Calculate the critical ratio from two sample means or a direct difference and standard error, with p-value and 95% confidence interval.
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Critical Ratio Formula
CR = (M1 - M2) / SED
The standard error of the difference (SED) depends on which spread values you have:
SED = sqrt(SD1^2/n1 + SD2^2/n2) (from SDs and sample sizes) SED = sqrt(SE1^2 + SE2^2) (from standard errors)
- CR — critical ratio (also called the z-score for the difference)
- M1, M2 — sample means of group 1 and group 2
- SD1, SD2 — sample standard deviations
- n1, n2 — sample sizes
- SE1, SE2 — standard errors of each mean
- SED — standard error of the difference between means
The critical ratio assumes the two samples are independent and that the sampling distribution of the difference is approximately normal. With small samples (roughly n < 30 per group), a t-statistic with appropriate degrees of freedom is more accurate than CR.
Reference Values
Compare your absolute CR to these thresholds to judge significance:
| Significance level | Two-tailed |CR| | One-tailed CR |
|---|---|---|
| 0.10 | 1.645 | 1.282 |
| 0.05 | 1.960 | 1.645 |
| 0.01 | 2.576 | 2.326 |
| 0.001 | 3.291 | 3.090 |
Quick interpretation of the result:
| |CR| | Approx. two-tailed p | Verdict |
|---|---|---|
| < 1.00 | > 0.32 | Difference looks like noise |
| 1.00 – 1.96 | 0.05 – 0.32 | Suggestive, not significant |
| 1.96 – 2.58 | 0.01 – 0.05 | Significant at 0.05 |
| > 2.58 | < 0.01 | Significant at 0.01 |
Example
Group 1: M1 = 82, SD1 = 10, n1 = 50. Group 2: M2 = 78, SD2 = 12, n2 = 60.
SED = sqrt(10²/50 + 12²/60) = sqrt(2 + 2.4) = sqrt(4.4) = 2.098
CR = (82 − 78) / 2.098 = 1.906
|CR| = 1.91 falls just under the 1.96 cutoff, so the difference is not significant at the two-tailed 0.05 level (p ≈ 0.057).
FAQ
Is the critical ratio the same as a z-score? Yes. CR is a z-score applied to the difference between two estimates.
When should you use a t-test instead? Use a t-test when sample sizes are small or population variances are unknown and you need exact p-values. CR treats the statistic as standard normal, which is fine for large samples.
Can CR be negative? Yes. The sign tells you which group is higher. For two-tailed tests, only the magnitude matters.
What if the two groups have very different variances? The Welch-style SED used here (SD1²/n1 + SD2²/n2) handles unequal variances. Do not pool the SDs.
