# How to do a one-way analysis of variance (ANOVA) (in Python, using SciPy)

## Task

If we have multiple independent samples of the same quantity (such as students’ SAT scores from several different schools), we may want to test whether the means of each of the samples are the same. Analysis of Variance (ANOVA) can determine whether any two of the sample means differ significantly. How can we do an ANOVA?

Related tasks:

- How to do a two-sided hypothesis test for two sample means (which is just an ANOVA with only two samples)
- How to do a two-way ANOVA test with interaction
- How to do a two-way ANOVA test without interaction
- How to compare two nested linear models
- How to conduct a mixed designs ANOVA
- How to conduct a repeated measures ANOVA
- How to perform an analysis of covariance (ANCOVA)
- How to do a Kruskal-Wallis test

## Solution

Let’s assume we have our samples in several different Python lists. (Although anything like a list is also supported, including pandas Series.) Here I’ll construct some made-up data about SAT scores at four different schools.

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school1_SATs = [ 1100, 1250, 1390, 970, 1510 ]
school2_SATs = [ 1010, 1050, 1090, 1110 ]
school3_SATs = [ 900, 1550, 1300, 1270, 1210 ]
school4_SATs = [ 900, 850, 1110, 1070, 910, 920 ]

ANOVA tests the null hypothesis that all group means are equal. You choose $\alpha$, the probability of Type I error (false positive, finding we should reject $H_0$ when it’s actually true). I will use $\alpha=0.05$ in this example.

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alpha = 0.05
# Run a one-way ANOVA and print out alpha, the p value,
# and whether the comparison says to reject the null hypothesis.
from scipy import stats
F_statistic, p_value = stats.f_oneway(
school1_SATs, school2_SATs, school3_SATs, school4_SATs )
reject_H0 = p_value < alpha
alpha, p_value, reject_H0

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(0.05, 0.0342311478489849, True)

The result we see above is to reject $H_0$, and therefore conclude that at least one pair of means is statistically significantly different.

Content last modified on 24 July 2023.

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Contributed by Nathan Carter (ncarter@bentley.edu)