XLSTAT - Comparison of one proportion

Test for the comparison of one proportion

XLSTAT uses the z-test to to compare one empirical proportion to a theoretical proportion.

Let n be the number of observations verifying a certain property among a sample of size N. The proportion of the sample verifying the property is defined by p = n / N. Let p0 be a known proportion with which we wish to compare p. Let D be the assumed difference (exact, minimum or maximum) between the two proportions p and p0. D is usually 0.

The two-tailed (or two-sided) test corresponds to testing the difference between p – p0 and D, using the null (H0) and alternative (Ha) hypotheses shown below:

  • H0 : p - p0 = D
  • Ha : p - p0 ≠ D

In the one-tailed case, you need to distinguish the left-tailed (or lower-tailed or lower one-sided) test and the right-tailed (or right-sided or upper one-sided) test. In the left-tailed test, the following hypotheses are used:

  • H0 : p - p0  = D
  • Ha : p - p0 < D

In the right-tailed test the following hypotheses are used:

  • H0 : p - p0  = D
  • Ha : p - p0 > D

Assumptions for the z-test

This z-test is based on the following assumptions:

  • The observations are mutually independent,
  • The probability p of having the property in question is identical for all observations,
  • The number of observations is large enough, and the proportions are neither too close to 0 nor to 1.

z statistic

One can find several ways to compute the z statistic in the statistical literature. The most used version is:

z = p – p0 - D ⁄ σ

The large sample approximation leads to the following estimate for its standard deviation: σ, σ

σ(π) = √ p (1- p) / N

The z statistic is asymptotically normally distributed. The larger N, the better the approximation. The p-value is computed using the normal approximation.

Confidence intervals for the comparison of one proportion

Many methods exist to compute confidence intervals on a proportion. XLSTAT offers the choice between four different versions: Wald, Wilson score, Clopper-Pearson, Agresti Coull.


This analysis is available in the XLStat-Basic addin for Microsoft Excel

About KCS

Kovach Computing Services (KCS) was founded in 1993 by Dr. Warren Kovach. The company specializes in the development and marketing of inexpensive and easy-to-use statistical software for scientists, as well as in data analysis consulting.

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