Binomial test and one-proportion Z-test
Proportions (counts) · One sample · reference distribution: Binomial or N(0, 1)
When to use it
Compare the proportion in a sample with a reference value, such as a pass rate against 50%.
Null hypothesis
The population proportion is p₀.
Assumptions
- Independent observations
- For the Z-test: np₀ and n(1 − p₀) at least 10; otherwise, use the exact binomial
Test statistic
z = \dfrac{\hat{p} - p_0}{\sqrt{p_0(1 - p_0)/n}}Effect size
Cohen’s h: the difference between the proportions on the arcsine scale. Benchmarks: 0.2, 0.5 and 0.8.
h = 2\arcsin\sqrt{\hat{p}} - 2\arcsin\sqrt{p_0}How to report it
62 of 100 (62%), binomial test against 50%, p = .021, h = 0.24
In R and Python
R
binom.test(62, 100, p = 0.5) # exact; prop.test() does the Z-testPython
stats.binomtest(62, 100, p=0.5)In Python, stats is scipy.stats and np is numpy.
Variants and alternatives
- Exact binomial test, for small samples
Where it sits in the catalog
Proportions (counts). For categorical answers, such as yes or no, hit or miss: the data are how often each category appears, and the test compares proportions.
One sample. A single group compared with a reference value set in advance, such as a target or a standard.
In the decision tree
- What do you want to do? Compare groups, or one group with a reference value
- What kind of response did you measure? Categorical: yes or no, or categories
- How many groups or measurements? One group, against a reference proportion