Funnel maths and adverse impact

The conversions on Apex's funnel are 30.0, 33.3, 30.0 and 83.3 percent, and each divides by the stage above it. The yields are 30.0, 10.0, 3.0 and 2.5 percent, and each divides by the 400 applicants. The screen and the yield agree at the first stage by construction, which is how the two get confused: the first row of either column reads 30.0, and from the second row on they part company.

In pandas the difference is one method. The conversion is total / total.shift(1), the stage before; the yield is total / total.iloc[0], the top. Write both columns, and write the base as its own column beside the conversion, so that "33.3 percent" can never be read without "of 120" next to it. A reader who sees 40 of 120 knows the size of the number; a reader who sees 33.3 alone has to trust the analyst about which 33.3 it is.

The last conversion, 83.3 percent, is the one that flatters: 10 of 12 offers accepted sounds like a strong funnel. It is ten people. The first conversion, 120 of 400, is the one that decides: 280 people were turned away at the screen, more than at every later stage put together. When a funnel is judged, it is judged at its widest stage, and that is where Part 2 splits the base.

The five counts only divide cleanly if they are the same people moving through: the 120 screened in are 120 of the 400 who applied. A monthly recruiting report usually counts events in the month instead: 400 applications received in March, 12 offers made in March, some of them to people who applied in January. The ratio of those two counts is not a conversion of anybody; it is two flows from different cohorts divided by each other, and it moves when the hiring volume moves even if nothing about the process changed.

The fix is the event-log discipline from deep-dive session 1 applied to candidates: one row per candidate per stage with a date, then count each stage for the same application cohort. Apex's constructed funnel is a cohort by construction. A real one has to be built that way on purpose, and the six decisions at the top of the script gain a seventh: which cohort, defined by which date.

The impact ratio at a stage is the lower group's selection rate over the highest group's selection rate, both on that stage's own base. With two groups it is one division; with a dozen ethnicity categories it is every group's rate over the top rate, which is why the build-along writes it as a function that takes a dictionary. It is a ratio of two rates, so it carries four counts behind it, and all four belong beside it: 40 of 160 over 80 of 240, not "0.75".

The US Uniform Guidelines on Employee Selection Procedures, 29 CFR 1607.4(D), 1978, give the rule of thumb: a selection rate less than four-fifths (80 percent) of the highest group's rate is generally regarded as evidence of adverse impact. That is the one sentence of law on this page, and it is a rule of thumb, not a verdict. What counts as a group, what to do with small numbers, how a screening tool moves the ratio and what each jurisdiction requires are taught with a working screening simulator and a legal map in the AI + HR course. Use that simulator to see a tool shift the ratio; tonight's job is the arithmetic that feeds it.

On Apex's invented split the screen gives 0.750 with women lower, and the offer stage gives 0.754 with men lower. Both are under four-fifths, in opposite directions. The interview (0.929) and the hire (0.933) are over it. That is a funnel with two stages worth a question and two that are not, and the next card shows why the headline ratio says none of this.

End to end, 6 of 240 men and 4 of 160 women were hired: 2.5 percent each, an impact ratio of exactly 1.000. A report that computes only this number declares the funnel even. It is even in outcome and uneven in process: the screen removed women at a lower rate of selection, the offer stage removed men at a lower rate, and the two stages cancelled on this particular draw of invented counts. Change either stage and the cancellation goes with it.

Each group's end-to-end rate is the product of its stage rates, so the end-to-end ratio multiplies the stage ratios together. It can hide a failing stage behind a lenient one, or compound two mild stages into a failing whole. Neither direction says anything about the stage where a decision was made. A screening tool acts at the screen; a hiring manager acts at the offer. The audit belongs at the stage where the decision is taken, with that stage's base, which is the whole reason the build-along loops over stages.

The offer stage is 7 of 26 men and 5 of 14 women. Offer one more man and the ratio goes from 0.754 to 0.862, from under four-fifths to over it. At the hire stage, 6 of 7 and 4 of 5: one fewer woman hired and 0.933 becomes 0.700. A ratio that crosses the line when one person changes is not a finding about a process; it is a finding about the size of the base, the same lesson deep-dive session 3 drew from HR's 8 of 75.

The screen is different. 40 of 160 women were screened in; the ratio reaches 0.8 only at 43 of 160, three more people on a base of 160. That is still a small number of people, but it is a stage with enough people for the ratio to mean something. The honest report prints the one-person width beside every ratio. Whether a difference is statistically distinguishable from zero is a question for a significance test, which the Statistics course teaches; the width is the version a board can read.

A published summary under Local Law 144 reports, by sex and by race/ethnicity, the selection rate of each category and the impact ratio of each category against the category with the highest rate. Those two columns are the moss block in the figure. They are the same two numbers the build-along prints at every stage, and they are only as good as the counts behind them: 0.750 on 40 of 160 women against 80 of 240 men, and 0.750 on 3 of 12 against 4 of 12, are the same number and not the same finding. Read a published audit with the counts columns first.

The audit covers one decision point, the tool, which is exactly the stage-by-stage view of Part 2. A tool that screens resumes sits at the screen; its audit says nothing about the interview panel downstream, and an end-to-end number would say less. The analyst's contribution is to produce the same table for every stage the company controls, not only the one the law names, with the one-person width beside every ratio.

Deep-dive session 9 returns to the same law alongside GDPR Article 22 and the EU AI Act, when the question is what a model may see at all. The legal map in depth, and what an employer does when a ratio comes back under four-fifths, stay with the AI + HR course.

Every impact ratio needs the group recorded for every person in the base. Applicants who decline to state their sex or ethnicity drop out of both the numerator and the denominator, and if they decline at different rates at different stages the stage bases shrink unevenly. Print, for every stage, how many people in the base had the category recorded, before any ratio. A screen with 400 applicants and 150 recorded categories is an impact ratio on 150 people, and the sentence should say so.

The other trap is the category list itself. Race/ethnicity with twelve categories gives twelve rates, several on single-digit counts, and a ratio of every one against the highest. The rates at the bottom of the list will swing by whole tenths on one person. Merging categories to get stable counts is a decision with consequences for who becomes invisible, and it belongs in the decisions block at the top of the script, written before any number.