Disparate impact ratio and the four-fifths rule

Step 1: selection rates per group

The selection rate is the share of a group that gets the favourable outcome: selected ÷ applicants. For a hiring screen that is the share shortlisted, for a loan model the share approved. In the example, 312 of 800 applicants in group A are selected, 39.0%, and 84 of 300 in group C, 28.0%. Across all four groups the rate is 598 ÷ 1,720 = 34.8%.

Count from the decisions, not from the model scores, and use the same population for every group: applicants who reached the same step, over the same period.

Step 2: impact ratios and the disparate impact ratio

Divide every group's rate by the highest rate: =D2/MAX($D$2:$D$5). Group A, the most-selected group, gets 1.000, group B 0.821, group C 0.718 and group D 0.897. The lowest of these, MIN ÷ MAX of the rates, is the disparate impact ratio of the model as a whole: 0.718. Fairness toolkits such as IBM's open-source AI Fairness 360 report the same quantity as "disparate impact", there usually for one unprivileged group against one privileged group.

The difference of the rates, 39.0% − 28.0% = 11.0 points, is the demographic parity difference. Ratio and difference tell the same story on different scales. A 3-point gap is large when rates are around 10% and small when they are around 80%, which is why the ratio is the more common screen.

Step 3: the four-fifths rule

The rule comes from the US Uniform Guidelines on Employee Selection Procedures of 1978 (29 CFR 1607.4(D)). A selection rate for any group that is less than four-fifths, 80%, of the rate of the group with the highest rate will generally be regarded by the federal agencies as evidence of adverse impact. In a sheet: =IF(E2<0.8,"Flag","OK"). Only group C is flagged.

Hiring is still the main legal context, and New York City's Local Law 144 requires bias audits of automated hiring tools that report exactly these selection rates and impact ratios. Machine learning teams also borrow the 0.80 line as a quick screen for other decisions, such as credit or insurance, where different laws and standards apply. Either way, a flag means "look closer", not "proven discrimination", and a pass does not prove fairness.

Worked example: four-fifths check for a screening model

Applicants and selections per group are the inputs. Rows 8 to 15 summarise the check and add a z-test of group C against group A. All data are fictional.

Worked example: four-fifths check for a screening model

	A	B	C	D	E	F
1	Group	Applicants	Selected	Selection rate	Impact ratio	Four-fifths check
2	A	800	312	39.0%	1.000	OK
3	B	500	160	32.0%	0.821	OK
4	C	300	84	28.0%	0.718	Flag
5	D	120	42	35.0%	0.897	OK
6	All groups	1,720	598	34.8%
7
8	Threshold (four-fifths)	80%
9	Highest selection rate	39.0%
10	Disparate impact ratio (lowest ÷ highest)	0.718
11	Groups flagged	1
12	Extra selections group C needs for 80%	10
13	Parity difference (highest − lowest rate)	11.0%
14	Pooled rate, groups A and C	36.00%
15	z statistic, A vs C	3.39

The four-fifths rule is used here as a screening heuristic, not as a legal test. Educational, not legal advice.

Step 4: how far below the line, in people

A ratio is hard to picture, so translate it into selections. To reach 80% of the top rate, group C needs ROUNDUP(0.8 × 39.0% × 300, 0) = 94 selections instead of 84, so 10 more. That number also shows how fragile the result is: in a small group, two or three decisions can move the ratio across the line, which is why the guidelines themselves say sample size and statistical significance matter.

Step 5: add a significance test

A two-proportion z-test asks whether the gap between two groups is larger than chance would produce. Pool the two groups, p = (312 + 84) ÷ (800 + 300) = 36.00%, and divide the difference of the rates by its standard error:

z = (39.0% − 28.0%) ÷ √(p × (1 − p) × (1/800 + 1/300)) = 3.39

US courts have often treated a gap of more than two or three standard deviations as significant, and 3.39 is above both. Here the ratio and the test agree. With 30 applicants in group C instead of 300, the same rates would give a z of about 1.2, a flag on the ratio but no statistical evidence. For the logic behind such tests, see the guide on the chi-square test in Excel, which is equivalent for a 2 × 2 table: the chi-square statistic without continuity correction equals z².

What the ratio does not tell you

Equal selection rates can hide unequal errors. A model may select the same share of every group but miss more qualified people in one of them. That is what equal opportunity (true positive rates) and equalised odds (true and false positive rates) measure, both built from a confusion matrix per group. Check intersectional subgroups too, such as group by age band, and flag any subgroup that is too small to judge. A fairness review uses several of these metrics side by side.

Common mistakes

• Dividing by the overall rate. The impact ratio compares each group with the most-selected group, not with the average of all applicants.
• Comparing rejections instead of selections. Ratios of rejection rates give different numbers. Use the rate of the favourable outcome.
• Reading 0.80 as a legal verdict. The rule is a rule of thumb. Small groups need a significance test, and other laws use other standards.
• Ignoring small samples. With a few dozen applicants, one or two decisions move the ratio across the line. Report the counts next to the ratio.
• Checking only the whole model. Look at every group and at intersectional subgroups, not just the lowest ratio overall.
• Stopping at selection rates. Equal rates can come with unequal error rates. Add equal opportunity and calibration checks.