The immunisation chart where the overall figure rises while both subgroups fall
A block health office publishes a two-page 'Immunisation at a glance' note. Page one is a bar chart titled 'Coverage is rising': five bars for 2019-2023 with the vertical axis starting at 88% and ending at 96%, so the 2022 bar looks about six times taller than the 2019 bar. Page two gives the underlying table. In 2022, 2,000 children were recorded in urban sub-centres with 95% coverage and 8,000 in rural sub-centres with 90% coverage, an overall figure of 91.0%. In 2023, 5,000 children were in urban sub-centres with 94% coverage and 5,000 in rural sub-centres with 89% coverage, giving 91.5%. The district officer reads the chart, tells the block to 'keep doing what you are doing', and forwards the note to the state review meeting. A community health worker points out that her rural sub-centre has fewer vaccinated children than last year and that her register was used to build the table.
1. Reconcile the figures: the overall number rose, yet coverage fell in both urban and rural sub-centres. Show the arithmetic and name the effect.
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The arithmetic. 2022: urban 0.95 x 2,000 = 1,900; rural 0.90 x 8,000 = 7,200; total 9,100 of 10,000 = 91.0%. 2023: urban 0.94 x 5,000 = 4,700; rural 0.89 x 5,000 = 4,450; total 9,150 of 10,000 = 91.5%.
So the overall rate genuinely went up by 0.5 percentage points, and it is equally true that urban fell 95% to 94% and rural fell 90% to 89%. Both statements are correct because the overall rate is a weighted average, and the weights changed.
Why. Urban sub-centres have the higher coverage, and the share of children recorded there rose from 20% (2,000 of 10,000) to 50% (5,000 of 10,000). Three thousand children moved into the higher-coverage stratum. That compositional shift is large enough to outweigh a one-point fall inside each stratum. This is Simpson's paradox, or more plainly a composition effect: the average improved because the mix improved, not because service delivery improved. On the ground, 100 children in a rural sub-centre now have 89 vaccinated instead of 90.
An extra check before believing the shift. A move from 2,000 to 5,000 urban children in one year is implausible as a real population change. More likely some rural sub-centres were reclassified as urban, or were reported under a different headcount. If so, the 2023 comparison is not like-for-like at all, and the 91.5% figure is partly an artefact of how registers were grouped.
2. Critique the chart and specify a better visualisation, including the axes, the marks and the accompanying numbers.
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What the chart does wrong. A truncated vertical axis starting at 88% is not automatically dishonest, but here the title ('Coverage is rising') plus bars cut at the baseline turns a plausible one-point movement into a six-fold visual difference. Bar length is being used to encode a ratio that the data does not contain: with a zero baseline, 91% and 91.5% are nearly the same height. The chart also shows only the overall series, which is exactly the series that hides the story, and it shows no denominators. A reader cannot tell whether 91.5% is 915 children out of 1,000 or 9,150 out of 10,000, and cannot see that the composition changed.
Better presentation.
- A line chart of urban and rural coverage over 2019-2023, two lines, y-axis either from 0 or clearly truncated with a labelled break and a note saying so. Two lines immediately show both falling.
- A second panel, same x-axis, showing number of children recorded in each stratum (stacked bars). This makes the 2,000 to 5,000 shift visible instead of invisible.
- A small table underneath: stratum, children, vaccinated, coverage, and change in percentage points for each year.
- Because the numbers come from registers, add the source and the date of extraction.
Rule to adopt. For any coverage claim, publish numerator, denominator and stratum alongside the percentage, and never let a title assert a direction that the stratified data contradicts. If a truncated axis is used for legibility, label it as truncated in the caption.
3. What should the block officer report to the state review meeting, and what follow-up should be triggered? Draft the substance of that report.
Read the worked answer
Report the stratified result, not the headline. State: overall recorded coverage moved from 91.0% to 91.5% between 2022 and 2023, but this is explained by a change in the mix of children recorded, from 20% urban to 50% urban. Coverage fell by one percentage point in both urban (95% to 94%) and rural (90% to 89%) sub-centres. Absolute numbers of vaccinated children: urban 1,900 to 4,700; rural 7,200 to 4,450. The rural absolute fall of 2,750 children is the operationally important figure and matches what the community health worker observed in her register.
Withdraw the instruction. 'Keep doing what you are doing' is not supported. The correct instruction is to investigate the rural fall.
Follow-up actions.
- Verify the 3,000-child reclassification: list which sub-centres moved between strata and whether boundaries or reporting practice changed. If this is a reporting artefact, restate both years on a consistent basis before any comparison is sent upward.
- For each rural sub-centre, compute coverage and absolute numbers against the 2022 figure and identify those with a real decline, not just a percentage dip caused by a smaller denominator.
- Check supply-side causes: session cancellations, vaccine stock-outs, cold-chain failures, worker vacancies, and whether the register itself was incomplete.
- Reissue the two-page note with the corrected chart and a one-paragraph correction to the state review, because the earlier version has already been forwarded.
- Add a standing check to the block's monthly template: no coverage percentage is published without its denominator and its urban/rural split.