The 20% Literacy Claim
A district education office issues a press note: "Literacy in our district has risen by 20% in five years." The underlying figures are these. A 2018 household survey found 45% of residents aged 15 and above literate; the population then was 1.20 million. A 2023 survey found 54%; the population then was 1.44 million. The two surveys did not use the same test. In 2018 a person counted as literate only if they could read a newspaper and write a short letter; in 2023 anyone who could read a short message in any language was counted. A reporter has three questions before running the story, and also wants to compare the district with a neighbouring district whose 2023 figure, measured on the older and stricter test, is 61%. The office has one day to respond and no budget for a new survey.
1. Is the "risen by 20%" claim defensible? State exactly what is and is not supported by the figures.
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The arithmetic is right and the inference is not.
The relative change is (54 − 45) / 45 = 0.20, so a rise of 20% is a correct description of the ratio between the two rates. What the note cannot support is the phrase "in five years", because the two rates do not measure the same thing. The 2023 test is looser: reading a short message is a lower bar than reading a newspaper and writing a letter. Some part of the nine-point movement is therefore a change in the ruler, not a change in the district.
The honest formulation is: "The share of adults counted as literate rose from 45% to 54%, nine percentage points, over five years; the survey method changed in 2023 and the two rates are not strictly comparable." Nine percentage points is a difference between two rates; 20% is a ratio. Both are true statements about the numbers, but only the first is safe when the definitions have moved. A press note that leads with 20% invites the reader to hear "one fifth more literate people", which is not what was measured.
2. Work out how many additional literate adults the district gained and separate, as far as the data allow, the population effect from the literacy effect.
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Number literate in 2018: 0.45 × 1,200,000 = 540,000. Number literate in 2023: 0.54 × 1,440,000 = 777,600. Gain in literate adults: about 237,600.
Now split the gain. Hold the rate fixed at 45% and apply it to the larger population: 0.45 × 1,440,000 = 648,000. That is what the district would have had in 2023 with no improvement in literacy at all, simply from having 240,000 more residents. The population effect is therefore 648,000 − 540,000 = 108,000 people, and the rate effect is 777,600 − 648,000 = 129,600 people.
So roughly 45% of the increase is demographic and 55% reflects the higher rate. This decomposition is arithmetically sound but should be flagged: because the 2023 definition is looser, part of the 129,600 is measurement artefact, so 129,600 is an upper bound on genuine improvement. The number to publish is "roughly 130,000 at most", not 237,600.
3. Can the district be compared with the neighbour's 61%? Explain what direction of conclusion, if any, survives the mismatch.
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The gap cannot be measured, but the direction of the comparison does survive, and the reason is the direction in which the definitions differ.
The neighbouring district used the stricter test and still reached 61%. Apply the looser 2023 test to that population and its figure can only rise, since more people pass an easier test. The neighbour's true rate under the district's own method is therefore at least 61%, not less.
Our district's 54% was obtained under the loose test. Under the strict test it would fall. Comparing 54% (loose) with 61% (strict) therefore understates the neighbour's advantage; on a like-for-like basis the gap is wider than nine points, not narrower.
What follows is a defensible sentence: "Even allowing for a more generous definition in our own survey, the district trails its neighbour." What does not follow is any specific figure for the gap, any ranking of improvement over time, or any claim that the district is closing the distance. The reporter should be told the comparison can be used for direction only, and the office should commit to one definition for the next round.