HookThe nurse who won an argument with a diagram
In 1858 Florence Nightingale, back from the Crimean War, needed to convince a sceptical Parliament and army of one uncomfortable fact: far more British soldiers were dying of preventable disease in the camps than of their battle wounds. Rows of figures had not moved anyone. So she drew them. Her famous polar-area diagram — a circle sliced into months, with the area of each wedge showing the death toll and the vast blue regions marking disease deaths dwarfing the red wound deaths — made the pattern impossible to ignore. That same year she became the first woman elected a Fellow of the Royal Statistical Society. She had discovered that the right picture of the right numbers can win an argument that raw data cannot.
Statistics is that craft: collecting numbers about a group, summarising them honestly, showing them clearly, and drawing conclusions you can defend. This section covers the three averages and the range, how to read those out of frequency tables, which chart suits which kind of data, why we sample a population instead of measuring all of it, and how scatter graphs reveal — or falsely suggest — a relationship between two things. The arithmetic is gentle; the marks are for judgement. AQA is less interested in whether you can compute a mean than in whether you can say what it means, compare two groups fairly, and refuse to be fooled by a coincidence dressed up as a cause.
ModelThe three averages — and when each one lies
An 'average' is a single number standing in for a whole set, and there are three, each measuring the middle differently. The mean is the total divided by how many values there are — the 'fair share'. The median is the middle value once the data is put in order. The mode is the most common value, the only average that works for non-numerical data like favourite colours.
They usually disagree, and the disagreement is the point. The mean uses every value, which makes it powerful but also vulnerable to outliers — one freak value drags it. The median ignores the sizes of the values and only cares about position, so a single billionaire walking into a room barely nudges the median income but sends the mean soaring. That is exactly why 'average UK salary' figures are usually quoted as a median: the Office for National Statistics reports median pay precisely because a handful of very high earners would make the mean misleadingly high.
Choosing the right average is a real exam skill. Use the median for skewed data like incomes or house prices; use the mode for the 'most popular' shoe size a shop should stock; use the mean when the data is fairly even and you want every value to count.
Find the mean, median and mode of 4, 7, 7, 9, 13. Mean: total 4 + 7 + 7 + 9 + 13 = 40, divided by 5 values = 8. Median: the data is already in order, and the middle of five is the third value, 7. Mode: 7 appears twice, more than anything else, so the mode is 7. Now watch an outlier bite — change the 13 to 43: the mean jumps to (4+7+7+9+43)/5 = 70/5 = 14, but the median stays 7. One value moved the mean by 6 and the median not at all.
MechanismRange and spread — and how to compare two groups
An average tells you the middle; it says nothing about how spread out the data is. The simplest measure of spread is the range: the largest value minus the smallest. A small range means the data is tightly bunched and consistent; a large range means it is scattered and unpredictable. Two classes can have identical mean test scores while one is all near the middle and the other is a mix of top marks and fails — the range is what separates them.
An outlier, a value far from the rest, inflates the range dramatically, so it is worth noting whether one is present. (At Foundation the range is the only spread measure you need; quartiles, the interquartile range and box plots are Higher-tier.)
The reason this matters is that AQA almost never asks you to compare two groups on an average alone. A good comparison quotes two things — one average and one spread — each interpreted in the context of the question. 'The girls' mean was higher, so on average they scored better, and their range was smaller, so they were also more consistent' is a full-mark comparison; 'the mean is higher' on its own is a partial one.
Class A scored 5, 6, 7, 7, 10 and Class B scored 2, 4, 7, 9, 13. Both have a mean of 7 (totals 35 ÷ 5). But Class A's range is 10 − 5 = 5, while Class B's is 13 − 2 = 11. Identical averages, very different stories: Class A is consistent, Class B is erratic. Writing both the average and the spread, each in context, is what turns two marks into four.
DataAverages from frequency tables — Σfx over Σf
Real data usually arrives already counted up in a frequency table, and you must find averages without listing every value out. The mode is the value with the highest frequency — read straight off the table. The median is found by position: for n values the middle is at position (n + 1) ÷ 2, and you count through the frequencies until you reach it. The mean needs a small trick: multiply each value by its frequency, add those up (that is Σfx, the grand total of all the data), then divide by the total frequency (Σf, how many values there are).
The most common error is dividing by the number of rows in the table instead of by the total frequency — but the mean must divide by how many pieces of data there actually are, not how many different values appeared.
When the data is grouped into class intervals (like heights from 150–160 cm), you no longer know the exact values, so you use the midpoint of each class as a best estimate, and the result is an estimated mean. You must call it an estimate, because the midpoints are assumptions. For grouped data you also give the modal class (the interval with the highest frequency) rather than a single mode.
A table records goals per match: 0 goals in 5 matches, 1 in 8, 2 in 4, 3 in 3. Find the mean. Σfx = (0×5) + (1×8) + (2×4) + (3×3) = 0 + 8 + 8 + 9 = 25 goals in total. Σf = 5 + 8 + 4 + 3 = 20 matches. Mean = 25 ÷ 20 = 1.25 goals per match. The mode is 1 goal (its frequency of 8 is the highest), and dividing by 4 rows instead of 20 matches would have given a nonsensical 6.25 — always divide by the total frequency.
ModelCharts — matching the picture to the data
Different data wants different pictures, and choosing wrongly is a real exam error. A bar chart compares separate categories, with gaps between the bars and the height showing frequency. A pictogram does the same with a symbol standing for a number of items, so you must read the key and handle part-symbols. A vertical line chart is a bar chart with thin lines, good for numerical data like shoe sizes. A pie chart shows how a whole splits into parts, with each slice's angle proportional to its share. A line graph joins points over time and is the right choice for time-series data — temperature through a day, sales across a year — because the line shows the trend between readings.
The rule of thumb: bar charts and pie charts for categories, line graphs for change over time. Nightingale's genius in the introduction was picking a form — area proportional to deaths — that made her comparison land; the wrong chart would have buried it.
Pie charts carry their own arithmetic. Because a full circle is 360°, each slice's angle is its frequency as a fraction of the total, times 360°. Reading one backwards, a slice's angle over 360 tells you its share of the whole.
In a survey of 40 people, 10 chose tea. What angle represents tea on a pie chart? The fraction choosing tea is 10 ÷ 40 = ¼, and a quarter of the full 360° is (10 ÷ 40) × 360 = 90°. If 15 chose coffee, that is (15 ÷ 40) × 360 = 135°. As a check, all the slice angles must add to 360° — if they do not, a frequency or a multiplication has gone wrong somewhere.
CaseSampling — why we don't measure everyone
A population is the whole group you want to know about — every voter, every light bulb from a factory, every tree in a forest. Measuring all of it is usually impossible, ruinously expensive, or destructive (you cannot test every bulb to destruction and still sell them). So we take a sample, a smaller part, measure that, and infer a conclusion about the whole population.
The catch is that a sample is only trustworthy if it is representative — a fair miniature of the population. A biased sample, one that systematically over- or under-represents part of the group, gives a wrong answer no matter how carefully you do the sums: surveying only your friends about a national election, or testing water quality only near the outflow pipe, builds the error in from the start. This is the key limitation AQA wants you to name: a sample can mislead, and a larger, more representative sample reduces the risk but never guarantees the truth.
So the honest statistician always states two things: what the sample suggests about the population, and how confident we can be given how the sample was chosen. Bigger and fairer beats bigger alone.
MechanismScatter graphs — correlation is not cause
A scatter graph plots two measurements for each individual — height against weight, revision hours against exam mark — as points on a grid, to see whether they are related. If the points trend upward together it is positive correlation; if one rises as the other falls, negative correlation; if the points are a shapeless cloud, no correlation. The strength shows in how tightly the points hug a line.
When there is correlation you can draw a line of best fit: a single straight line following the trend, with roughly as many points above it as below, ignoring any clear outlier. That line lets you predict: read across and up to estimate one value from the other. Predicting within the range of your data (interpolation) is fairly safe; predicting beyond it (extrapolation) is unreliable, because you are assuming a pattern continues where you have no evidence.
The deepest idea, and the one AQA tests explicitly, is that correlation does not prove causation. Ice-cream sales and drowning deaths rise together every summer, but ice cream does not drown anyone — hot weather drives both. It was exactly by refusing the obvious-but-wrong cause (bad air) that John Snow traced Soho's 1854 cholera outbreak to a single water pump on Broad Street. When a question asks you to comment on a relationship, name the correlation, but never claim one thing causes the other without evidence.
A scatter graph of revision hours (x) against test score (y) shows strong positive correlation. A line of best fit passes through (2, 40) and (8, 82). Estimate the score for a student who revised 5 hours. Reading up from x = 5 to the line and across, the line sits at about 61 marks (halfway along, halfway up between those points). That is interpolation — 5 is inside the data range — so it is a reasonable estimate. Predicting the score for 20 hours of revision would be extrapolation: off the end of the data, where the trend may not hold, so any such prediction must be flagged as unreliable.
VocabularyKey terms the mark scheme pays for
TrapsMisconceptions that cost marks
ExamWhat examiners want
Statistics spans all three papers, and the calculator papers carry the arithmetic while the marks reward interpretation. When you find an average, be ready to say which one and why it suits the data; when you compare two groups, quote two measures — one average and one spread (the range) — and interpret each in the context of the question, because a bare 'the mean is higher' scores well below a comparison that also mentions consistency. Naming the measures without a sentence of meaning is the commonest way strong students underscore.
For a mean from a frequency table, compute Σfx and divide by Σf (the total frequency), not by the number of rows — the single biggest table error. For grouped data, use class midpoints and call the answer an estimated mean, and give the modal class rather than a mode. On pie charts, every slice angle is (frequency ÷ total) × 360°, and the angles must sum to 360° — a free check. Read chart scales carefully; a misread axis loses easy marks.
On scatter graphs, draw a single ruled line of best fit through the trend with roughly equal points either side and ignore outliers, use it for predictions, and label whether you are interpolating (safe, inside the data) or extrapolating (unreliable, beyond it). Whenever a question mentions a relationship, state that correlation does not imply causation unless there is evidence of a genuine link — it is almost always worth a mark. And for sampling questions, name a specific limitation: a small or biased sample may not represent the population.