Ztest Proportions Calculator
Free Ztest proportions Calculator for hypothesis tests. Enter values to get step-by-step solutions with formulas and graphs.
Reviewed by Daniel Agrici, Founder & Lead Developer
Formula
z = (pฬโ - pฬโ) / โ(pฬ(1-pฬ)(1/nโ + 1/nโ))
The z-statistic is calculated by dividing the difference in sample proportions by the pooled standard error. The pooled proportion p-hat combines both samples under the null hypothesis assumption that the true proportions are equal.
Worked Examples
Example 1: A/B Test for Website Conversion
Problem:Version A: 120 conversions out of 1,500 visitors. Version B: 155 conversions out of 1,500 visitors. Test at ฮฑ = 0.05 (two-tailed).
Solution:p1 = 120/1500 = 0.0800, p2 = 155/1500 = 0.1033\nPooled p = (120+155)/(1500+1500) = 0.0917\nSE = โ(0.0917 ร 0.9083 ร (1/1500 + 1/1500)) = 0.01055\nz = (0.0800 - 0.1033) / 0.01055 = -2.209\np-value = 0.0272 (two-tailed)\n0.0272 < 0.05 โ Reject Hโ
Result:z = -2.209 | p-value = 0.027 | Significant โ Version B has higher conversion rate
Example 2: Clinical Trial Drug Effectiveness
Problem:Treatment group: 85 recoveries out of 200 patients. Control group: 60 recoveries out of 200 patients. Test at ฮฑ = 0.01 (right-tailed).
Solution:p1 = 85/200 = 0.425, p2 = 60/200 = 0.300\nPooled p = (85+60)/(200+200) = 0.3625\nSE = โ(0.3625 ร 0.6375 ร (1/200 + 1/200)) = 0.04808\nz = (0.425 - 0.300) / 0.04808 = 2.600\np-value = 0.0047 (right-tailed)\n0.0047 < 0.01 โ Reject Hโ
Result:z = 2.600 | p-value = 0.005 | Significant โ Treatment group has higher recovery rate
Frequently Asked Questions
How do I interpret the p-value in a proportions z-test?
The p-value represents the probability of observing a difference in sample proportions as extreme as (or more extreme than) the one calculated, assuming the null hypothesis is true (that the population proportions are actually equal). A small p-value, typically below 0.05, indicates that such an extreme result would be very unlikely if there were truly no difference, leading us to reject the null hypothesis. For example, a p-value of 0.03 means there is only a 3% chance of seeing the observed difference (or a larger one) if the proportions were actually equal. Importantly, the p-value does not tell you the probability that the null hypothesis is true, the magnitude of the practical effect, or whether the result is meaningful in a real-world context. Always consider the confidence interval and effect size alongside the p-value, as statistically significant results with very large samples may represent trivially small practical differences.
References
Reviewed by Daniel Agrici, Founder & Lead Developer ยท Editorial policy