Statistics – Overview

1 The Statistical Cycle

Statistics is about turning data into understanding. Every statistical investigation follows the same four stages.

The two big questions statistics answers:
• What is typical? — the averages: mean, median, mode.
• How spread out is it? — the range, interquartile range and the shape of the distribution.
A good comparison of two data sets always mentions both.

Worked Example — The whole cycle in one question

A teacher asks: "Do students in Year 11 sleep less than students in Year 7?" Describe how the four stages would be carried out.

①Question: already clear, and it names both the populations and the variable (hours of sleep).

②Collect: take a stratified random sample from each year group, asking how many hours they slept last night.

③Represent and analyse: draw box plots for the two years, and calculate the median and interquartile range for each.

④Conclude: compare the medians (the average) and the IQRs (the spread), quoting the actual figures and stating any limitations of the sample.

2 Types of Data

Type	Meaning	Example
Qualitative	Words or categories	Eye colour, favourite sport
Quantitative	Numbers	Height, number of siblings
Discrete	Counted — only certain values	Number of goals, shoe size
Continuous	Measured — any value in a range	Mass, time, length
Primary	You collected it yourself	Your own survey
Secondary	Collected by someone else	Census data, a website

The type of data decides the diagram. Bar charts for categories, vertical line charts for discrete numbers, histograms for grouped continuous data. Choosing the wrong one loses marks.

3 The Key Formulae

Measure	How to find it
Mean	$\dfrac{\text{sum of values}}{\text{number of values}}$
Median	The middle value when in order; position $\dfrac{n+1}{2}$
Mode	The most common value
Range	Largest $-$ smallest
Lower quartile $Q_1$	Position $\dfrac{n+1}{4}$
Upper quartile $Q_3$	Position $\dfrac{3(n+1)}{4}$
Interquartile range	$Q_3 - Q_1$
Mean from a table	$\dfrac{\sum fx}{\sum f}$
Frequency density	$\dfrac{\text{frequency}}{\text{class width}}$

The interquartile range is a single number. If $Q_1 = 12$ and $Q_3 = 20$, the IQR is $8$ — not "$12$ to $20$".

Worked Example — Picking the right formula

Twelve students scored: $4, 9, 6, 15, 7, 9, 11, 3, 9, 12, 8, 10$. Find the mean, median, mode, range and IQR.

①Order first: $3, 4, 6, 7, 8, 9, 9, 9, 10, 11, 12, 15$

②Mean: total $= 103$, so $\dfrac{103}{12} = 8.58$ (2 d.p.)

③Median: position $\dfrac{13}{2} = 6.5$, so average the 6th and 7th: $\dfrac{9+9}{2} = 9$

④Mode: $9$ (appears three times)

⑤Range: $15 - 3 = 12$

⑥Quartiles: $Q_1$ at position $\dfrac{13}{4} = 3.25$, between $6$ and $7$, so $Q_1 = 6.5$

⑦$Q_3$ at position $\dfrac{39}{4} = 9.75$, between $10$ and $11$, so $Q_3 = 10.5$

⑧IQR $= 10.5 - 6.5 = 4$

Notice the IQR ($4$) is much smaller than the range ($12$) — the extremes $3$ and $15$ stretch the range but not the IQR.

4 The 6 Subtopics

The National Curriculum divides Statistics into six statements, usually labelled S1 to S6. Each has its own page with explanations, worked examples and ten practice questions.

S1Sampling and PopulationsRandom and stratified samples, bias, and the limitations of sampling.
S2Tables, Charts and DiagramsBar charts, pie charts, pictograms, vertical line charts and time series.
S3Histograms and Cumulative FrequencyGrouped and continuous data, unequal class widths, and estimating from a curve.
S4Averages, Spread and Box PlotsMean, median, mode, range, quartiles, outliers and comparing distributions.
S5Using Statistics to Describe a PopulationDrawing valid conclusions, estimating totals, and spotting misleading claims.
S6Scatter Graphs and CorrelationLines of best fit, interpolation, extrapolation, and why correlation is not causation.

5 Common Mistakes to Avoid

Mistake 1 — Forgetting to order the data before finding the median. The median of $7, 2, 9$ is $7$, not $2$.

Mistake 2 — Comparing only averages. A full comparison needs a measure of average and a measure of spread, both interpreted in context.

Mistake 3 — Using frequency as the height of a histogram bar. The height must be the frequency density whenever the class widths differ.

Mistake 4 — Treating grouped data as exact. The mean from a grouped table is only an estimate, because you use midpoints rather than the real values.

Mistake 5 — Assuming correlation means causation. Ice cream sales and drownings both rise in summer, but one does not cause the other.

Mistake 6 — Extrapolating far beyond the data. A line of best fit is only reliable within the range of the points you actually have.

6 Quick Reference

Mean

Total $\div$ how many. Affected by extreme values.

Median

Middle value in order. Not affected by outliers.

Mode

Most common. The only average for categories.

Range

Largest $-$ smallest. Easily distorted by one odd value.

IQR

$Q_3 - Q_1$. A more robust measure of spread.

Grouped mean

$\dfrac{\sum fx}{\sum f}$ using midpoints — an estimate.

Histogram

Height is frequency density; area represents frequency.

Cumulative frequency

Plot at the upper class boundary; read the median at $\tfrac{n}{2}$.

Comparing

Always give an average and a spread, in context.

Correlation

Never implies causation.

7 Practice Questions

These ten questions sample the whole topic. If one type catches you out, follow the link in Section 4 to the page that covers it.

Question 1

Find the mean, median, mode and range of: $4, 7, 3, 7, 9, 2, 7$.

▶ Show solution

Ordered: $2, 3, 4, 7, 7, 7, 9$

Mean $= \dfrac{39}{7} = 5.57$ (2 d.p.)

Median: the $4$th of $7$ values $= 7$

Mode $= 7$ (appears three times)

Range $= 9 - 2 = 7$

Question 2

Classify each as discrete or continuous: (a) number of cars in a car park, (b) the time to run 100 m, (c) shoe size, (d) the mass of a parcel.

▶ Show solution

(a) Discrete — you count cars.

(b) Continuous — time can take any value.

(c) Discrete — sizes come in fixed steps.

(d) Continuous — mass is measured.

Question 3

Estimate the mean from this grouped table.

Height $h$ (cm)	$140 \leq h \lt 150$	$150 \leq h \lt 160$	$160 \leq h \lt 170$
Frequency	$6$	$14$	$10$

▶ Show solution

Midpoints: $145$, $155$, $165$.

$\sum fx = 6(145) + 14(155) + 10(165) = 870 + 2170 + 1650 = 4690$

$\sum f = 30$

Estimated mean $= \dfrac{4690}{30} = 156.3$ cm (1 d.p.)

Question 4

A pie chart shows $180$ people. The sector for "bus" has an angle of $84^\circ$. How many travelled by bus?

▶ Show solution

$\dfrac{84}{360} \times 180 = 42$ people

(Alternatively, each person is $360 \div 180 = 2^\circ$, so $84 \div 2 = 42$.)

Question 5

Find the interquartile range of: $3, 5, 6, 8, 11, 13, 14, 18, 20$.

▶ Show solution

$n = 9$, already in order.

$Q_1$ at position $\dfrac{9+1}{4} = 2.5$, so halfway between $5$ and $6$: $Q_1 = 5.5$

$Q_3$ at position $\dfrac{3(10)}{4} = 7.5$, so halfway between $14$ and $18$: $Q_3 = 16$

IQR $= 16 - 5.5 = 10.5$

Question 6

A histogram has a bar for $10 \leq t \lt 30$ with frequency density $2.5$. How many values are in this class?

▶ Show solution

Class width $= 30 - 10 = 20$

Frequency $= $ density $\times$ width $= 2.5 \times 20 = 50$

Question 7

A school has $600$ students in the ratio Year 10 : Year 11 $= 7 : 5$. A stratified sample of $60$ is taken. How many Year 11 students should be in the sample?

▶ Show solution

Total parts $= 12$, so Year 11 has $\dfrac{5}{12} \times 600 = 250$ students.

Sample fraction $= \dfrac{60}{600} = 0.1$

Year 11 in sample $= 0.1 \times 250 = 25$ students

Question 8

A scatter graph of hours revised against test score shows strong positive correlation. Explain what this means, and why it does not prove that revising causes higher scores.

▶ Show solution

Strong positive correlation means that as revision hours increase, test scores tend to increase too, and the points lie close to a straight line.

It does not prove causation because a third factor could explain both — for example, more motivated students may both revise more and pay more attention in lessons. Correlation shows an association, not a cause.

Question 9

Two classes take the same test. Class A: median $62$, IQR $8$. Class B: median $58$, IQR $19$. Compare the two classes.

▶ Show solution

Average: Class A has the higher median ($62$ vs $58$), so on average Class A scored better.

Spread: Class A has a much smaller IQR ($8$ vs $19$), so Class A's scores are more consistent, while Class B's are far more varied.

Overall: Class A performed better and more consistently.

Question 10

The cumulative frequency table shows the times of $80$ runners.

Time $t$ (min)	$\leq 20$	$\leq 25$	$\leq 30$	$\leq 35$	$\leq 40$
Cumulative frequency	$5$	$22$	$50$	$71$	$80$

(a) How many took between $25$ and $30$ minutes? (b) Estimate the median. (c) Estimate the IQR.

▶ Show solution

(a) $50 - 22 = 28$ runners

(b) Median is at $\dfrac{80}{2} = 40$th value. That falls between $25$ min (cf $22$) and $30$ min (cf $50$).

Reading from the curve gives roughly $\mathbf{28}$ minutes.

(c) $Q_1$ at the $20$th value $\approx 24$ min; $Q_3$ at the $60$th value $\approx 32$ min.

IQR $\approx 32 - 24 = \mathbf{8}$ minutes.

Statistics · GCSE Maths Revision · Created with MathJax