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Imagine you have 2 ethnic groups on the planet Zorblax. Let us call them the A's and the B's.

Stealing on Zorblax can have 3 different penalties:

  • A fine
  • A jail term
  • Death penalty

In a certain year, there were 1100 convictions for theft.

Out of this 1100, 100 A's were convicted, and 1000 B's were convicted

The A's received the following sentences:

  • 50 Fines
  • 25 Jail terms
  • 25 Death penalties

The B's received the following sentences

  • 700 Fines
  • 200 Jail terms
  • 100 Death penalties

This data seems to suggest the A's are unfairly treated, getting harsher sentences. But is this difference significant, considering the overall sentencing, the sizes of the groups, etc? How do you measure this, and which statistical tests / techniques should be used?

The null-hypothesis here (I believe) would be: 'The A sentences and the B sentences are drawn from the same population'. So I guess what I'm asking is along the lines of 'What is the probability of a deviation this large or more happening in the A group assuming there is no difference in the population?' How would you calculate this?

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    $\begingroup$ A P-value is not " the probability of a deviation this large or more from the average happening in the A group by random chance?". It is the probability of observing the difference observed (or a larger on) IF there is actually no difference in the population. Is this what you are looking for ? $\endgroup$
    – Robert Long
    Commented Mar 3, 2021 at 9:32
  • $\begingroup$ @RobertLong: I've updated the wording in the question to be more precise. Thank you, that is what I am looking for. $\endgroup$
    – Baldrick
    Commented Mar 3, 2021 at 9:46

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You might try a chi-square test of homogeneity, with the null hypothesis that the distribution of sentences is the same for the A's and the B's. Just 2 statements in R. Thanks for the cute example; I'm swiping it (attributed of course) for my introductory stats students.

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  • $\begingroup$ Thanks very much for the suggestion. I'm flattered that you want to swipe my example :) Would that homogeneity test take into account the difference sizes of the samples? $\endgroup$
    – Baldrick
    Commented Mar 3, 2021 at 10:47
  • $\begingroup$ The chi-square test of homogeneity implicitly controls for sample size for each expected-observed score when the terms are calculated: $\frac{(E_i - O_i)^2}{E_i}$ $\endgroup$
    – Mike Anderson
    Commented Jul 12, 2022 at 20:54

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