2.02StatisticsCore

Representing Data

Q: Why use frequency density?

With unequal widths, a wide class would look bigger just because it is wide. Using density makes area proportional to frequency, so the picture is fair.

Q: Which sampling method is best?

It depends on context. Stratified random sampling is usually most representative when subgroups differ, but it needs a full sampling frame.

Q: What makes a value an outlier on a box plot?

Usually being more than 1.5 IQR beyond a quartile, unless the question defines it differently.

Q: What does a strong correlation tell me?

That the points lie close to a straight line. It does not tell you that one variable causes changes in the other.

A good diagram shows the shape of a data set in a way no single statistic can: whether it is symmetric or skewed, where it clusters, whether two variables move together. A-Level focuses on the diagrams that carry real information — histograms with unequal class widths, cumulative frequency graphs, box plots and scatter diagrams — and on reading them critically, starting with how the data were collected.

45 min Data Analysis

The big picture

Every statistical conclusion is only as good as the data behind it, which is why this topic starts with sampling: a biased sample makes even perfect calculations meaningless. From there, each diagram answers a particular question. A histogram shows a distribution’s shape, with area — not height — representing frequency. A cumulative frequency graph gives medians, quartiles and percentiles for large grouped data. Box plots make comparisons between groups fast and highlight outliers. Scatter diagrams show relationships between two variables, and the language of correlation — with its crucial caveat that correlation does not imply causation — carries straight into the regression and hypothesis tests later in the course. OCR regularly sets these questions in the context of its large data set, so interpretation in context is always part of the answer.

What you'll be able to do

Describe populations, samples and common sampling methods, with their advantages and disadvantages
Draw and interpret histograms using frequency density
Use cumulative frequency graphs and box plots to find and compare quartiles and outliers
Interpret scatter diagrams, correlation and regression lines, including the limits of interpretation

Populations and samples

A $\textbf{population}$ is the whole group of interest; a $\textbf{census}$ measures all of it, while a $\textbf{sample}$ measures part of it. Samples are cheaper and quicker, but may not represent the population.

$\textbf{Simple random sampling}$ gives every member an equal chance of selection. $\textbf{Systematic}$ sampling takes every $k$ th item from a list. $\textbf{Stratified}$ sampling samples from each subgroup in proportion to its size. $\textbf{Quota}$ sampling fills targets for each group without random selection. $\textbf{Opportunity}$ sampling uses whoever is available.

Random methods reduce bias but need a sampling frame (a full list). Quota and opportunity sampling are convenient but prone to bias.

Example 1A school has 360 Year 12 and 240 Year 13 students. A stratified sample of 50 is needed. How many from each year?

1Total

600

2Year 12:

\tfrac{360}{600} \times 50 = 30

. Year 13:

\tfrac{240}{600} \times 50 = 20

Answer30 from Year 12, 20 from Year 13

Tip — When criticising a sampling method, name the specific source of bias in context — "only people at the station at 8am are sampled, so non-commuters are excluded".

Histograms

In a histogram of grouped continuous data, bars touch and $\textbf{area is proportional to frequency}$ . With unequal class widths, heights must be $\textbf{frequency density}$ , not frequency.

Frequency density $= \dfrac{\text{frequency}}{\text{class width}}$ . To read a frequency back, multiply density by width — or, if the scale is not labelled, work out how many units of frequency one unit of area represents.

Histograms reveal shape: symmetric, positively skewed (a long tail to the right) or negatively skewed (a long tail to the left).

\text{frequency density} = \dfrac{\text{frequency}}{\text{class width}}

Heights of histogram bars.

Example 1Classes

0 \le t < 5

5 \le t < 10

10 \le t < 20

20 \le t < 40

have frequencies

15, 25, 30, 20

. Find the frequency densities, and estimate how many values lie between 15 and 25.

1Densities:

\tfrac{15}{5} = 3

\tfrac{25}{5} = 5

\tfrac{30}{10} = 3

\tfrac{20}{20} = 1

2From 15 to 20: width 5 at density 3, giving 15. From 20 to 25: width 5 at density 1, giving 5.

AnswerDensities

3, 5, 3, 1

; about

20

values between 15 and 25.

Cumulative frequency and box plots

A $\textbf{cumulative frequency}$ graph plots running totals against the $\textbf{upper class boundaries}$ , joined by a smooth curve or straight lines. Read the median at $\tfrac{n}{2}$ , quartiles at $\tfrac{n}{4}$ and $\tfrac{3n}{4}$ , and any percentile similarly.

A $\textbf{box plot}$ shows the minimum, $Q_{1}$ , median, $Q_{3}$ and maximum on a scale. Outliers are marked separately with crosses, and the whiskers then extend to the most extreme non-outlier values.

Box plots are ideal for comparing groups. The position of the median within the box, and the lengths of the whiskers, indicate skew.

Example 1Two box plots of commute times: town A has median 24 min and IQR 10 min; town B has median 31 min and IQR 22 min. Compare them.

1Location: town B’s median is 7 minutes higher, so its commutes are longer on average.

2Spread: town B’s IQR is more than double, so its commute times are much more variable.

AnswerCommutes in B are typically longer and far less consistent than in A.

Why this works

When the median sits closer to $Q_{1}$ than $Q_{3}$ , the upper half of the middle 50% is more spread out — the data are positively skewed. Box plots let you see skew at a glance.

Scatter diagrams and correlation

A $\textbf{scatter diagram}$ plots paired data $(x, y)$ to show whether two variables are related. $\textbf{Positive correlation}$ : $y$ tends to increase with $x$ . $\textbf{Negative correlation}$ : $y$ tends to decrease. No clear pattern: no correlation.

The $\textbf{explanatory}$ (independent) variable goes on the $x$ -axis and the $\textbf{response}$ (dependent) variable on the $y$ -axis.

$\textbf{Correlation does not imply causation}$ : ice-cream sales and drownings are positively correlated because both rise in hot weather.

A regression line $y = a + bx$ can be used to predict $y$ from $x$ within the range of the data ( $\textbf{interpolation}$ ). Predicting outside that range ( $\textbf{extrapolation}$ ) is unreliable.

Example 1For data on engine size

x

(litres, from 1.0 to 3.0) and fuel economy

y

(mpg), the regression line is

y = 68 - 11x

. Interpret the gradient and comment on predicting

y

for

x = 5

1Gradient

-11

: each extra litre of engine size is associated with about 11 fewer miles per gallon.

x = 5

is well outside the data range of 1.0 to 3.0 litres.

AnswerFuel economy falls by about 11 mpg per litre; a prediction at 5 litres would be extrapolation and unreliable (it gives 13 mpg, with no evidence the linear trend continues).

Tip — Interpret a regression gradient as "for each one-unit increase in $x$ , $y$ changes by $b$ on average", with the actual units.

Think like an examiner

Use frequency density for histograms with unequal widths, and use area to find frequencies.
Plot cumulative frequency at upper class boundaries.
Compare distributions with one comment on location and one on spread, in context.
Avoid claiming causation from correlation, and flag extrapolation.

Common misconceptions

Using frequency as bar height with unequal class widths.

Heights must be frequency density.

Plotting cumulative frequency at class midpoints.

Plot at upper class boundaries.

Saying one variable causes the other because they are correlated.

Correlation shows association only; a third factor may be responsible.

Using a regression line far outside the data range.

Extrapolation is unreliable; say so.

Representing data

\text{f.d.} = \tfrac{\text{frequency}}{\text{class width}}

Frequency density.

Q_{1}: \tfrac{n}{4},\ Q_{2}: \tfrac{n}{2},\ Q_{3}: \tfrac{3n}{4}

Positions on a cumulative frequency graph.

n_{\text{stratum}} = \tfrac{\text{stratum size}}{\text{population}} \times n

Stratified sample size.

Stretch yourself

In a histogram of reaction times, the bar for $0.20 \le t < 0.25$ seconds is 4 cm wide and 6 cm tall and represents 36 people. The bar for $0.25 \le t < 0.40$ seconds is 3.2 cm tall. How many people does it represent, and what is its frequency density?

Hint — Find how many people 1 cm² represents from the first bar. The horizontal scale is 4 cm per 0.05 s.

Reveal solution

SolutionFirst bar area

= 4 \times 6 = 24

cm² for 36 people, so 1 cm² represents

1.5

people. The horizontal scale is 4 cm per 0.05 s, i.e. 80 cm per second, so the second class (width 0.15 s) is

0.15 \times 80 = 12

cm wide. Its area is

12 \times 3.2 = 38.4

cm², representing

38.4 \times 1.5 = 57.6

, so about 58 people. (A non-integer result indicates rounding in the drawing; the calculation method is what matters.) Frequency density

= \dfrac{57.6}{0.15} = 384

people per second. Check the first bar: density

\tfrac{36}{0.05} = 720

, and

\tfrac{3.2}{6} \times 720 = 384

✓.

\blacksquare

Questions students ask

Why use frequency density?

With unequal widths, a wide class would look bigger just because it is wide. Using density makes area proportional to frequency, so the picture is fair.

Which sampling method is best?

It depends on context. Stratified random sampling is usually most representative when subgroups differ, but it needs a full sampling frame.

What makes a value an outlier on a box plot?

Usually being more than

1.5 \times

IQR beyond a quartile, unless the question defines it differently.

What does a strong correlation tell me?

That the points lie close to a straight line. It does not tell you that one variable causes changes in the other.

Key takeaways

Samples should be chosen to minimise bias; know the main sampling methods and their drawbacks.
Histograms use frequency density; area represents frequency.
Cumulative frequency graphs give quartiles and percentiles; box plots summarise and compare.
Scatter diagrams show correlation, which is not causation.
Regression lines are for interpolation, not extrapolation.

How this fits the course

Build on