Calculator Inputs
Example Data Table
This table shows sample situations where Fisher z transformation is useful for stable interval estimation and reliable comparison of independent correlations.
| Study | Correlation (r) | Sample Size | Approx. Fisher z | Use Case |
|---|---|---|---|---|
| Customer Satisfaction vs Retention | 0.65 | 30 | 0.7753 | Estimate interval for observed relationship |
| Training Hours vs Performance | 0.42 | 50 | 0.4477 | Compare effect size across teams |
| Ad Spend vs Sales Lift | 0.30 | 40 | 0.3095 | Test if association is meaningful |
| Stress vs Sleep Quality | -0.58 | 45 | -0.6625 | Build confidence interval on r |
Formula Used
1. Fisher z transformation
z = 0.5 × ln((1 + r) / (1 - r))
2. Inverse Fisher transformation
r = (e2z - 1) / (e2z + 1)
3. Standard error of Fisher z
SE = 1 / √(n - 3)
4. Confidence interval on z scale
z ± zcritical × SE
5. Convert interval back to correlation scale
Apply the inverse Fisher formula to both z limits.
6. Comparing two independent correlations
Z = (z₁ - z₂) / √(1/(n₁ - 3) + 1/(n₂ - 3))
How to Use This Calculator
- Select the calculation mode that fits your analysis.
- Enter the needed correlation values, sample sizes, or z value.
- Set the confidence level when interval estimation is required.
- Press the calculation button to generate the output.
- Review the result box shown below the header and above the form.
- Inspect the summary table and the Plotly graph.
- Export the displayed result as CSV or PDF when needed.
Frequently Asked Questions
1. What does Fisher z transformation do?
It converts a correlation coefficient into a value with a more nearly normal sampling distribution. That makes interval estimation and hypothesis testing more stable, especially when correlations are moderately large.
2. Why not analyze r directly?
The sampling distribution of r is skewed, especially near -1 or 1. Fisher z reduces that skewness, making standard errors and confidence intervals easier to handle with normal theory methods.
3. What sample size rule matters here?
The usual standard error formula uses n - 3 in the denominator. Because of that, the calculator requires at least 4 observations for Fisher-based interval and comparison calculations.
4. Can I compare two correlations from different studies?
Yes. Use the comparison mode for two independent correlations. The calculator transforms each correlation to z, computes the difference, and standardizes it using the combined standard error.
5. Does the calculator support negative correlations?
Yes. Negative correlations are fully supported as long as the value remains greater than -1 and less than 1. The transformation works symmetrically on both negative and positive sides.
6. Why are exact values of -1 and 1 rejected?
The logarithmic Fisher formula becomes undefined at those endpoints. Values extremely close to them can be transformed, but exact perfect correlations cannot be processed mathematically.
7. What confidence level should I choose?
Most users choose 95% for standard reporting. Higher levels create wider intervals, while lower levels create narrower intervals. The right choice depends on your reporting standard and risk tolerance.
8. What does the graph show?
The graph displays how Fisher z changes across possible correlation values. It helps you see that the relationship is nonlinear, especially as the correlation approaches the extremes.