Pooled ranking reversal: compare the same mix

A Pooled ranking reversal can occur when two alternatives have different mixtures of groups. In the fictional example below, route B has a higher completion fraction within each group, while route A has the higher fraction across its own pooled records. The counts support both statements. The next question is which population the comparison is meant to describe.

Every record and outcome here is invented. Easy and hard are simply the two group names stipulated for this exercise, without a measured difficulty scale or claim about actual work. Route labels name two columns of fictional observations. They do not identify a product, employer, treatment, service or tested operating method.

Declare the four count pairs

Route A has 90 records in the easy group, of which 81 are marked complete. It has ten records in the hard group, of which one is complete. Route B has 20 easy records with 19 complete, and 80 hard records with 16 complete. Those four pairs are the complete starting data.

Within each route, every record belongs to exactly one of the two groups. Every record is marked either complete or incomplete, and none has a missing outcome. The group counts therefore add without overlap. These supplied boundaries make the following arithmetic possible; they are conditions of the example rather than discoveries from a percentage.

First check the complementary counts. A has nine incomplete easy records and nine incomplete hard records. B has one incomplete easy record and 64 incomplete hard records. Each completed count plus its incomplete count recovers the corresponding group total. Keep those totals beside the fractions so the population behind each result stays visible.

Compare within each named group

Among A's easy records, the completion fraction is 81 divided by 90, or 90 percent. Among B's easy records it is 19 divided by 20, or 95 percent. B's fraction is higher by five percentage points in this group. That comparison retains the same group name on both sides.

Among A's hard records, one out of ten is complete, or ten percent. Among B's hard records, 16 out of 80 are complete, or 20 percent. B's fraction is higher by ten percentage points here. Thus B has the larger fraction in both separately named groups of the stipulated data.

These comparisons provide no causal explanation for the outcomes. The exercise provides counts and labels without an assignment procedure, independent observations about circumstances or evidence explaining why records took a route. Describing the fractions accurately leaves those questions unresolved. A larger arithmetic result cannot supply the missing causal evidence.

Pool each route's own records

A contains 100 records altogether: 90 easy plus ten hard. Its completed records total 82: 81 easy plus one hard. The pooled completion fraction for A's own population is therefore 82 divided by 100, or 82 percent. No outcome changes when the two groups are added.

B also contains 100 records altogether: 20 easy plus 80 hard. Its completed records total 35: 19 easy plus 16 hard. B's pooled completion fraction is 35 divided by 100, or 35 percent. Comparing these two pooled populations places A above B, even though B's fraction is higher within each group.

There is no disagreement between the additions and the earlier divisions. A's pooled population contains mostly easy records, whose completion fractions are higher for both routes. B's population contains mostly hard records. The overall figures incorporate those different mixtures as well as the fractions within each group.

Do not repair the reversal by moving records, replacing outcomes or choosing whichever denominator produces the preferred result. All four original count pairs remain fixed. Instead, label the comparison: 82 versus 35 percent describes each route's own observed mixture in this fictional data. It does not hold group composition constant.

Write the mixtures explicitly

A's easy group supplies 90 of its 100 records, so its easy share is 0.9. Its hard share is 0.1. Weighting the two completion fractions gives 0.9 times 90 percent plus 0.1 times ten percent. The contributions are 81 and one percentage points, giving the same 82 percent.

B's easy share is 0.2 and its hard share is 0.8. Its corresponding calculation is 0.2 times 95 percent plus 0.8 times 20 percent. Those contributions are 19 and 16 percentage points, giving 35 percent. Each route uses its own group shares because each pooled result describes its own supplied records.

The weights in these expressions sum to one separately for A and B. They record each group's share of that route's records. Writing them out reveals why a pooled comparison can answer a different question from the comparisons within the two groups.

Ask a separate same-mix question

Now stipulate a different arithmetic question: what results follow if each route's given subgroup fractions receive half the weight from easy and half from hard? This explicitly requested mixture is a hypothetical comparison. It does not replace the original counts or establish that an actual future population will have equal group shares.

For A, half of 90 percent plus half of ten percent gives 45 plus five, or 50 percent. For B, half of 95 percent plus half of 20 percent gives 47.5 plus ten, or 57.5 percent. At this common mixture, B has the higher calculated fraction by 7.5 percentage points.

These are weighted calculations from the stipulated subgroup fractions, not newly observed records. The original A population still has 82 completed records and the original B population still has 35. A note presenting the equal-mix results should identify the half-and-half assumption instead of calling 50 and 57.5 the original pooled percentages.

Equal weighting is chosen because this question requests it. The example does not establish that half-and-half is the right mixture for another comparison. A real request would need to state the population of interest and justify the weights it uses. Arithmetic alone cannot choose that population on the reader's behalf.

Check another shared mixture

Try a shared mix of one quarter easy and three quarters hard. A then gives one quarter of 90 percent plus three quarters of ten percent: 22.5 plus 7.5, or 30 percent. B gives one quarter of 95 plus three quarters of 20: 23.75 plus 15, or 38.75 percent.

The values change because the common question changed, while B remains higher. You can check every shared mix at once. Let w be the easy share, between zero and one, and let one minus w be the hard share. Retain exactly the same w for both routes before comparing their results.

A's percentage becomes 90w plus ten times one minus w, which simplifies to ten plus 80w. B's becomes 95w plus 20 times one minus w, or 20 plus 75w. Subtracting A's expression from B's gives ten minus five w, measured in percentage points.

For w between zero and one, that difference lies between five and ten percentage points and stays positive. Thus B's calculated fraction exceeds A's for every shared mixture allowed by this particular two-group example. This conclusion follows from the specified fractions and shared weights, without a search over calculator outputs.

The original pooled comparison is outside that shared-weight calculation because it uses different easy shares: 0.9 for A and 0.2 for B. Substituting a different w into each expression restores 82 and 35. Check whether the weights are shared before interpreting that comparison.

Read the actual calculator offers separately

The NumWorks Graphing Calculator offer names a graphing calculator. Its sole image shows one complete white device with NumWorks labeling, screen, keys and directional controls. App icons are pictured context, not a verified execution of this exercise. No operated device, exam eligibility or measured runtime is established here.

The Texas Instruments TI-30XIIS five-pack offer retains five-pack wording. Its sole image shows five complete black and silver calculators with TI-30XIIS model labels. Displayed expressions are illustrations, not observed results for these count pairs. The picture does not independently verify a delivered quantity, maker authenticity or tested performance.

Neither selected account supplies the fictional records or a measured route comparison. This worksheet requires no purchase. The gallery observations describe the selected accounts; they do not turn the example into a test of either calculator or establish the complete contents of a delivered parcel.

Keep the comparison's boundary visible

The Fibonacci parity-pattern guide follows consecutive parity states to explain a recurring position. Here the records are fixed count pairs. No recurrence generates their outcomes, and the pooled reversal is not a claim about a sequence repeating.

The last digit power-sum guide checks a position in a repeating last-digit cycle. A route's group share is instead a weight in a specified population. Transferring a cycle position into this worksheet would supply no missing record count or comparison rule.

Before handing over the worksheet, record the four original count pairs, the question being answered and the mixture attached to each result. For the observed pooled populations, report A at 82 percent and B at 35 percent with their unequal group shares. For the stipulated equal mix, report A at 50 percent and B at 57.5 percent.

Keep an actual purchase outside the worksheet

If you separately consider a calculator order on republeq.com and want to ask about USDT, begin with the exact offer and intended quantity. The fictional records, route names and completion percentages in this worksheet are not prices, cryptocurrency amounts or payment instructions. Correct arithmetic establishes no checkout method or actual order status. Read the payment options and conditions presented for the intended transaction, and ask the responsible support contact to clarify essential missing information before deciding whether to proceed. If a cryptocurrency option is offered, establish the specified asset and network, requested amount, any stated validity period and the reference associating payment with the order. Do not infer a transfer destination from a product image, worksheet heading or calculator display. Resolve material product questions separately, including actual supplied contents and any intended use you need the item to support. Ask about applicable delivery, cancellation, return and refund conditions, including the denomination of any approved refund and relevant deductions. Keep each answer with the offer it concerns. Information about one account does not automatically describe another quantity or purchase, and a changed order may require reviewing the resulting terms again. This article promises no present USDT availability, fee level, conversion result, confirmation time, refund amount or delivery outcome. You can inspect every fictional count pair without purchasing anything. Preserve any eventual order acknowledgement and transaction record separately from the worksheet. Those records concern an actual transaction under its documented conditions; the calculations compare expressly stated populations. A preferred payment asset or a plausible percentage cannot fill a gap in actual order information.

Retain the population beside the ranking

The Pooled ranking reversal comes from comparing different mixtures in the original pooled populations. B's fraction is higher within each named group and under every shared mixture in this defined example, while A's own pooled fraction is higher. Preserve both the question and its weights when reporting either ordering.