S1, S2 - Samples, tables and charts | AQA GCSE Mathematics

Use a sample to learn about a larger group, while keeping its limits in view. Organise data in frequency and two-way tables, choose suitable charts, and construct and interpret bar charts, pictograms, vertical line charts, pie charts and time-series graphs.

Samples and populations

A school has $800$ pupils. Of $40$ pupils asked, $26$ want a later lunch. Is that enough to say that about $520$ pupils want it? The arithmetic is easy. Whether the answer is useful depends on who was asked.

You already know

A relative frequency is the number of times an outcome occurs divided by the number of trials. Fractions and percentages can scale a proportion from one total to another.

Asking every pupil would be a census. Asking some saves time, but their views may not represent everyone else's. Asking only pupils still in the lunch queue is especially risky: they may have a particular reason to want lunch moved.

Population and sample

The population is the whole group about which you want information. A sample is the smaller group actually investigated. A sample is biased if its selection systematically favours some members or views over others. For example, surveying a sports club about the whole school's favourite activity is likely to over-represent sport.

Choose pupils from a complete list using a random selection, rather than choosing friends or volunteers. A larger unbiased sample usually gives a more reliable estimate, but it still has sampling variation. Doubling a biased sample does not repair its selection. Even asking every member of one class leaves the other classes out.

Estimating a population count

\text{estimated count}=\frac{\text{sample count}}{\text{sample size}}\times\text{population size}

Use this when the sample reasonably represents the population. The result is an estimate, not a guaranteed count.

Not on the formulae sheet: learn it.

First separate a calculation from the assumption that makes it useful.

Worked example 1: estimating from a random sample

Question. A random sample of $40$ out of $800$ pupils contains $26$ who want a later lunch. Estimate how many pupils want a later lunch. Non-calculator.

Plan. Apply the sample fraction to the whole school.

\begin{aligned} \text{sample fraction}&=\frac{26}{40} &&\text{use all sampled pupils}\\ \text{estimated count}&=\frac{26}{40}\times800 &&\text{scale to the population}\\ &=26\times20=520 &&\text{each sampled pupil represents 20} \end{aligned}

Why this step? The fraction describes support, while $800$ supplies the size of the group being estimated.

Answer. About $520$ pupils, assuming the random sample represents the school reasonably well.

Check. $26$ is more than half of $40$ ; $520$ is more than half of $800$ .

Change only the selection method. The same arithmetic can now hide a serious weakness.

Worked example 2: the same figures from a biased sample

Question. The $40$ pupils are instead the first $40$ people at the lunch queue. Again $26$ want a later lunch. Work out the estimate and explain one limitation. Non-calculator.

Plan. Calculate as before, then connect the limitation to the people selected.

\begin{aligned} \text{estimated count}&=\frac{26}{40}\times800 &&\text{same observed proportion}\\ &=520 &&\text{same numerical estimate} \end{aligned}

Why this step? A correct calculation cannot remove bias in the data that went into it.

Answer. The estimate is $520$ . It may be unreliable because early queue users may have different lunch preferences from pupils who arrive later or do not queue.

Check. Both surveys give $65\%$ , but only the first deliberately gave all pupils a chance of selection.

Here is a useful surprise: two surveys can produce exactly the same table and still deserve different levels of trust. A chart cannot show a selection bias that was never recorded.

Watch out

Wrong: “The sample is unreliable because it does not include everyone.” That describes every sample. Right: identify a particular missing or over-represented group and explain why its responses might differ.

Keep the inference and its limitation together in the following question.

Frequency tables, bar charts and pictograms

A list of favourite lunch choices is hard to scan. A frequency table counts each category once, then a chart makes the comparison visible.

Frequency and categorical data

Frequency means how many times a value or category occurs. Categorical data are labels, such as soup, pasta and salad. Their order need not have a numerical meaning. A pupil's height is numerical data, not categorical data.

For the responses soup, pasta, salad, pasta, soup, pasta, the frequencies are soup $2$ , pasta $3$ , salad $1$ . Tally each response in one row. The total frequency must match the number of responses.

A bar chart has equal-width bars with gaps, labelled categories and a frequency axis starting at zero. Categories can be reordered. A pictogram uses equal-sized symbols and a key; a half symbol means half the key's value. Do not enlarge a symbol to show a larger count, because its area then misleads.

[DIAGRAM: asset_name: Bar chart and pictogram of lunch choices; asset_slug: aqa8300_s01_bar_pictogram; caption: The same frequencies can be shown by bar heights or by symbols with a key; description: Left panel has separate equal-width bars for soup 12, pasta 18 and salad 6 with a frequency scale from zero. Right panel uses a full circle for four pupils and a half-circle for two, giving three circles for soup, four and a half for pasta and one and a half for salad.]
The same frequencies can be shown by bar heights or by symbols with a key

Choose a bar chart for comparing several categories accurately; a pictogram is friendly for a few counts when a simple key works. These chart conventions are not on the formulae sheet: learn them.

Worked example 3: constructing a bar chart

Question. A frequency table gives soup $12$ , pasta $18$ , salad $6$ . Describe how to draw its bar chart and work out the total. Non-calculator.

Plan. Choose a frequency scale covering the largest value, then draw three separate bars.

\begin{aligned} \text{largest frequency}&=18 &&\text{choose a scale reaching 20}\\ \text{bar heights}&=12,18,6 &&\text{copy the table accurately}\\ \text{total}&=12+18+6=36 &&\text{add all categories} \end{aligned}

Why this step? A zero baseline lets the heights show the actual sizes, rather than exaggerating differences.

Answer. Label the horizontal categories and vertical axis “Frequency”. Use equal-width, separated bars of heights $12$ , $18$ , $6$ . There are $36$ responses.

Check. Pasta's bar is three times salad's height, matching $18=3\times6$ .

Change only the representation: keep the frequencies, but use a pictogram key.

Worked example 4: constructing a pictogram

Question. Show the same frequencies using one full circle for $4$ pupils. Non-calculator.

Plan. Divide each frequency by the value of one symbol.

\begin{aligned} \text{soup symbols}&=12\div4=3 &&\text{use the key}\\ \text{pasta symbols}&=18\div4=4.5 &&\text{include a half-symbol}\\ \text{salad symbols}&=6\div4=1.5 &&\text{include a half-symbol} \end{aligned}

Why this step? Half a circle means $2$ pupils because the full circle means $4$ .

Answer. Draw $3$ , $4\frac12$ and $1\frac12$ equal-sized circles in the three labelled rows, and write the key.

Check. The $9$ full-symbol equivalents represent $9\times4=36$ pupils.

Always read the key before counting pictures.

Watch out

Wrong: $4\frac12$ symbols means $4.5$ pupils. Right: with a key of $4$ , it means $4.5\times4=18$ pupils. It is the symbol count, not the pupil count, that can include halves.

The next key has a different value, so a half-symbol represents a different count.

Vertical line charts for discrete data

A number of pets can be $0$ , $1$ , $2$ or $3$ . It cannot be $1.4$ . The numerical positions matter, so leave their spacing intact.

Discrete numerical data

Numerical data taking separate possible values, often counts. A vertical line chart shows each value at its correct position on a numerical axis, with a line whose height is its frequency. For example, number of pets is discrete; measured mass is continuous.

Include a missing value's position even if its frequency is zero. Unlike a categorical bar chart, moving “three pets” to the left of “one pet” would destroy the scale. Do not join the tops: the lines show frequencies at separate values, not a change through time.

[DIAGRAM: asset_name: Vertical line chart for pets; asset_slug: aqa8300_s01_vertical; caption: Each possible pet count keeps its numerical position, including zero frequency at three pets; description: Numerical horizontal axis shows 0 to 4 pets at equal spacing, frequency axis starts at zero. Vertical lines have heights 5, 8, 4, 0 and 3 respectively, with an open marker at the zero-height value three.]
Each possible pet count keeps its numerical position, including zero frequency at three pets

The method is a plotting convention, not on the formulae sheet: learn it.

Worked example 5: a numerical frequency chart

Question. Pet counts $0,1,2,3,4$ have frequencies $5,8,4,0,3$ . Describe the vertical line chart and work out how many people have at least $2$ pets. Non-calculator.

Plan. Plot each frequency above its number, then add the qualifying frequencies.

\begin{aligned} \text{plotted heights at }0,1,2,3,4&=5,8,4,0,3 &&\text{keep numerical spacing}\\ \text{at least 2 pets}&=4+0+3=7 &&\text{include 2, 3 and 4} \end{aligned}

Why this step? “At least” includes the boundary value $2$ .

Answer. Draw the vertical lines shown. There are $7$ people with at least $2$ pets.

Check. There are $20$ people in total and $13$ with fewer than $2$ ; $20-13=7$ .

Change only the boundary: “more than” excludes the named value.

Worked example 6: reading a strict boundary

Question. Using the same table, work out how many people have more than $2$ pets. Non-calculator.

Plan. Add only frequencies above $2$ on the horizontal axis.

\begin{aligned} \text{allowed counts}&=3,4 &&\text{exclude exactly 2}\\ \text{number of people}&=0+3=3 &&\text{add frequencies} \end{aligned}

Why this step? The horizontal values are pet counts; the heights are people. Add heights, not $3+4$ .

Answer. $3$ people.

Check. Removing the $4$ people with exactly $2$ pets from the previous $7$ leaves $3$ .

Read both axes before using a chart.

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S1, S2 - Samples, tables and charts

Samples and populations

Frequency tables, bar charts and pictograms

Vertical line charts for discrete data

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