Data presentation and interpretation

Section S2

4 spec leafs

Notes and three levels of exam-style practice for each registered specification leaf in this section.

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S2.1

Interpret diagrams for single-variable data, including understanding that area in a histogram represents frequency; connect to probability distributions.

In a histogram, frequency is proportional to bar area and frequency density is $\frac{\text{frequency}}{\text{class width}}$ ; unequal class widths make bar height alone misleading.
Frequency polygons show class patterns, box plots summarise centre and spread, and cumulative frequency diagrams support estimates of medians, quartiles and counts below a value.
For a continuous probability density histogram, total area is $1$ and the area above an interval is the probability of an observation in that interval.
Read class boundaries and axis scales before calculating. A common error is to use frequency density as though it were frequency.

Tier 1 · Easy

2 marks

ORIGINAL

A histogram class is $12\leq t<17$ and has frequency density $3.6$ . Find the frequency in this class.

Tier 2 · Standard

3 marks

ORIGINAL

A cumulative frequency diagram represents $80$ observations. The cumulative frequencies at $x=10$ and $x=20$ are $18$ and $54$ respectively. Estimate the number of observations satisfying $10<x\leq20$ .

Tier 3 · Hard

5 marks

ORIGINAL

The probability density histogram for a continuous random variable $X$ has constant heights $0.12$ on $0\leq x<3$ , $0.08$ on $3\leq x<8$ , and $0.12$ on $8\leq x\leq10$ . Verify that it defines a probability distribution and find $P(1<X<6)$ .

S2.2

Interpret scatter diagrams and regression lines for bivariate data, including recognising distinct sections of the population (regression calculations excluded); interpret correlation informally; correlation does not imply causation.

A scatter diagram shows paired bivariate data; describe the direction and strength of its association and note any outliers or clusters.
A regression line estimates the mean response for a given explanatory-variable value and is most defensible within the observed data range.
Distinct clusters may represent different sections of the population, so one overall correlation or regression line can hide different within-group patterns.
Correlation measures association, not causation; a lurking variable, reverse causation or coincidence may explain the observed relationship.

Tier 1 · Easy

2 marks

ORIGINAL

A scatter diagram of daily ice-cream sales against temperature shows a strong positive correlation. Interpret this and explain why it does not prove that higher temperature causes every increase in sales.

Tier 2 · Standard

3 marks

ORIGINAL

A scatter diagram of journey distance against journey time contains one cluster for bicycles and a separate cluster for cars. Explain why fitting one regression line to all journeys may be misleading.

Tier 3 · Hard

4 marks

ORIGINAL

For trees aged between $3$ and $18$ years, a regression line of trunk diameter $d$ cm on age $a$ years is $d=1.8a+4.2$ . A $12$ -year-old tree has diameter $29$ cm. Interpret the difference between the observed and predicted values and critique using the line to predict the diameter of a $40$ -year-old tree.

S2.3

Interpret measures of central tendency and variation, extending to standard deviation; be able to calculate standard deviation, including from summary statistics.

The mean uses every value, while the median is resistant to extremes; choose a measure that suits the distribution and context.
Range and interquartile range measure spread using endpoints or quartiles; standard deviation measures typical spread about the mean using all observations.
For $n$ data values, use $\sigma=\sqrt{\frac{\sum x^2}{n}-\left(\frac{\sum x}{n}\right)^2}$ unless a different convention is stated.
When groups are combined, add $n$ , $\sum x$ and $\sum x^2$ before recalculating; averaging separate standard deviations is not valid.

Tier 1 · Easy

3 marks

ORIGINAL

Calculate the mean and population standard deviation of $4,7,7,8,9$ . Give the standard deviation to $3$ significant figures.

Tier 2 · Standard

3 marks

ORIGINAL

For $20$ observations, $\sum x=310$ and $\sum x^2=5020$ . Calculate the population standard deviation to $3$ significant figures.

Tier 3 · Hard

5 marks

ORIGINAL

Group A has $12$ values with mean $18$ and $\sum x^2=3996$ . Group B has $8$ values with mean $24$ and $\sum x^2=4736$ . Find the mean and population standard deviation of all $20$ values.

S2.4

Recognise and interpret possible outliers in data sets and statistical diagrams; select or critique data presentation techniques in context; clean data, including dealing with missing data, errors and outliers.

A common outlier rule flags values below $Q_1-1.5\operatorname{IQR}$ or above $Q_3+1.5\operatorname{IQR}$ , but context should guide the final decision.
Investigate a suspicious value against the original record before correcting or removing it; an unusual valid observation is not automatically an error.
Handle missing data transparently: record how many values are missing, avoid inventing unsupported values and consider whether missingness could bias conclusions.
Choose displays to suit the data and purpose: histograms for continuous grouped data, box plots for comparing distributions, and scatter diagrams for paired variables.

Tier 1 · Easy

2 marks

ORIGINAL

For a data set, $Q_1=14$ and $Q_3=22$ . Use the $1.5\operatorname{IQR}$ rule to determine whether the value $36$ is a possible outlier.

Tier 2 · Standard

4 marks

ORIGINAL

A table of package masses contains one blank entry and one value $482$ among values near $48.2$ grams. Describe a defensible way to clean these two entries before analysis.

Tier 3 · Hard

5 marks

ORIGINAL

A data set of $40$ readings was summarised as $\sum x=504$ and $\sum x^2=6856.9$ . One reading was entered as $31$ but the source record confirms it should be $13$ . Calculate the corrected mean and population standard deviation, and state the likely effect of the error on the original spread.