Calculator
Formula Used
The Terry-Hoeffding approach pools both samples, ranks the combined observations, and replaces each pooled rank with an expected normal score.
Default score approximation: p(r) = (r - 0.375) / (N + 0.25) and a(r) = Φ-1(p(r)).
If tied values occur, the calculator averages the normal scores across the tied rank positions and assigns the same score to each tied observation.
The main score sum for the first sample is A₁ = Σ a(r). The standardized statistic is Z = (A₁ - n₁ā) / √(s² n₁ n₂ / N).
Here, ā is the mean of all assigned normal scores, s² is the sample variance of those scores, n₁ and n₂ are group sizes, and N = n₁ + n₂.
The calculator reports χ² = Z², the selected p value, and the effect size r = |Z| / √N.
How to Use This Calculator
Enter one independent sample in the first box and the second sample in the next box. You can separate values with commas, spaces, or line breaks.
Set the group labels, choose the significance level, select the alternative hypothesis, and pick the expected normal score approximation you want.
Click the test button. The result appears immediately above the form, under the header area, as requested.
Review the z statistic, p value, effect size, summary table, detailed score table, and Plotly graph. Download CSV for complete rows or PDF for a compact report.
Example Data Table
| Observation | Sample A | Sample B |
|---|---|---|
| 1 | 18.2 | 16.9 |
| 2 | 17.9 | 17.2 |
| 3 | 19.1 | 17.4 |
| 4 | 18.7 | 16.8 |
| 5 | 20.3 | 17.7 |
| 6 | 18.5 | 17.1 |
| 7 | 19.4 | 16.6 |
| 8 | 18.8 | 17.5 |
Interpretation Notes
This test is useful when you want a rank-based comparison of two independent samples but still want scoring that reflects normal order statistics instead of raw ranks.
Larger positive score sums for the first sample usually indicate higher observed values in that sample. Larger negative values indicate the reverse direction.
The effect size r helps compare practical magnitude across studies. It complements the hypothesis test instead of replacing it.
FAQs
1. What does this test measure?
It checks whether two independent samples differ in location after replacing pooled ranks with normal scores. It is a nonparametric location test with strong efficiency under many distributions.
2. When should I use it?
Use it when you want a two-sample comparison that is less tied to strict distribution assumptions than a classical mean test, yet still uses normal-score weighting.
3. Can I enter decimals and negative values?
Yes. The parser accepts integers, decimals, and signed numbers. Separate them by commas, spaces, semicolons, or line breaks.
4. How are ties handled?
Tied observations share the same pooled rank block. The calculator averages the normal scores across that tied block and assigns the same score to every tied value.
5. What is the difference between Blom, Tukey, and Rankit?
They are different approximations for expected normal order scores. Blom is the default here because it is widely used and gives stable results for many sample sizes.
6. Is the graph required for the test?
No. The graph is a visual aid. It helps you inspect how pooled observations map to assigned normal scores and how the two groups spread across the score scale.
7. What does effect size r mean?
It is a compact standardized effect summary based on the test statistic. Smaller values suggest weaker evidence of separation, while larger values indicate stronger practical differences.
8. Which export should I choose?
Use CSV when you want every detailed row for spreadsheet work. Use PDF when you want a short report containing the headline outputs and summary rows.