Graphs, tables and justified comparisons

Establish what each value measures

A graph or table compresses information, but the labels determine what its numbers mean. Before calculating, identify the groups or time points, the measured quantity, its unit and any qualification in a caption. A column headed people is different from one headed hundreds of people. A percentage can describe a share of all respondents, a share within one subgroup or a change from an earlier value. Locate the denominator before treating two percentages as directly comparable.

In a table, use row and column labels together to select a cell. Totals can refer to different dimensions: a row total might sum categories within one centre, while a column total sums a category across centres. Trace the required intersection with a finger or write down its labels. This small step prevents answering with a nearby total that looks plausible. If categories overlap, their counts cannot automatically be added to produce a number of distinct people.

Read the visual scale before judging the size of a change

A bar chart commonly compares separate categories. A line chart often shows a sequence, such as measurements over time. The visual height or slope must be interpreted through the axis values. A truncated vertical axis can make a small numerical difference look large; uneven spacing on the horizontal axis can make equal-looking slopes correspond to different rates. Read the actual values and time intervals before describing growth as rapid or constant.

A pie chart shows shares of a whole, so its categories should correspond to a defined total. A quarter of the circle represents one quarter of that total, not necessarily one quarter of another chart's larger total. Two charts can have the same percentage slice and different category counts. When moving from share to count, multiply the fraction by the relevant whole. When moving from count to share, divide by that same whole. Keep the unit attached throughout the calculation.

Compare like quantities

A count answers how many; a rate compares that count with an exposure, population or time. A centre with more successful outcomes may also have many more participants. To compare success rates, divide successes by participants within each centre. To compare output per hour, divide each output by its own operating time. Do not use one centre's denominator for another centre's numerator. A larger count can accompany a smaller rate, so state precisely which comparison the question requests.

Percentage change uses the earlier or stated reference value as its denominator. If a quantity rises from 40 to 50, the absolute increase is 10 and the percentage increase is 10 divided by 40, or 25%. A fall from 50 to 40 has the same absolute magnitude but is a 20% decrease because the reference is now 50. When a percentage itself rises from 40% to 50%, the increase is ten percentage points; the relative increase is still 25%.

Combining groups requires their sizes. An overall rate is a weighted average of group rates, with weights proportional to the relevant denominators. Averaging two percentages equally is valid only when the group sizes are equal or the question explicitly requests an unweighted average of those percentages. A useful independent route is to add the underlying successes and divide by the combined participants. This often exposes an incorrectly averaged result immediately.

Restrict conclusions to the available evidence

A graph can show that two variables changed together without proving that one caused the other. Other factors might influence both, or the direction of influence might differ from the proposed explanation. A before-and-after comparison without an appropriate comparison group does not by itself isolate the effect of an intervention. This does not make the data useless; it limits the conclusion that follows from them.

Interpolation estimates a value between observed points using a stated assumption about the intervening pattern. Extrapolation extends beyond the observed range and needs additional caution. A line joining two points is not evidence that the underlying quantity followed an exactly straight path at every intermediate moment. In a short reasoning problem, distinguish an exact value read from the chart, a value calculated from supplied data and an estimate dependent on an assumption. Choose the claim with the support actually provided.