Statistics Overview – A Level Maths Year 1

1 The Big Idea

Statistics is the science of learning from data. Real questions — Is this town getting warmer? Do students who revise longer score higher? — are answered by collecting data, summarising it, and interpreting what it shows, while being honest about what it does not show.

The whole of A Level statistics follows a cycle: plan (what question, what population?) → collect (how to sample?) → process and present (diagrams and summary measures) → interpret (in context, with limitations). Year 1 covers every stage; Year 2 adds the normal distribution and more hypothesis testing.

What is new compared with GCSE

Systematic, stratified, quota and opportunity sampling · frequency density
Variance and standard deviation · interpolation for grouped data
Formal outlier rules · cleaning data · a large data set

DfE content areas K · Statistical sampling and L · Data presentation and interpretation. Every board also gives you a large data set to explore with technology during the course. Expect questions that ask you to interpret and criticise — "in context" answers earn the marks, bare numbers rarely do. Calculators with statistics modes are expected; know how to get the mean and standard deviation from yours.

2 The Language You Need

Word	Meaning	Example
Population	Everything or everyone of interest.	All adults in a town
Sample	A subset of the population.	$100$ of those adults
Sampling frame	The list the sample is chosen from.	The electoral register
Qualitative	Non-numerical data.	Favourite sport
Discrete	Numerical; only particular values.	Number of pets
Continuous	Numerical; any value in a range.	Time, mass
Bivariate	Two variables measured on each item.	Height and arm span
Outlier	A value far from the rest of the data.	A $95$ among values $20$–$64$
Bias	A systematic distortion in a sample or result.	Surveying only gym users about exercise

3 The Six Core Techniques

1 — Stratified sample sizes

$\dfrac{\text{size of stratum}}{\text{size of population}} \times \text{sample size}$

2 — Frequency density

$\text{frequency density} = \dfrac{\text{frequency}}{\text{class width}}$; area of bar $\propto$ frequency

3 — Interpolation for grouped data

lower boundary $+ \dfrac{\text{position} - \text{c.f. before}}{\text{class frequency}} \times \text{class width}$

4 — Mean and standard deviation

$\bar{x} = \dfrac{\sum fx}{\sum f}$, $\sigma = \sqrt{\dfrac{\sum fx^2}{\sum f} - \bar{x}^2}$

5 — Outliers

Beyond $Q_1 - 1.5\times\text{IQR}$ or $Q_3 + 1.5\times\text{IQR}$; or beyond $\bar{x} \pm 2\sigma$ (or $3\sigma$)

6 — Regression line $y = a + bx$

$b$: change in $y$ per unit of $x$ · interpolate, don't extrapolate · correlation $\neq$ causation

Worked Example — Several techniques in one question

For $16$ values, $\sum x = 192$ and $\sum x^2 = 2560$. The largest value is $22$. (a) Find the mean and standard deviation. (b) Is $22$ an outlier using $\bar{x} \pm 2\sigma$?

①$\bar{x} = \dfrac{192}{16} = 12$.

②$\sigma^2 = \dfrac{2560}{16} - 12^2 = 160 - 144 = 16$, so $\sigma = 4$.

③Upper limit $12 + 2(4) = 20$.

④$22 > 20$, so $22$ is an outlier. Before removing it, check whether it is an error or a genuine value.

4 The Statistics Subtopics

Everything in DfE content areas K and L, and the page that covers it.

KStatistical SamplingPopulations, samples and sampling frames; random and non-random sampling methods; their advantages, disadvantages and sources of bias.
LPresenting DataHistograms with frequency density, cumulative frequency, box plots and scatter diagrams; comparing distributions.
LMeasures of Location and SpreadMean, median, mode, quartiles and percentiles; interpolation; variance and standard deviation; choosing measures.
LCorrelation and OutliersScatter diagrams and regression lines, correlation and causation, outliers and cleaning data.

Statistics continues in Probability and Distributions and Hypothesis Testing, where a sample is used to make a formal decision about a population.

5 Common Mistakes to Avoid

Plotting frequency on a histogram. With unequal classes the vertical axis must be frequency density. It is the area that shows frequency.

Confusing quota and stratified sampling. Both fill groups in proportion, but only stratified sampling chooses randomly within each group.

$(\sum fx)^2$ instead of $\sum fx^2$. Square each $x$ first, then multiply by its frequency, then add.

Comparisons without context. "B has a higher median" is not enough; "on average, group B scored higher" is.

Claiming causation. Strong correlation never proves that one variable causes the other.

Deleting every outlier. Only errors should be removed. Genuine extreme values are real data.

6 Quick Reference

Random methods

Simple random, systematic, stratified (need a frame).

Non-random

Quota, opportunity (no frame).

Histogram

Frequency density $=$ frequency $\div$ width.

Grouped median

Interpolate at $\tfrac{n}{2}$.

Variance

Mean of squares $-$ square of mean.

Skewed data

Median and IQR.

Symmetric data

Mean and standard deviation.

Outlier rules

$1.5 \times$ IQR beyond quartiles; or $2\sigma$/$3\sigma$ from mean.

Extrapolation

Outside the data — unreliable.

7 Practice Questions

Ten questions drawing on the whole Statistics topic. Work each one through before opening the solution.

Question 1

Classify each variable: (a) number of siblings; (b) time to run $100$ m; (c) eye colour.

▶ Show solution

(a) Quantitative, discrete. (b) Quantitative, continuous. (c) Qualitative.

Question 2

Describe how to take a systematic sample of $50$ from a numbered list of $1000$ customers.

▶ Show solution

Interval $k = \dfrac{1000}{50} = 20$. Choose a random start $r$ from $1$ to $20$, then take customers $r, r + 20, r + 40, \ldots$ up to $50$ customers.

Question 3

A school has $360$ students in Year 12 and $240$ in Year 13. Find the numbers from each year in a stratified sample of $40$.

▶ Show solution

Total $600$. Year 12: $\dfrac{360}{600} \times 40 = 24$. Year 13: $\dfrac{240}{600} \times 40 = 16$.

Question 4

Find the mean, median and mode of $3, 7, 8, 8, 10, 12, 15$.

▶ Show solution

$\sum x = 63$, so $\bar{x} = 9$. Median (4th value) $= 8$. Mode $= 8$.

Question 5

For $10$ values, $\sum x = 120$ and $\sum x^2 = 1650$. Find the mean and standard deviation.

▶ Show solution

$\bar{x} = 12$. $\sigma^2 = 165 - 144 = 21$, so $\sigma = \sqrt{21} = 4.58$ (3 s.f.).

Question 6

For the data $4, 6, 7, 8, 9, 10, 11, 12, 13, 30$, find the quartiles and decide whether $30$ is an outlier using the $1.5 \times \text{IQR}$ rule.

▶ Show solution

Median $= \dfrac{9 + 10}{2} = 9.5$. Lower half $4, 6, 7, 8, 9$ gives $Q_1 = 7$; upper half $10, 11, 12, 13, 30$ gives $Q_3 = 12$.

IQR $= 5$; upper fence $12 + 7.5 = 19.5$. Since $30 > 19.5$, it is an outlier.

Question 7

In a grouped frequency table, the class $10 \leqslant t < 20$ has frequency $30$ and $20 \leqslant t < 40$ has frequency $24$. Find the frequency densities, and explain why the second bar is shorter even though its frequency is similar.

▶ Show solution

$30 \div 10 = 3$ and $24 \div 20 = 1.2$.

The second class is twice as wide, so its values are spread more thinly; area, not height, represents frequency.

Question 8

For cars aged $1$ to $8$ years, the regression line of value $y$ (£1000s) on age $x$ (years) is $y = 18.4 - 1.6x$. (a) Interpret $-1.6$. (b) Explain why the line should not be used for a $15$-year-old car.

▶ Show solution

(a) For each extra year of age, a car's value falls by about £$1600$.

(b) $15$ years is outside the data range: extrapolation. Indeed the line gives $18.4 - 24 = -5.6$, a negative value, which is impossible.

Question 9

Daily maximum temperatures in June: Town A mean $15.2\,^\circ$C, standard deviation $3.1\,^\circ$C; Town B mean $12.8\,^\circ$C, standard deviation $5.4\,^\circ$C. Compare the two towns.

▶ Show solution

Town A had the higher mean, so on average it was warmer.

Town A had the smaller standard deviation, so its temperatures were more consistent; Town B's varied more from day to day.

Question 10

The number of books $x$ read last month by $40$ students is shown.

$x$	$0$	$1$	$2$	$3$	$4$	$5$
$f$	$4$	$10$	$12$	$8$	$4$	$2$

(a) Find the mean and standard deviation. (b) Find the median and interquartile range. (c) Is $x = 5$ an outlier using (i) $\bar{x} \pm 2\sigma$; (ii) the $1.5 \times \text{IQR}$ rule? (d) The students were the members of a school reading club. Explain why the results should not be used to estimate the number of books read by all students in the school.

▶ Show solution

(a) $\sum fx = 0 + 10 + 24 + 24 + 16 + 10 = 84$, so $\bar{x} = 2.1$.

$\sum fx^2 = 0 + 10 + 48 + 72 + 64 + 50 = 244$, so $\sigma^2 = \dfrac{244}{40} - 2.1^2 = 6.1 - 4.41 = 1.69$ and $\sigma = 1.3$.

(b) Cumulative frequencies: $4, 14, 26, 34, 38, 40$.

Median: $20$th and $21$st values are both $2$, so median $= 2$.

$Q_1$: $10$th and $11$th values are both $1$. $Q_3$: $30$th and $31$st values are both $3$. IQR $= 2$.

(c)(i) $2.1 + 2(1.3) = 4.7$; $5 > 4.7$, so $5$ is an outlier by this rule.

(ii) $Q_3 + 1.5 \times 2 = 6$; $5 < 6$, so $5$ is not an outlier by this rule. Different rules can give different conclusions.

(d) Reading-club members are not representative of all students — they are likely to read more than average — so the sample is biased and would overestimate the school's mean.

Statistics Overview · A Level Maths Year 1 · Created with MathJax