Reading data: association, causation and changing denominators

By UK GCSE Sociology · independent study · myAudience

Researchers with blank coding sheets observe speaking, writing and listening; an interview inset invites comparison between consistency and meaning. No measure of understanding is depicted.

Original OpenAI imagegen conceptual teaching illustration, generated 4 October 2026; not observational evidence. · AI-generated original asset for this independent guide; no third-party artwork intentionally reproduced.

A percentage expresses a count relative to a denominator. In a fictional homework study, 60 of 100 respondents with reported quiet space complete all tasks, compared with 40 of 100 without it. The completion proportions are 60% and 40%, a difference of 20 percentage points. The relative increase from 40% to 60% is 50%, because the increase of 20 is divided by the original 40. Percentage points and percentage change answer different questions. Always state the outcome, group, denominator and period. The counts here are invented teaching data, not UK attainment estimates or findings from a classic study.

An association means variables vary together in the observed data. Correlation usually refers to a statistical relationship between variables; in ordinary discussion it is also used for association. These fictional group comparisons do not supply a correlation coefficient. Causation means changing one factor produces a change in another through a mechanism. To support a causal claim, consider temporal order, alternative explanations and how cases were selected and measured. A common cause can influence both variables. Prior attainment or available time might relate to study conditions and completion. Reverse causation is another possibility in some questions, such as successful learners obtaining more opportunities.

Disaggregating data can change the apparent pattern. A second constructed dataset has two prior-attainment strata. Higher prior attainment: quiet space, 72 completions among 90 learners; no quiet space, 9 among 10. Lower prior attainment: quiet space, 1 completion among 10; no quiet space, 18 among 90. Within the higher group the rates are 80% and 90%; within the lower group they are 10% and 20%. Yet combining the strata gives 73/100 = 73% for quiet space and 27/100 = 27% for no quiet space. The aggregate association reverses the direction of both within-stratum comparisons because the group compositions differ.

This reversal, often called Simpson's paradox, illustrates why a pooled comparison can mislead. It does not prove that quiet space harms completion: the data are observational and deliberately constructed. Comparing within a relevant stratum addresses one observed compositional difference, but other unmeasured differences may remain. Nor should analysts mechanically control every variable: a variable may lie within the very causal pathway being investigated. At this level, justify why the proposed comparison is relevant and keep conclusions conditional. Counting accurately is necessary, but interpretation requires an account of the process that generated the observations.

Patterns over time also require care. Counts can rise because the population grows, a definition changes or recording improves. A rate uses an appropriate population denominator, but the choice of denominator still matters. A graph should show units, period, categories and axis scale; a truncated axis can exaggerate a small difference. A chart can be arithmetically correct while excluding inconvenient years or groups. Tables with missing data should show the missing category rather than silently changing the base. A small apparent change does not establish statistical significance without information about sampling and uncertainty. We do not calculate or claim significance for these teaching datasets.

Data criticism should lead to a more precise claim, not dismissal of all numbers. The first study permits a statement about completion proportions among its respondents. The second shows how group composition affects the pooled comparison. Neither establishes a national effect or individual destiny. Compare quantitative patterns with appropriately selected qualitative accounts to examine meaning and possible mechanisms, while keeping each evidence type's limits visible. The conceptual images contain no calibrated numerical information: use the written counts for arithmetic. This lesson supplies numerical reasoning for later work on divorce, achievement and crime statistics without inventing real trends.

Worked example: Original constructed dataset: higher prior attainment, quiet space 72/90 complete and no quiet space 9/10 complete; lower prior attainment, quiet space 1/10 and no quiet space 18/90 complete. A headline says, 'Quiet space raises completion by 46 percentage points.' Calculate the aggregate and stratified comparisons, then evaluate the headline.

Sum completions and denominators separately: quiet space has 72 + 1 = 73 completions from 90 + 10 = 100; no quiet space has 9 + 18 = 27 from 10 + 90 = 100.

The aggregate difference is 73% − 27% = 46 percentage points. Within higher prior attainment, quiet space is 80% versus 90%; within lower prior attainment, 10% versus 20%.

Each within-stratum difference is −10 percentage points for quiet space. Higher prior attainment is much more common in the quiet-space group, generating the pooled reversal.

Replace causal raises with an association among these constructed cases. Explain that stratification challenges the headline but does not identify a causal effect in either direction.

Answer: The headline's numerical difference is correct for the pooled fictional data, but 'raises' is an unjustified causal verb. The quiet-space group contains many more higher-prior-attainment learners. Within both prior-attainment strata its completion proportion is ten percentage points lower, so the pooled positive association is not stable across relevant comparisons. That reversal demonstrates a compositional problem, not evidence that quiet space causes worse outcomes. A sound report gives denominators and both levels of comparison, discusses selection and measurement, and reserves causal judgment pending an appropriate design and mechanism evidence.

Check your understanding: Transfer task: in a fictional town, 30 of 50 survey respondents in Year A and 48 of 80 in Year B say the library hours suit them. A report claims satisfaction rose by 60% because positive responses rose from 30 to 48. Evaluate the arithmetic and interpretation, then identify two design questions. Indicative rubric: 12 points: counts/change 3, proportions 3, inference critique 3, design questions 3.

Answer: Positive-response counts rose by 18, or 60% relative to 30. But the respondent total also rose by 60%; satisfaction is 30/50 = 60% and 48/80 = 60%, so the observed proportion did not change. Ask whether the same population, sampling and wording were used and whether response patterns changed. These repeated samples do not show individual changes in satisfaction.

Why: Reward recognition that the count claim is mathematically true but the satisfaction claim changes the meaning of the denominator. Two useful design questions should concern comparability, not simply demand more data. A statement that 'nothing changed in the town' goes beyond the evidence: the samples may conceal changes, and uncertainty has not been assessed.