0580 · 9.2Interpreting statistical data

Reading data, averages and spread

0:00Two bus services can have similar average journey times, yet give passengers very different experiences. One might arrive predictably, while the other sometimes takes much longer. An average alone cannot show that difference. In this video, we will interpret tables and graphs, compare typical values and spread, and recognise when a conclusion goes beyond the evidence. Statistics helps us turn collections of observations into useful information. Reading data means finding what is shown directly.

0:28Interpreting means explaining what those values mean in their context. Drawing an inference means using the evidence to reach a conclusion that is not simply written in the table. Before comparing anything, identify what was measured, who or what was included, and the units used. Also check whether the figures represent counts, percentages or measurements. A frequency is the number of observations in a category. These details determine which comparisons are meaningful, even when the arithmetic looks straightforward.

0:56We also need a brief reminder about averages and spread. The mean shares the total equally across all observations, while the median is the middle value in ordered data. The mode is the most frequent value. These measures describe a typical value, but they do not describe how much the observations vary. The range is the largest value minus the smallest value. The interquartile range measures the spread of the middle half of the data.

1:20A smaller spread means the measured values are more closely grouped, which often indicates greater consistency. The video on averages and spread shows how to calculate them.

Comparing proportions fairly

1:30Our first exam-style example uses practice survey data about revision preferences. The question asks which group has the greater proportion of students preferring video. Group P contains twenty students, while Group Q has forty. Because the group sizes differ, comparing only the numbers choosing video could give a misleading impression. Read the table headings before choosing your figures. Twelve students in Group P prefer video, compared with eighteen in Group Q.

1:56Group Q therefore has the greater number choosing video. However, that does not yet answer a question about the proportion within each group. For Group P, 12 ÷ 20 gives zero point six. Multiplying by one hundred converts this to sixty per cent. Group Q's proportion is 18 ÷ 40, which is zero point four five. Converting that proportion gives forty-five per cent. Subtract forty-five from sixty to obtain a difference of fifteen percentage points.

2:24Group P has the greater proportion preferring video, despite having fewer students choosing it. Percentages make the comparison fair by expressing each result out of one hundred. The denominator must be the relevant group total, not the total number surveyed. An appropriate conclusion is that video was more popular proportionally in Group P among the students surveyed. We cannot conclude that video produces better examination results. The table records preferences, not achievement.

2:52We also cannot automatically extend these results to every student, because we do not know how the participants were selected.

Interpreting bar chart frequencies

2:58Our second example shows waiting times for two routes, with ten observations for each. Each bar represents the frequency of one waiting time. The horizontal axis shows minutes, while the vertical axis shows how often that waiting time occurred. A tall bar therefore means many observations, not a longer wait. The question asks which route had the greater proportion of waits shorter than six minutes. Shorter than six includes the bars at two and four minutes.

3:25The bar at six minutes is excluded, because those waits equal the limit rather than falling below it. Careful reading of the boundary matters just as much as accurate calculation. Adding Route P’s frequencies gives ten observations. The four frequencies for Route Q also total ten. For Route P, adding two and six gives eight waits shorter than six minutes. 8 ÷ 10, multiplied by one hundred, gives eighty per cent.

3:50For Route Q, the two-minute bar gives five qualifying waits, and the four-minute bar adds none. 5 ÷ 10, multiplied by one hundred, gives fifty per cent. Route P therefore had the greater proportion of waits shorter than six minutes. This conclusion answers the stated question using the bars and their totals. With only ten waits per route, though, a few unusual waits could change these percentages noticeably. The graph also shows a difference in consistency.

4:17Route P’s observations are concentrated around four minutes, while Route Q’s are split between two and eight minutes. Route Q sometimes gives a shorter wait, but also has several longer waits. Saying one route is simply better would hide the criterion being used. Whenever you compare graphs, check that their scales and units match. A bar can look much taller simply because its graph uses a different vertical scale. An axis beginning above zero can exaggerate a difference visually.

4:44Read the labelled values rather than judging only the appearance.

Comparing averages, spread and limitations

4:48Our third example asks us to compare two services using their journey-time summaries. Both were observed over twenty journeys, and every time measure is in minutes. Service P has the lower mean and median. Service Q has the smaller range and interquartile range, so the measures point to different advantages. Subtract eighteen from twenty to find that Service P’s mean is two minutes lower. Comparing the medians gives 20 − 17, which is three minutes.

5:15Service P therefore has shorter journey times on average in these records. Both measures of typical value support that comparison. Service Q’s range is four minutes, compared with twelve for Service P. Its interquartile range is also smaller, at two minutes compared with four. Service Q therefore has more consistent journey times in these records. The interquartile range specifically supports this conclusion for the middle half of the observations. A strong comparison states both the typical value and the spread, with their meanings in context.

5:46Service P is quicker on average, while Service Q is more consistent. Neither statement means that every journey follows that pattern. A passenger who needs a predictable arrival time might choose Service Q, while one who wants the quickest typical journey might choose Service P. The range uses only the two extreme values, so one unusually long journey can make it much larger. The interquartile range describes the middle half instead.

6:10These measures can therefore reveal different aspects of variation. A smaller spread is not automatically better either, because the preferred pattern depends on what is being measured. Restrictions on conclusions are part of interpreting data, not an optional warning afterwards. A sample is the group actually observed, while the population is the whole group we want to understand. A small or unrepresentative sample may not reflect that population well. Even a large survey can be misleading if it systematically excludes relevant people.

6:39When two quantities increase together, that link alone does not prove that one causes the other. For these journey records, we would want to know whether both services were observed under comparable conditions. Peak-time journeys might take longer than journeys on quiet roads. Data from one particular week may not describe performance throughout the year. Comparing summaries cannot prove that the service itself caused every difference. In an examination, commands such as read, interpret and compare require different uses of the information.

7:08A full answer here says that Service P has the lower median, so its journeys are generally quicker. It then says that Service Q has the smaller interquartile range, so its journey times are more consistent. Stating only which number is bigger, without its meaning in context, loses marks. If asked about restrictions, give a specific reason the records might not represent other journeys. These interpretations and simple percentage calculations must work without a calculator on Paper two.

7:36Paper four allows a calculator, but the reasoning remains essential. Keep intermediate values unrounded when a calculation is longer. For calculated answers that require rounding, use three significant figures unless instructed otherwise. The exact results here need no rounding. The central idea is to match every conclusion to the evidence supporting it. Read headings, units, scales and totals before comparing values. Use averages to discuss typical outcomes and measures of spread to discuss variation.

8:06Finally, recognise the limits of the data, so your conclusion says what is supported rather than what merely sounds convincing.