How can a batter beat a rival in each of two seasons yet trail on the combined average?
answer
- totals, not the average of averages
- each season is weighted by its at-bats
- the good season had few chances
- ratio of sums versus sum of ratios
- equal splits make reversal impossible
basics
~20 sA combined batting average is total hits over total at-bats, so each season is weighted by its at-bats rather than counted equally. If a batter's strong season carries few at-bats and the rival's strong season carries many, the ordering flips.
solid answer
~40 sA batting average is a ratio, and combining two seasons means dividing total hits by total at-bats — a ratio of sums, not the average of the two season averages. That makes the combined figure a weighted average whose weights are the at-bat counts. Concretely: batter A goes 3 for 10 (.300) then 20 for 100 (.200); batter B goes 26 for 100 (.260) then 3 for 20 (.150). A wins both seasons. Combined, A is 23 for 110 = .209 and B is 29 for 120 = .242, so B wins. A's good season carried only 10 at-bats while B's carried 100, so A's average is pulled toward his weak season and B's toward his strong one. The reversal is pure base-rate imbalance: nothing about the players changed between the two tables.
go deeper
Recall that a combined average divides total hits by total at-bats, never averages two percentages. Being able to say that out loud is most of the answer at this level.
Write the pooled figure as w times one season's average plus one minus w times the other's, name the weights as at-bat counts, and show which weight configuration produces the flip.
Show what you would do about it: fix a common weighting for both subjects, report the standardised figure with the weighting named, and keep the stratified table visible next to any headline number.
Decide the organisation's default: which denominators are published, whether headline rates are composition-adjusted, and who owns the reference weighting so comparisons across teams and periods stay meaningful.
## Ratio of sums, not sum of ratios A batting average is hits divided by at-bats. When you combine two seasons you add the hits and add the at-bats and divide once: ``` combined = (hits_1 + hits_2) / (AB_1 + AB_2) ``` That is a **ratio of sums**. It is *not* the average of the two season averages, and the difference between those two quantities is where the reversal lives. Rewrite the ratio of sums as a weighted average of the season averages: ``` combined = w * avg_1 + (1 - w) * avg_2, where w = AB_1 / (AB_1 + AB_2) ``` The weights are the at-bat counts. Two batters with different at-bat splits are having their season averages combined with **different weights**, and different weights can reverse an ordering that holds term by term. ## A concrete pair of seasons | | Season 1 | Season 2 | Combined | |---|---|---|---| | Batter A | 3 / 10 = .300 | 20 / 100 = .200 | 23 / 110 = .209 | | Batter B | 26 / 100 = .260 | 3 / 20 = .150 | 29 / 120 = .242 | A beats B by 40 points in season 1 and by 50 points in season 2, and loses the combined comparison by 33 points. Look at the weights: A's weight on season 1 is 10/110 = 0.09, B's is 100/120 = 0.83. Season 1 is the high-scoring season for both players, and B spends most of his combined average there while A spends almost none of his there. ## The two ingredients A reversal needs both of the following, and neither alone is enough. 1. **A spread in the stratum rates.** Both batters hit better in season 1 than season 2 — season is doing something to everybody's average, perhaps a change of park, of health, of role. 2. **A spread in the weights, in the opposite pattern.** A's at-bats pile up in the low-average season, B's in the high-average season. If both batters had identical at-bat splits — say each had 50 at-bats per season — the weights would match, and a term-by-term win would be a combined win with certainty. That is the cleanest way to see that the reversal comes from the *design of the exposure*, not from the batters. ## Why "average the averages" is not the fix A common instinct is to sidestep the problem by averaging the two season averages: A gets (.300 + .200)/2 = .250 and B gets (.260 + .150)/2 = .205, restoring A's win. That is a legitimate quantity, but it is not a batting average — it silently declares that the two seasons should count equally regardless of how much either player actually played. That is a *choice of reference weighting*, and it must be stated as one. If you want a single comparable number, the disciplined version is to fix a common set of weights for both players (equal seasons, or the league's at-bat split) and apply it to both — which is standardisation, not an average of averages. ## The general form Nothing here is about baseball. Replace "season" with any stratifying variable, "at-bats" with exposure or denominator, and "hits" with events, and you have the general statement: whenever two groups have different denominators across strata that themselves differ in base rate, the pooled ratio can reverse the stratified ratios. Hospital survival rates across case severities, click rates across placements, defect rates across production lines — all the same arithmetic. ## What an interviewer is checking Three things. First, that you know a pooled rate is a weighted average and can name the weights out loud. Second, that you do not reach for "one of these numbers must be wrong". Third — the part that separates a middle answer from a junior one — that you can say what would have prevented the reversal: equal weights across the compared groups, which in an experiment is exactly what balanced assignment buys you. ## Reporting it honestly If you must publish one number, publish the stratified table alongside it, and if a single figure is demanded, standardise: apply one common weighting to both players' season averages and label the weighting. A reader who sees only .209 versus .242 has been handed a true number and a false impression, and the fix is not a better number but a visible denominator.
- What would guarantee that no reversal is possible?Identical weights for both batters. If each has the same at-bat split across the two seasons, the two combined averages are weighted averages with the same weights, so winning both terms forces winning the combination. In experimental terms this is what balanced assignment buys: equal exposure across strata makes the pooled comparison agree with the stratified one.
- Is this the same phenomenon as Simpson's paradox with proportions?Yes — it is the same weighting arithmetic, dressed in a sports example rather than a treatment example. What differs is the stakes of interpretation. Here nobody claims a season causes hits; in a treatment comparison the same reversal decides which table has a causal reading, which is why the paradox gets its name in that setting.
- How would you report a single fair comparison across the two seasons?Fix one weighting and apply it to both batters — equal seasons, or the league-wide at-bat split — then report that standardised average and name the weighting you used. Show the season-by-season table next to it. A single unlabelled number invites exactly the misreading the paradox produces.
saying these in an interview costs you the question
- Says the two season averages should be averaged to get the total
- Claims the numbers contain an arithmetic mistake
- Argues the small-sample season should simply be discarded
- Adds rates together as if they were counts