 0.4894).
2. In cell B7, calculate the upper $z$-bound:

   =B3 + B5

   (Returns 1.0612).

Step 4: Invert the Bounds Back to Correlation with FISHERINV

1. In cell B8, convert the lower bound back to $r$:

   =FISHERINV(B6)

   (Returns 0.4537).
2. In cell B9, convert the upper bound back to $r$:

   =FISHERINV(B7)

   (Returns 0.7861).

The analyst can conclude with 95% statistical confidence that the true population correlation coefficient lies between 0.45 and 0.79.

Troubleshooting Common Errors with FISHER and FISHERINV

If your calculation fails, check these common diagnostic points:

• #NUM! Error in FISHER: Triggered if the supplied input is equal to or outside the boundary $[-1, 1]$. A perfect correlation of exactly $1.0$ produces division by zero in the logarithm, yielding an infinite $z$-score.
• #VALUE! Error: Occurs if the argument references text strings or formatted cells that cannot be parsed as valid numeric values.
• Sample Size Constraint: Note that standard error formulas for Fisher transformations require a sample size of at least four ($n > 3$); smaller samples produce negative square roots in error calculations.