One-Sample Z-Test for a Proportion Calculator
Written by Thierno Sadou Diallo, formula verified per our methodology • Last checked on 9/9/2026
The z-test for a proportion is calculated with z = (p̂ − p0) ÷ √(p0(1−p0) ÷ n), where p̂ is the observed proportion and p0 the reference proportion. For an observed proportion of 55% in 100 people, tested against a reference of 50%, the z-statistic is exactly 1.
Explanation
The z-test for a proportion answers a different question than our confidence interval for a proportion calculator: rather than estimating a plausible range for the true proportion, it tests whether a proportion observed in a sample (p̂) significantly deviates from a known or hypothesized reference value (p0) — for example, checking whether an observed 55% conversion rate genuinely differs from 50% (the equal-odds hypothesis), or whether an observed defect rate differs from an expected manufacturing standard. The important statistical nuance compared to the confidence interval lies in how the standard error is calculated in the denominator: the hypothesis test uses p0 (the reference value, under the so-called null hypothesis that it's the true proportion), while the confidence interval uses p̂ (the actually observed proportion) — both approaches share the same mathematical structure but answer distinct questions, and this difference in convention isn't a mere typo but a deliberate methodological distinction in inferential statistics. The resulting z-statistic measures how many standard errors separate the observed proportion from the expected reference under the tested hypothesis: the larger its absolute value, the less plausible it is that the observed gap is due to sampling chance alone. Like our other inferential statistics calculators, this calculator deliberately stops at the z-statistic itself, with no automatic significance conclusion: comparing this result to the usual critical scores (1.645 for a 90% threshold, 1.96 for 95%, already used in our z-score calculator) lets you conclude whether the difference is statistically significant.
Example: an observed proportion of 55% against a reference of 50%, in 100 people
Inputs
Observed proportion: 0.55. Reference proportion: 0.5. Sample size: 100.
Calculation
z = (0.55 − 0.5) ÷ √(0.5 × 0.5 ÷ 100) = 0.05 ÷ √0.0025 = 0.05 ÷ 0.05 = 1.
Result
The z-statistic is exactly 1, a moderate difference from the expected reference.
Frequently asked questions
Why does this calculation use p0 rather than p̂ in the standard error?
Because this hypothesis test deliberately places itself under the null hypothesis, meaning it provisionally assumes the true proportion actually is p0: under that framework, the theoretical standard error should logically be calculated from p0, the assumed true value, rather than from p̂, which is only an estimate observed in a particular sample. This is a standard methodological convention in hypothesis testing, distinct from the standard error calculation used to build a confidence interval.
Does this calculator directly give a statistical significance conclusion?
No, deliberately: this calculator only gives the z-statistic, with no automatic conclusion. Comparing this statistic to the usual critical scores of the normal distribution (1.645 for a 90% confidence test, 1.96 for 95%, 2.576 for 99%, the same values as our z-score calculator) lets you determine whether the observed gap is statistically significant at the chosen threshold.
What is the difference from the confidence interval for a proportion calculator?
The confidence interval starts from the observed proportion alone to estimate a plausible range for the true proportion in the population, with no external reference value to test. The z-test, by contrast, explicitly compares the observed proportion to a precise reference value (p0) fixed in advance, to answer the question "is this observed proportion different from this expected value?" rather than "what's the likely range of the true proportion?"