Statistics

Section S

6 spec leafs

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

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Infer properties of populations or distributions from a sample, while knowing the limitations of sampling

A population is the whole group of interest and a sample is the smaller group from which data are collected.
Use a representative sample proportion or mean to estimate the corresponding population value, scaling by the population size when needed.
For example, if $18$ of a random sample of $60$ have a feature, estimate that $18/60=30\%$ of the population has it.
An inference is an estimate, not a certainty; small samples, non-response, convenience sampling and under-coverage can all make it unreliable.

Tier 1 · Easy

2 marks

ORIGINAL

In a random sample of $50$ library users, $30$ prefer later opening. Estimate how many of the library's $1000$ users prefer later opening.

Tier 2 · Standard

3 marks

ORIGINAL

A random sample of $80$ items from a production run of $800$ contains $18$ items with a surface mark. Estimate the number in the whole run with a surface mark and state one limitation of the estimate.

Tier 3 · Hard

4 marks

ORIGINAL

A service invites a random sample of $450$ customers to answer a survey. Only $270$ reply, and $189$ of the replies support a change. Use the replies to estimate the number of supporters among all $12000$ customers, then explain a serious limitation.

Interpret and construct frequency tables, bar charts, pie charts and pictograms (categorical data), vertical line charts (ungrouped discrete numerical data), tables and line graphs (time series)

Choose a display to match the data: separated bars for categories, a vertical line chart for discrete numerical values, and joined time-ordered points for a time series.
For a pie chart, calculate sector angle using $\text{angle}=\dfrac{\text{category frequency}}{\text{total frequency}}\times360^\circ$ .
A correct graph uses a labelled, evenly spaced scale and plots each frequency or value accurately; bar widths and gaps should be consistent.
Do not join unrelated categories with lines, and do not interpret a truncated or uneven axis as though visual lengths were proportional.

Tier 1 · Easy

2 marks

ORIGINAL

The data are $2,3,2,5,4,3,2,4,5,2$ . Construct a frequency table for the values $2$ , $3$ , $4$ and $5$ .

Tier 2 · Standard

2 marks

ORIGINAL

In a survey of $240$ journeys, $54$ are made by bicycle. Work out the angle of the bicycle sector in a pie chart.

Tier 3 · Hard

4 marks

ORIGINAL

Quarterly sales are Q₁: $320$ , Q₂: $410$ , Q₃: $390$ , Q₄: $520$ , and the following Q₁: $350$ . State the five points to plot on a time-series line graph. Work out the percentage change from the first Q₁ to the following Q₁ and explain why comparing Q₄ directly with the following Q₁ could mislead.

Construct and interpret diagrams for grouped discrete and continuous data, i.e. histograms with equal and unequal class intervals and cumulative frequency graphs [Higher only]

In a histogram, bar area represents frequency and frequency density is $\dfrac{\text{frequency}}{\text{class width}}$ .
For unequal class intervals, calculate every class width from its boundaries before finding bar heights; a greater height means greater density, not necessarily greater frequency.
A cumulative frequency graph plots upper class boundaries against running totals and can estimate the median at $n/2$ and quartiles at $n/4$ and $3n/4$ .
Do not use frequency as histogram height unless all class widths are equal, and do not read quartiles from the ordinary frequency axis.

Tier 1 · Easy

2 marks

ORIGINAL

A histogram class is $15<x\leq20$ and has frequency $20$ . Work out the frequency density.

Tier 2 · Standard

3 marks

ORIGINAL

Grouped data have classes $0<x\leq10$ , $10<x\leq25$ and $25<x\leq40$ with frequencies $12$ , $30$ and $18$ . Calculate the three frequency densities and identify the tallest histogram bar.

Tier 3 · Hard

5 marks

ORIGINAL

A cumulative frequency graph for $80$ values passes through $(10,8)$ , $(20,26)$ , $(30,50)$ , $(40,70)$ and $(50,80)$ . Use linear interpolation between the given points to estimate the median and the interquartile range.

Interpret, analyse and compare data-set distributions via graphs (incl. box plots), central tendency (median, mean, mode, modal class) and spread (range, outliers, quartiles, inter-quartile range)

Central tendency describes a typical value: use the mean for all values, the median for the ordered middle, and the mode or modal class for the most frequent outcome.
Spread describes variation: range is maximum minus minimum. At Higher tier, box plots also show quartiles and the interquartile range $Q_3-Q_1$ .
Compare distributions in context using one statement about centre and one about spread, such as a higher median together with a smaller range.
Do not conclude that one set is 'better' from spread alone, and remember that outliers can strongly affect the mean and range.

Tier 1 · Easy

2 marks

ORIGINAL

For the data $4,6,6,9,10$ , work out the mean and the range.

Tier 2 · Standard

2 marks

ORIGINAL

Data set A has median $24$ and range $33$ . Data set B has median $25$ and range $29$ . Compare the two distributions.

Tier 3 · Hard

4 marks

ORIGINAL

The data are $12,13,13,14,15,15,16,52$ . Work out the mean, median and range. Decide which of the mean or median better describes a typical value for these data. Give a reason.

Apply statistics to describe a population

Population statistics summarise a whole group, while sample statistics can be scaled or weighted to estimate population features.
For groups of different sizes, use a weighted mean: add each group size multiplied by its mean, then divide by the total population.
To estimate a population count, apply the appropriate sample proportion to each population subgroup before adding the estimates.
Do not average group means without accounting for group sizes, and do not report an estimate with unjustified precision.

Tier 1 · Easy

2 marks

ORIGINAL

A representative sample of parcels has mean mass $2.4\,\text{kg}$ . Estimate the total mass of $250$ parcels in the population.

Tier 2 · Standard

3 marks

ORIGINAL

A population contains $120$ junior members with mean attendance $6.5$ sessions and $80$ senior members with mean attendance $8$ sessions. Work out the mean attendance for the whole population.

Tier 3 · Hard

5 marks

ORIGINAL

A town has $1800$ residents in the north and $1200$ in the south. In representative samples, $15$ of $60$ northern residents and $20$ of $50$ southern residents cycle to work. Estimate the total number and percentage of the town's residents who cycle to work.

Use and interpret scatter graphs of bivariate data; recognise correlation, know it does not indicate causation; draw estimated lines of best fit; make predictions; interpolate/extrapolate with caution

A scatter graph shows paired values: upward patterns indicate positive correlation, downward patterns negative correlation, and no clear pattern no correlation.
Draw a line of best fit through the centre of the points with a roughly balanced spread above and below, ignoring an outlier only when justified.
Use the line to estimate one variable from the other; interpolation stays within the observed range, while extrapolation extends beyond it and is less reliable.
Correlation does not prove causation because a third variable, reverse causation or coincidence may explain the association.

Tier 1 · Easy

2 marks

ORIGINAL

A scatter graph shows that as daily temperature increases, ice-cream sales usually increase. State the type of correlation and explain why the graph alone does not prove that temperature is the only cause of higher sales.

Tier 2 · Standard

3 marks

ORIGINAL

For data with observed $x$ -values from $5$ to $20$ , a line of best fit is $y=1.8x+6$ . Estimate $y$ when $x=14$ . Explain why using the line at $x=35$ is less reliable.

Tier 3 · Hard

5 marks

ORIGINAL

A scatter graph compares a puppy's age $a$ months with mass $m$ kg for ages from $2$ to $12$ months. Its estimated line of best fit is $m=0.42a+1.8$ . Estimate the mass at $9$ months and at $16$ months. Comment on the reliability of both estimates and on whether the graph proves that age alone causes the change in mass.