Notes · MathematicsUK · A-Levels

Data presentation and interpretation

This topic covers the description of data: measures of central tendency and spread including variance and standard deviation, the representation of data by histograms, cumulative frequency graphs and box plots, the identification of outliers, and the analysis of bivariate data through scatter diagrams, correlation and linear regression. Interpretation in context, often using the large data set, is central.

4 sections~11 min reading time3 competenciesLevel Standard 3 · Advanced 1

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Exam profile

AO1 · Calculate measures of location and spread and draw statistical diagramsAO2 · Interpret diagrams, correlation and regression linesAO3 · Draw and justify conclusions from data in context

Operators:calculateinterpretdescribeestimatecommentcompare

basic level

AS-Level covers measures of location and spread, histograms, box plots, correlation and regression.

higher level

The full A-Level expects confident interpretation, comparison of distributions and use of the large data set.

Depth

Text

Contents · 4 sections

Data presentation and interpretation

Measures of location and spread#

StandardDfE GCE Mathematics — dataAQA 7357 (Statistics)

A discrete frequency distribution

Fig. 1The distribution has mode 2; the mean and median are computed from the frequencies.

Measures of location summarise where the centre of a data set lies. The mean

\bar{x} = \dfrac{\sum x}{n}

uses every value and is the most informative when the data are roughly symmetric, but it is sensitive to extreme values. The median is the middle value when the data are ordered and is resistant to outliers, and the mode is the most frequent value. Choosing the appropriate average for the shape of the data is itself an interpretive skill.

Measures of spread describe how dispersed the data are. The range is max minus min, and the interquartile range

\text{IQR} = Q_3 - Q_1

measures the spread of the middle half, ignoring extremes. The most important measure is the standard deviation, the root-mean-square deviation from the mean; its square, the variance, is

\sigma^2 = \dfrac{\sum x^2}{n} - \bar{x}^2

, a formula usually more convenient than the definition for calculation.

For data given in a frequency table the formulae carry a frequency

f

\bar{x} = \dfrac{\sum fx}{\sum f}

and the variance is

\dfrac{\sum fx^2}{\sum f} - \bar{x}^2

. Grouped data use the mid-interval value as

x

, which introduces a small approximation. Being systematic — tabulating

x

f

fx

and

fx^2

— keeps these calculations reliable.

Coding (a linear transformation such as

y = \dfrac{x - a}{b}

) simplifies awkward numbers: the mean transforms the same way,

\bar{x} = a + b\bar{y}

, and the standard deviation scales by

|b|

\sigma_x = |b|\,\sigma_y

(a shift

a

does not affect spread). Coding is a standard device for handling large or awkwardly-scaled values, and reversing it correctly is examined.

\bar{x} = \frac{\sum x}{n}, \qquad \sigma^2 = \frac{\sum x^2}{n} - \bar{x}^2, \qquad \sigma = \sqrt{\sigma^2}

Mean, variance and standard deviation

The variance is the mean of the squares minus the square of the mean.

Worked example

Mean and standard deviation

Find the mean and standard deviation of $2, 4, 4, 6, 9$ .

Mean
$\bar{x} = \dfrac{2 + 4 + 4 + 6 + 9}{5} = \dfrac{25}{5} = 5$ .
Sum of squares
$\sum x^2 = 4 + 16 + 16 + 36 + 81 = 153$ .
Variance
$\sigma^2 = \dfrac{153}{5} - 5^2 = 30.6 - 25 = 5.6$ .
$\sigma^2 = \frac{153}{5} - 25 = 5.6$
Standard deviation
$\sigma = \sqrt{5.6} = 2.37$ (3 s.f.).

Result: Mean $5$ , standard deviation $\sqrt{5.6} \approx 2.37$ .

Typical mistakes

Forgetting to subtract $\bar{x}^2$ , or squaring the mean of $x$ instead of using the mean of $x^2$ .
Applying the shift $a$ to the standard deviation when coding (it affects the mean only).

Active revision

For the data $4, 7, 7, 9, 13$ , find the mean, the standard deviation, and the interquartile range.

Active recall

Recall the key points — then reveal.

Sources: Mathematics: AS and A level content (GCE subject content) (Department for Education) · AQA A-level Mathematics 7357 specification (AQA)

Representing data: histograms and box plots#

StandardDfE GCE Mathematics — dataAQA 7357 (Statistics)

A histogram with unequal class widths

Fig. 2With unequal widths, the bar heights are frequency densities so that each area equals its class frequency.

A histogram represents grouped continuous data, and its defining feature is that area, not height, represents frequency. When class widths are unequal you must plot frequency density

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

on the vertical axis, so that the area of each bar equals the frequency of that class. Reading a histogram means reading areas, and a common question asks you to estimate a frequency from a bar's area.

A cumulative frequency graph plots the running total of frequencies against the upper class boundaries, producing an increasing S-shaped curve. From it you read off the median (at half the total frequency), the quartiles (at one quarter and three quarters) and hence the interquartile range, and you can estimate the number of values below any given figure. These readings are estimates because the original values within each class are not known.

A box plot (box-and-whisker diagram) displays the five-number summary: minimum, lower quartile

Q_1

, median, upper quartile

Q_3

and maximum. The box spans the interquartile range with the median marked inside, and the whiskers reach to the extreme values (or to the outlier boundaries, with outliers plotted separately). It shows location, spread and skew at a glance.

Box plots are especially useful for comparing two distributions: aligning them lets you compare medians (location), interquartile ranges (spread) and the symmetry of each. A comparison answer should refer to the context and to at least a measure of location and a measure of spread — 'the boys' median time is higher and their times are more spread out' — rather than merely describing one diagram.

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

Histogram vertical axis

Area represents frequency, so unequal-width classes require frequency density.

Worked example

Frequency from a histogram

In a histogram, the class $10 \leq x < 25$ is drawn with frequency density $4$ . Find the frequency of this class.

Class width
Width $= 25 - 10 = 15$ .
Frequency = density x width
$\text{frequency} = 4 \times 15$ .
$\text{frequency} = 4 \times 15 = 60$
State
The class contains 60 data values.

Result: The frequency of the class is $60$ .

Typical mistakes

Plotting frequency instead of frequency density for unequal class widths.
Comparing box plots by the median alone, without mentioning spread or the context.

Active revision

A histogram has a class $20 \leq x < 30$ of frequency density $1.5$ and a class $30 \leq x < 50$ of frequency density $0.8$ . Find the frequency of each class and the total.

Active recall

Recall the key points — then reveal.

Sources: Mathematics: AS and A level content (GCE subject content) (Department for Education) · AQA A-level Mathematics 7357 specification (AQA)

Outliers and cleaning data#

StandardDfE GCE Mathematics — dataAQA 7357 (Statistics)

A box plot with an outlier

Fig. 3The five-number summary is shown as a box and whiskers; the value 55 lies beyond the upper fence and is plotted as an outlier.

An outlier is a value that is far from the rest of the data. A common rule identifies a value as an outlier if it lies more than

1.5 \times \text{IQR}

beyond a quartile, that is below

Q_1 - 1.5\,\text{IQR}

or above

Q_3 + 1.5\,\text{IQR}

. An alternative rule flags values more than two standard deviations from the mean. The rule to use is usually stated in the question, and applying it correctly gives the outlier boundaries (fences).

Not every outlier should be removed. Some are genuine extreme observations that carry real information; others are recording or measurement errors. The decision to keep or discard a value must be justified in context — a recorded human height of 3 metres is certainly an error, whereas an unusually tall person is a genuine value — and any value removed should be noted, not silently dropped.

Because the mean and standard deviation use every value, they are pulled towards outliers, while the median and interquartile range are resistant. This is why a skewed distribution or one with outliers is often better summarised by the median and IQR. Recognising when the mean is being distorted, and choosing resistant measures instead, is part of interpreting data honestly.

Cleaning data — identifying errors, handling missing values and deciding on outliers — is an explicit part of working with real data such as the large data set. A well-argued answer states the outlier rule, applies it to find the fences, identifies the outliers, and then comments on whether each is likely an error or a genuine extreme value.

\text{outlier if } x < Q_1 - 1.5\,\text{IQR} \ \text{or} \ x > Q_3 + 1.5\,\text{IQR}

The 1.5 x IQR rule

Values beyond these fences are flagged as outliers; the alternative rule uses mean plus or minus two standard deviations.

Worked example

Applying the outlier rule

For data with $Q_1 = 20$ and $Q_3 = 32$ , use the $1.5 \times \text{IQR}$ rule to find the fences and test the value $52$ .

Interquartile range
$\text{IQR} = 32 - 20 = 12$ .
Fences
Lower: $20 - 1.5(12) = 2$ ; upper: $32 + 1.5(12) = 50$ .
$32 + 1.5 \times 12 = 50$
Test the value
$52 > 50$ , so 52 lies beyond the upper fence and is an outlier.

Result: The fences are 2 and 50; the value 52 is an outlier.

Typical mistakes

Removing every outlier automatically, without considering whether it is a genuine value.
Using the mean and standard deviation to summarise clearly skewed data instead of the median and IQR.

Active revision

A data set has $Q_1 = 30$ , $Q_3 = 50$ . Using the $1.5 \times \text{IQR}$ rule, find the outlier boundaries and decide whether the values $5$ and $78$ are outliers.

Active recall

Recall the key points — then reveal.

Sources: Mathematics: AS and A level content (GCE subject content) (Department for Education) · AQA A-level Mathematics 7357 specification (AQA)

Correlation and regression#

AdvancedDfE GCE Mathematics — dataAQA 7357 (Statistics)

Scatter diagram with a regression line

Fig. 4The points show strong positive correlation; the least-squares regression line is drawn through them.

Bivariate data — paired observations of two variables — are displayed on a scatter diagram, which reveals any association between the variables. Correlation measures the strength and direction of a linear relationship: positive correlation means one variable tends to increase with the other, negative that one decreases as the other increases, and near-zero correlation means little linear association. The product moment correlation coefficient (PMCC), denoted

r

, quantifies this on a scale from

-1

+1

A value of

r

close to

+1

-1

indicates strong linear correlation, while

r

near

0

indicates weak or no linear correlation. Crucially, correlation does not imply causation: a strong correlation between two variables may be due to a third underlying factor, or be coincidental. Interpreting

r

in context, and resisting the leap to a causal claim, is a core AO2/AO3 skill.

When there is reasonable linear correlation, a regression line can be fitted, usually by the method of least squares, giving an equation

y = a + bx

(with

y

the dependent variable regressed on the independent

x

). The gradient

b

is interpreted as the change in

y

per unit change in

x

, and the intercept

a

as the value of

y

when

x = 0

; both interpretations should be given in the context of the data.

The regression line is used to predict

y

from a value of

x

. Prediction within the range of the data (interpolation) is reasonably reliable, but prediction outside it (extrapolation) is unreliable, because the linear relationship may not continue. You should be able to state whether a given prediction is an interpolation or extrapolation and comment on how much it can be trusted.

-1 \leq r \leq 1, \qquad y = a + bx

PMCC and the regression line

$r$ measures linear correlation; the least-squares line predicts $y$ from $x$ , with gradient $b$ .

Worked example

Interpreting a regression line

The regression line of a plant's height $h$ cm on the number of days $d$ is $h = 1.5 + 0.8d$ , based on data for $d$ from 0 to 30 days. Interpret the line and predict the height at $d = 20$ .

Interpret the gradient
The gradient $0.8$ means the plant grows on average $0.8$ cm per day.
Interpret the intercept
The intercept $1.5$ is the height at $d = 0$ , i.e. the initial height of $1.5$ cm.
Predict at d = 20
$h = 1.5 + 0.8(20) = 17.5$ cm; since $20$ is within the data range $0$ to $30$ , this interpolation is reliable.
$h = 1.5 + 0.8 \times 20 = 17.5$

Result: Growth $0.8$ cm/day, initial height $1.5$ cm; predicted height at 20 days is $17.5$ cm (a reliable interpolation).

Typical mistakes

Claiming one variable causes the other from correlation alone.
Using the regression line to extrapolate far outside the data and trusting the result.

Active revision

A regression line of ice-cream sales $y$ (hundreds) on temperature $x$ (degrees C) is $y = -2.1 + 0.4x$ . Interpret the gradient, and comment on using the line to predict sales at $40\,^{\circ}\text{C}$ when the data ran from $10$ to $28\,^{\circ}\text{C}$ .

Active recall

Recall the key points — then reveal.

Sources: Mathematics: AS and A level content (GCE subject content) (Department for Education) · AQA A-level Mathematics 7357 specification (AQA)

References & sources

Sources

Department for Education

Mathematics: AS and A level content (GCE subject content)

AQA

AQA A-level Mathematics 7357 specification

Data presentation and interpretation

4 sections~11 min reading time3 competenciesLevel Standard 3 · Advanced 1

Data presentation and interpretation

Key points

Mean and standard deviation

Key points

Frequency from a histogram

Key points

Applying the outlier rule

Key points

Interpreting a regression line

Sources

Data presentation and interpretation

Key points

Mean and standard deviation

Key points

Frequency from a histogram

Key points

Applying the outlier rule

Key points

Interpreting a regression line

Sources