The numbers don't lie

What do you think?

A county is 79% Mexican-American. Over 11 years, 870 people are called for jury duty. If selection were random, about how many would be Mexican-American?

But here's the real question: could 339 happen by pure chance?

What do you think?

We expected 688 Mexican-American jurors but got 339. Is this gap statistically significant?

Explore what different statistical deviations look like:

Jury Pool

Fairness Test

Expected: 688 of 870 jurors

(79.1% population)

Actually selected: 339

288788

-29.1σ

339 observed (349 fewer)

What does "29 standard deviations" mean?

Z-Score Scale

Standard deviations (z-score): 2

1σ10σ

Probability beyond 2σ

2.3%

If a result is 2σ from expected, about what percentage of the time does this happen by chance? (whole number)

1/3

The Supreme Court agreed. The conviction was overturned.

The math behind the verdict

How do we calculate whether a result is statistically significant? Here's the Partida case step by step.

Calculating Statistical Significance

X \sim \text{Binomial}(n=870, p=0.791)

Each juror selection is random with probability p = 0.791. With n = 870 trials, this follows a [binomial distribution](/lessons/bernoulli-binomial).

Step 1 of 4

A company has 1000 employees. The applicant pool is 40% female, but only 280 employees are female. What's the expected number under random hiring? (whole number)

With n=1000 and p=0.4, the standard deviation is √(1000×0.4×0.6) ≈ 15.5. What's the z-score for observing 280? (decimal to 2 places, e.g. -3.45)

True or False: A z-score of -7.74 provides strong evidence of bias in hiring.

Statistics cannot prove intent, it cannot say why selection was biased. But it can prove the bias exists. That's often enough.

The legal standard

The Supreme Court adopted a rule: if the disparity exceeds 2-3 standard deviations, it establishes a prima facie case of discrimination. The burden then shifts to the state to explain.

What do you think?

A company has 200 employees from a region that's 60% Black. Only 80 employees are Black (40%). Is this statistically significant evidence of bias?

The same framework applies to employment discrimination, school admissions, police stops, and medical trials.

A word of caution

Statistical evidence is strong, but interpretation requires care.

Simpson's paradox

Consider a company with two departments:

Department	Women Applied	Women Promoted	Men Applied	Men Promoted
Dept A (Competitive)	80	24 (30%)	20	6 (30%)
Dept B (Less Competitive)	20	16 (80%)	80	64 (80%)
Overall	100	40 (40%)	100	70 (70%)

Within each department, promotion rates are equal. But overall, men are promoted at nearly double the rate. The paradox: women disproportionately applied to the more competitive department, dragging down their aggregate numbers.

Always ask: could a lurking variable explain this pattern? Aggregate statistics can mislead when subgroups behave differently.

Toggle between the aggregate view and the department breakdown to see how the same data tells two different stories:

Simpson's Paradox

A company has equal promotion rates within each department. But overall, men get promoted more. How is this possible?

Women

38%

promoted overall

Men

70%

promoted overall

Looks like discrimination! Men are promoted at almost double the rate. But look at the department breakdown...

What do you think?

A company has equal promotion rates within each department. Can it still have unequal rates overall?

Test your understanding

For Binomial(n=500, p=0.3), what is E[X]? (whole number)

For Binomial(n=500, p=0.3), what is the standard deviation? (σ = √(np(1-p))) (decimal to 2 places, e.g. 5.23)

If you observe X=120 when E[X]=150 and σ≈10.25, what's the z-score? (decimal to 2 places, e.g. -3.45)

True or False: The 2-3σ legal threshold means we can prove intentional discrimination.

A fair coin is flipped 100 times. You expect 50 heads. σ = 5. You get 65 heads. What's the z-score? (whole number)