Coaching Money Across Four Blocks
A state recruitment board releases block-wise data on a constable recruitment test. Population is in lakh; appeared and qualified are candidate counts for the two cycles. 'Qualified' means a candidate who cleared both the written test and the physical measurement test in that cycle. Population figures are from the 2021 census and are treated as unchanged across the two cycles.
| Block | Population (lakh) | 2022 appeared | 2022 qualified | 2023 appeared | 2023 qualified |
|---|---|---|---|---|---|
| North | 12 | 8,400 | 1,008 | 9,600 | 1,344 |
| South | 9 | 6,300 | 630 | 7,200 | 936 |
| East | 15 | 7,500 | 450 | 8,250 | 660 |
| West | 10 | 5,000 | 500 | 6,000 | 720 |
The board has ₹40 lakh for pre-examination coaching and must place it in the two blocks with the weakest outcomes per head of population, using 2023 figures. A block officer from East argues that East should get the largest share because it has the most candidates appearing and the lowest success rate. The chair wants a short note that ranks the blocks, defends the ranking rule and answers the East officer.
1. Compute each block's success rate in both cycles and the change in percentage points. State which block improved fastest and which added the most qualified candidates in absolute terms.
Read the worked answer
Success rate = qualified ÷ appeared.
| Block | 2022 | 2023 | Change (pp) | Qualified added |
|---|---|---|---|---|
| North | 1,008/8,400 = 12.0% | 1,344/9,600 = 14.0% | +2.0 | +336 |
| South | 630/6,300 = 10.0% | 936/7,200 = 13.0% | +3.0 | +306 |
| East | 450/7,500 = 6.0% | 660/8,250 = 8.0% | +2.0 | +210 |
| West | 500/5,000 = 10.0% | 720/6,000 = 12.0% | +2.0 | +220 |
Fastest improver: South, +3.0 percentage points - it closed most of its gap with North and is now within one point of it.
Largest absolute addition: North, +336 qualified candidates.
The two answers differ, and that is the point of the question. North adds the most candidates simply because it is the biggest pool; South's gain is a change in the quality of preparation. Any ranking must state which of the two it is using, because a board that rewards absolute additions will always fund the largest block.
2. Compute qualified candidates per lakh of population for 2023 and rank the blocks. Does this ranking agree with the success-rate ranking? Explain the difference.
Read the worked answer
Qualified per lakh (2023):
- North: 1,344 ÷ 12 = 112
- South: 936 ÷ 9 = 104
- West: 720 ÷ 10 = 72
- East: 660 ÷ 15 = 44
Ranking by outcome per head: North > South > West > East. The success-rate ranking (14% > 13% > 12% > 8%) is in the same order, so the two measures agree on the order but differ sharply on the distance: East is 1.75 times behind North on rate but 2.5 times behind on candidates per lakh.
The gap widens because the per-head measure folds in participation as well as conversion. Appeared per lakh in 2023: North 800, South 800, West 600, East 550. East starts with a smaller share of its population even sitting the test. So East's deficit has two parts - fewer people attempt, and fewer of those who attempt succeed - and the per-head measure captures both, which is exactly why the chair chose it for placing coaching money. Coaching can move conversion quickly; participation usually needs outreach into schools and colleges.
3. Decompose East's shortfall against the state average into a participation effect and a conversion effect, and answer the East officer's argument for the largest share.
Read the worked answer
State totals (2023): appeared 31,050; qualified 3,660; overall success rate 3,660 ÷ 31,050 = 11.79%; participation 31,050 ÷ 46 lakh = 675 appeared per lakh.
Step 1 - participation effect. Lifting East to 675 per lakh: 15 × 675 = 10,125 appeared. At East's own 8% conversion that yields 810 qualified, against 660 today: +150 candidates from participation alone.
Step 2 - conversion effect. Holding that higher pool, lifting conversion to 11.79%: 10,125 × 0.1179 ≈ 1,194 qualified. That is a further +384 over the 810.
So of the roughly 534 candidates East is short of a state-average outcome, about 28% is participation and about 72% is conversion. Coaching targeted at candidates who already appear attacks the larger half first, which justifies East's claim to a large share - but the ranking rule still puts East (44) and West (72) as the two weakest per head, so East gets a large share, not 'the largest, because it has the most candidates'. Appearing candidates is a measure of size; per-head outcomes measure need. The officer's second point, the lowest success rate, is the stronger half of his case and should be conceded in the note.