Lesson 9: Statistics — Averages and Spread

Statistics is how the UK understands itself: the ONS measures inflation, unemployment, and life expectancy; the NHS tracks waiting times; schools publish GCSE results. Averages and spread are the first tools of that trade. This lesson covers the three averages (mean, median, mode), the range and interquartile range, and — most importantly — choosing and comparing: GCSE increasingly asks not 'calculate' but 'which average, and what do the numbers tell you?'

The three averages

The mean uses every value: mean = (sum of values) ÷ (number of values). For data 4, 7, 7, 9, 13: mean = 40 ÷ 5 = 8. The mean is representative but vulnerable to extremes: one Premiership footballer's £300,000 weekly wage hugely inflates the 'average salary' of a squad — which is why the ONS usually reports median earnings.

The median is the middle value when the data is ordered. For 4, 7, 7, 9, 13 the median is 7. For an even count, average the two middle values once ordered: 3, 5, 8, 10 → median = (5 + 8)/2 = 6.5. The median resists outliers, which is why house prices are reported as medians: half of homes cost less, half more.

The mode is the most frequent value — the only average that works for non-numerical data (the modal shoe size in a shop, the modal eye colour). Data can be bimodal (two modes) or have no mode. The range = highest − lowest measures spread: for 4, 7, 7, 9, 13 the range is 9.

Worked example: temperatures in York over a week: 11, 14, 15, 15, 17, 19, 40 (°C — the 40 is clearly a faulty sensor). Mean with the fault: 131 ÷ 7 = 18.7. Without it: 91 ÷ 6 = 15.2. Median (ordered, fault included): 15 — unchanged by the error. Moral: the median shrugs off outliers; the mean does not. Exam questions about 'which average best represents the data' almost always hinge on one rogue value.

Grouped data and estimates

When data is grouped into classes (e.g. heights 140 < h ≤ 150), individual values are lost, so the mean can only be estimated using class midpoints. Worked example: test marks for a class of 30: 0–20 (3 pupils), 20–40 (7), 40–60 (12), 60–80 (6), 80–100 (2). Midpoints: 10, 30, 50, 70, 90. Estimated total = 3(10) + 7(30) + 12(50) + 6(70) + 2(90) = 30 + 210 + 600 + 420 + 180 = 1,440. Estimated mean = 1,440 ÷ 30 = 48 marks. Say 'estimate' — the true mean is unknowable from grouped data; using endpoints instead of midpoints is the standard error.

The modal class is the class with the highest frequency (40–60 above). The median class is the class containing the middle value (here the 15th–16th value lies in 40–60 too). GCSE asks for all three from grouped tables — read the question to see which.

Worked example (cumulative frequency thinking): a gym in Bristol records members' ages: 10–20 (4), 20–30 (11), 30–40 (17), 40–50 (9), 50–60 (5). Total = 46 members. The median (23rd member) falls in 30–40. The estimated mean: midpoints 15, 25, 35, 45, 55 → total = 60 + 275 + 595 + 405 + 275 = 1,610 → mean ≈ 1,610 ÷ 46 = 35.0 years.

Quartiles and the interquartile range

Quartiles split ordered data into quarters. The lower quartile Q1 is the median of the lower half (25% of data below it); the upper quartile Q3 is the median of the upper half (75% below it). The interquartile range, IQR = Q3 − Q1, measures the spread of the middle 50% — ignoring the extremes entirely, so it (like the median) resists outliers.

Worked example: data 2, 5, 6, 8, 9, 11, 13, 15, 18 (nine values). Median = 9 (the 5th value). Lower half: 2, 5, 6, 8 → Q1 = (5 + 6)/2 = 5.5. Upper half: 11, 13, 15, 18 → Q3 = (13 + 15)/2 = 14. IQR = 14 − 5.5 = 8.5. Range = 18 − 2 = 16. If an extreme value of 60 replaced 18, the range would jump to 58 but the IQR would barely move — exactly why analysts prefer the IQR.

Box plots display the five-number summary (minimum, Q1, median, Q3, maximum) — the standard way to compare distributions visually (Lesson 11 develops this). Sketch a box plot for the data above: a box from 5.5 to 14 with a line at 9, whiskers to 2 and 18.

Worked example: two classes take the same maths test. Class X: median 58, IQR 12. Class Y: median 64, IQR 20. Interpretation: Y's typical (median) score is higher, but X's scores are more consistent (smaller IQR). 'Higher on average, less consistent' — that comparative language is what GCSE comparison questions reward.

Choosing the right average

• Mean: best all-rounder, uses all data, but distorted by outliers. Use for roughly symmetric data (heights, test marks without errors).
• Median: robust to outliers; use for incomes, house prices, anything skewed.
• Mode: for categories and for stock decisions (a shoe shop stocks modal sizes).

Comparison questions need two points — one about average ('on average, higher/lower') and one about spread ('more/less consistent') — each backed by a statistic. Worked example: two bus routes in Glasgow. Route A delays (min): 1, 2, 2, 3, 12. Route B: 4, 5, 5, 6, 5. Means: A = 20/5 = 4; B = 25/5 = 5. Medians: A = 2; B = 5. Route A 'looks better' by median but has a 12-minute disaster (roadworks); route B is steadier (range 2 vs 11). A good answer discusses both centre and spread — and may suggest the median and IQR as the fairer comparison here.

Worked example (missing value): the mean of five numbers is 14 and four of them are 9, 12, 17, 20. The total must be 5 × 14 = 70; the known four sum to 58, so the missing number is 12. This 'work backwards from the mean' pattern appears every year at GCSE.

Worked example — comparing two data sets in full

Two Year 11 classes record their weekly revision hours. Class P: 2, 3, 3, 4, 5, 5, 6, 20 (one pupil's 20 is almost certainly an error — or a very keen weekend). Class Q: 4, 4, 5, 5, 5, 6, 6, 7. For P: mean = 48 ÷ 8 = 6 hours; ordered median = (4 + 5)/2 = 4.5; range = 18. For Q: mean = 42 ÷ 8 = 5.25; median = 5; range = 3. Comparison in exam language: 'On average, P revised more by the mean (6 > 5.25) but less by the median (4.5 < 5); the mean for P is inflated by the outlier 20. Q's revision was far more consistent (range 3 vs 18). Overall the median suggests Q is the more reliably revising class.' That answer quotes four statistics and discusses the outlier — the full-marks pattern.

The lesson generalises: whenever mean and median disagree, an outlier or a skew is responsible, and the median is usually the honest average. Whenever two groups are compared, one group will typically win on average and the other on consistency — report both, or your comparison is half-finished.

Frequency tables — averages from tallies

Not all data arrives as a list. Worked example: the number of siblings of 20 pupils: 0 siblings (4 pupils), 1 (9), 2 (5), 3 (2). Mean = (0×4 + 1×9 + 2×5 + 3×2) ÷ 20 = (0 + 9 + 10 + 6) ÷ 20 = 25 ÷ 20 = 1.25. Mode = 1 (highest frequency). Median: the 10th–11th values both lie in the '1 sibling' group (cumulative frequencies 4, 13, ...) → median = 1. Range = 3. The Σfx method — multiply each value by its frequency, sum, divide by total frequency — is the same machinery as the grouped estimate, just without midpoints.

Common mistakes

• Median of unordered data. Always order the values first — the middle of the raw list is meaningless.
• Endpoints instead of midpoints in grouped means. The estimate uses the class midpoint, and the answer is an estimate.
• Comparing with averages only. 'Class Y is better because its mean is higher' is half an answer. Always pair centre with spread (range or IQR).
• Range as an average. The range is a measure of spread, never a centre. A B-mark is often lost by calling it one.
• Forgetting the even-count median. With 46 values, the median lies between the 23rd and 24th — use the median class or the average of the two middle values.
• 'Mean' when an outlier rules. If one value is absurd (the 40°C sensor), the median is the representative average — say so explicitly.

Exam Experience

Question 1. The mean of five numbers is 14. Four of the numbers are 9, 12, 17 and 20. Work out the fifth number. [2 marks]

Question 2. The table shows the heights of 40 students. Estimate the mean height. [3 marks]

Question 3. Two classes take the same test. Class X has median 58 and interquartile range 12. Class Y has median 64 and interquartile range 20. Compare the test results of the two classes. [3 marks]

Mark scheme.

• Q1: M1 for using total = 5 × 14 = 70 (or summing the four known values, 58). A1 for 70 − 58 = 12.
• Q2: M1 for using midpoints 145, 155, 165, 175. M1 for Σfx = 145(6) + 155(14) + 165(13) + 175(7) = 870 + 2,170 + 2,145 + 1,225 = 6,410. A1 for mean = 6,410 ÷ 40 = 160.25 cm (accept 'estimate' language; approx. 160 cm).
• Q3: B1 for a correct comparison of average (on average Class Y scored higher — median 64 > 58). B1 for a correct comparison of spread (Class X was more consistent — IQR 12 < 20). B1 for both points made in context with figures quoted. (Writing only 'Y is better' scores at most 1 of 3.)