GCSE Maths Statistics revision

A population is everyone or everything a question is about; a sample is the part you actually measure. Conclusions are transferred from sample to population, so the sample must represent it: every kind of member needs a fair chance of appearing. A survey of school lunches answered only by the football team over-represents hungry athletes; a phone poll over-represents people who answer phones. Size matters too — asking two people proves nothing — but a large biased sample is still biased, a point examiners love to test.

Sizing a sample is a percentages job: a 10% sample of 450 customers is 0.10 × 450 = 45 customers. Keeping groups in proportion is the seed of stratified sampling: a school of 1,200 with 720 juniors and 480 seniors, sampled 60-strong, should contain 720⁄1200 × 60 = 36 juniors and 24 seniors, not 60 of whichever pupils happen to be near the gate. When you read a statistic (“9 out of 10 dentists…”), train yourself to ask: sampled from whom, how many, and who paid for the question?

Bar charts and vertical line charts suit discrete data: bars (or single lines) stand over each category with heights giving frequencies, and gaps between them signal that the categories are separate. Pictograms trade bars for icons — the key might say one book icon = 4 books, so 5½ icons means 5.5 × 4 = 22 books, and half-icons are fair game. Pie charts carve 360° in proportion to frequency: for 30 pupils with 12 walking to school, the walking sector is 12⁄30 × 360° = 144°. Working backwards, a 45° sector in a survey of 240 people represents 45⁄360 × 240 = 30 people.

Time-series graphs plot a value against time (months, quarters, years) and are read for two things at once: the trend (the long drift — sales rising year on year, say) and seasonality (the repeating wiggle — ice-cream peaking every summer quarter). A time series can hold both at once: overall growth and an annual dip. Comparison questions ask you to name both, quoting values as evidence rather than gesturing at the picture.

The mean shares the total out equally: add everything, divide by how many. Five scores of 2, 3, 7, 8 and 10 total 30, so the mean is 30 ÷ 5 = 6. Run it backwards to find missing values: if six numbers have mean 8, their total is 6 × 8 = 48, and any five known values leave the sixth as the leftover. The median is the middle value once the list is ordered — for an even count, average the two middle ones. The mode is the most frequent value (a list can have more than one, or none that help), and from grouped data you can only name the modal class, the class with the highest frequency.

The range — largest minus smallest — is the GCSE spread workhorse: for 3, 11, 8, 17 and 5 it is 17 − 3 = 14. Its weakness is that one extreme value hijacks it, which is exactly why Higher compares distributions with the interquartile range instead. Frequency tables need the total rebuilt: goals per match with frequencies 0 goals (5 matches), 1 (8), 2 (3), 3 (4) give 0×5 + 1×8 + 2×3 + 3×4 = 26 goals over 20 matches, a mean of 26⁄20 = 1.3. Mean always “1.3 goals” — decimals are fine; averages are averages, not scores that happened.

Comparisons need a pair of sentences: one about centre, one about spread. “Café P and Café Q both have a mean queue of 6 minutes, but Q’s range is 9 minutes against P’s 5, so Q is less consistent.” An average without a spread hides chaos; a spread without an average hides level. Adding one very large value drags the mean up strongly but shifts the median hardly at all — the reason “average salary” press releases prefer the median.

A histogram looks like a bar chart with the brakes off: when class widths differ, the height is the frequency density — frequency ÷ class width — and the area is the frequency. A class 10 ≤ x < 30 holding 8 items spans width 20 and rises to a density of 8⁄20 = 0.4. Unequal widths are the whole point: doubling a class width halves the height needed for the same frequency. Reading back is the reverse: frequency = frequency density × class width, so a bar of density 1.2 over a width of 20 holds 24 items — and half a bar’s area holds half its frequency, which is how “at least 50” questions work when 50 sits inside a class.

A cumulative frequency graph plots the running total against the upper class boundary of each class. For n values, the median is read at n⁄2 up the frequency axis (50th of 100), the lower quartile at n⁄4 and the upper quartile at 3n⁄4, each read across to the curve and down to the data axis. The interquartile range is UQ − LQ, and it compares spreads without letting outliers vote. An outlier by the standard rule sits below LQ − 1.5 × IQR or above UQ + 1.5 × IQR.

A box plot packs minimum, LQ, median, UQ and maximum into one picture, with the box spanning the middle half of the data. Two box plots invite the same two-sentence comparison: medians for level, IQRs (or box widths) for consistency. For 100 pupils with LQ 32, median 47 and UQ 60, the IQR is 28 and the outlier fences sit at 32 − 1.5 × 28 = −10 and 60 + 42 = 102 — so a score of 8 is unusual but not an outlier, and neither is 78.

A scatter graph plots one variable against another, point by point, and the cloud’s shape tells the story: positive correlation (both rise together — plant height against leaf count), negative correlation (one rises as the other falls — a car’s age against its value) or no correlation (scatter with no drift). The trap examiners set every year: correlation is not causation. Ice-cream sales and drownings rise together each summer not because ice cream drowns anyone but because hot weather drives both. A third variable is called a confounding variable.

A line of best fit is drawn by eye through the cloud’s centre of mass, passing near as many points as possible with roughly equal numbers either side. It turns scatter into a usable rule: find a value on one axis, track to the line, read across. The reliability depends on where you read it. Interpolation — estimating inside the plotted data range — leans on real evidence. Extrapolation — estimating outside it — assumes the pattern carries on into territory nobody measured; plants taller than any measured may simply stop growing leaves. Exam phrasing: “explain why your estimate may be unreliable” almost always wants “extrapolation / outside the range of the data”.