Calculate the critical ratio from two sample means or a direct difference and standard error, with p-value and 95% confidence interval.

Means, differences and spread values must use the same measurement unit.

Normal approximation for independent estimates, with a zero null difference. Sample-SD inference may require Welch’s t test; this tool does not calculate t probabilities.

Critical Ratio Formula

CR = (M1 - M2) / SED

The standard error of the difference (SED) depends on which spread values you have:

SED = sqrt(SD1² / n1 + SD2² / n2) (from SDs and sample sizes); SED = sqrt(SE1² + SE2²) (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 two-mean modes assume independent estimates. The p-values and two-sided 95% interval use a normal approximation with a zero null difference. When population variances are estimated, Welch’s t test uses appropriate degrees of freedom; there is no universal sample-size cutoff that makes normal inference exact.

Reference Values

For two-sided tests, compare |CR| with the two-sided thresholds. For one-sided tests, use the signed CR in the direction specified before inspecting the data:

Significance level Two-tailed |CR| One-tailed CR
0.101.6451.282
0.051.9601.645
0.012.5762.326
0.0013.2913.090

Quick interpretation of the result:

|CR| Approx. two-tailed p Verdict
< 1.00> 0.32p > 0.32 under the normal model
1.00 – 1.960.05 – 0.32p ≥ 0.05 under the normal model
1.96 – 2.580.01 – 0.05Significant at 0.05
> 2.58< 0.01Significant 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 estimating population variances and the t-model assumptions apply. This calculator reports approximate standard-normal probabilities; a large sample alone does not validate independence, sampling, or distribution assumptions.

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 unpooled SED used here is √(SD1²/n1 + SD2²/n2). It allows unequal spreads for independent groups, but the reported probabilities remain normal approximations and do not use Welch degrees of freedom.