Maths · 10 Statistics

10.6 Statistical charts and diagrams

Before you start

Counting data values into a tally table (10.1).
The difference between categorical, discrete and continuous data (10.3).
Finding a fraction of an amount, such as $\frac{3}{8}$ of $40$ .
A full turn is $360 °$ , and measuring or drawing an angle with a protractor to the nearest degree.
Finding the median, mode and range of a list of values (10.4).

Syllabus objectives covered (1)

Draw and interpret: (a) bar charts (b) pie charts (c) pictograms (d) stem-and-leaf diagrams (e) simple frequency distributions.

Part 1 of 7

Frequency tables: where every chart starts

A frequency distribution lists every value (or category) the data can take, next to its frequency: the number of times that value occurs. Every chart in this lesson is a picture of one, so you build the table first and draw from it.
To build one from a raw list, go through the list once, in order. For each value, put a tally mark in its row of the table and cross the value off the list. Crossing off is what stops you counting a value twice or missing one.
Write tallies in groups of five, four strokes with a fifth across them, so you can count them quickly. Then write each count in the frequency column.
Twenty students were asked how many brothers and sisters they have. The answers were $1, 2, 0, 1, 3, 1, 2, 0, 1, 1, 2, 4, 0, 1, 2, 1, 3, 0, 1, 2$ . Tallying them gives this frequency table.
$\begin{matrix} Number of siblings & 0 & 1 & 2 & 3 & 4 \\ Frequency & 4 & 8 & 5 & 2 & 1 \end{matrix}$
Check the table before you use it: the frequencies must add up to the number of data values. Here $4 + 8 + 5 + 2 + 1 = 20$ , which matches the $20$ students. If the total is wrong, every chart drawn from the table is wrong too.
Keep the two rows apart in your head. The top row is the value (number of siblings); the bottom row is how many students have that value. The modal number of siblings is $1$ , the value with the highest frequency. It is not $8$ : that is the frequency.
Read the wording before you add frequencies. 'At least $2$ siblings' means $2$ , $3$ or $4$ , so $5 + 2 + 1 = 8$ students. 'More than $2$ ' means $3$ or $4$ only, so $2 + 1 = 3$ students. The value $2$ is in the first count but not the second.
The total number of siblings is not the total number of students. Each of the $5$ students with $2$ siblings has $2$ , so multiply each value by its frequency and add.
$0 \times 4 + 1 \times 8 + 2 \times 5 + 3 \times 2 + 4 \times 1 = 28 siblings$
When discrete data have many different values, group them into classes such as $0$ – $4$ , $5$ – $9$ , $10$ – $14$ , and give each class a frequency. Continuous data need classes written as inequalities, such as $10 \leq t < 20$ , so that no value falls into a gap (10.3).

Part 2 of 7

Bar charts: drawing them and reading them

A bar chart shows each category's frequency as the height (or length) of a bar. Use it for categorical data, such as favourite colour, or for discrete data with a few values, such as number of siblings.
Draw every bar the same width. Only the height carries the frequency, so a wider bar would look bigger without being bigger.
Leave equal gaps between the bars. The gaps show that the categories are separate: there is nothing between '2 siblings' and '3 siblings'.
Start the frequency axis at $0$ and go up in equal steps. If the axis started at $4$ , the bar for frequency $8$ would be $4$ units tall and the bar for frequency $5$ would be $1$ unit tall, so $8$ would look four times as big as $5$ .
Choose a scale that is easy to read, such as $1$ , $2$ , $5$ or $10$ for each labelled step, and that lets the chart fill most of the grid. Label the horizontal axis with the categories and the vertical axis 'Frequency'.
graph diagram — not drawn yet
A vertical bar chart of the siblings data. Horizontal axis labelled 'Number of siblings' with categories 0, 1, 2, 3, 4. Vertical axis labelled 'Frequency', from 0 to 9 in steps of 1. Five bars of equal width with equal gaps between them, reaching heights 4, 8, 5, 2 and 1.
Equal widths, equal gaps, a frequency axis that starts at 0, and both axes labelled. Only the heights differ.
To read a bar chart, find what one small square is worth before you read any bar. If the labels go $0, 10, 20, 30$ and there are $5$ small squares between two labels, each small square is $10 \div 5 = 2$ , not $1$ .
Then read across from the top of the bar to the frequency axis. A bar chart can be drawn with horizontal bars instead; you read it the same way, from the end of the bar to the axis.

Part 3 of 7

Dual and composite bar charts: two groups on one chart

Sometimes each category is split into two groups. A school records how many boys and how many girls are in each club.
$\begin{matrix} Chess & Drama & Football & Music \\ Boys & 9 & 5 & 14 & 7 \\ Girls & 6 & 12 & 8 & 10 \end{matrix}$
There are two ways to show both groups on one chart. Both need a key that says which shading is boys and which is girls. Without the key, nobody can read the chart.
A dual bar chart (side-by-side bar chart) gives each club two bars standing next to each other, one for boys and one for girls. Leave a gap between one club's pair and the next club's pair.
A composite bar chart (stacked bar chart) gives each club one bar, split into sections. The boys' section goes from $0$ up to the boys' frequency, and the girls' section sits on top of it. For Drama, the boys' section goes from $0$ to $5$ and the girls' section goes from $5$ to $5 + 12 = 17$ .
graph diagram — not drawn yet
Two bar charts side by side with the same vertical 'Frequency' axis from 0 to 24 and the clubs Chess, Drama, Football, Music along the bottom. Left, labelled 'Dual bar chart': for each club a light bar for boys (9, 5, 14, 7) touching a dark bar for girls (6, 12, 8, 10), with a gap between clubs. Right, labelled 'Composite bar chart': one bar per club, light boys' section from 0 to 9, 5, 14, 7 and dark girls' section stacked on top, ending at 15, 17, 22, 17. A key shows light = boys, dark = girls.
The same data twice. In the dual chart every bar starts at 0; in the composite chart the girls' sections start where the boys' sections stop, and the top of each bar is the club's total.
The height of a whole composite bar is the total for that club. Drama's bar stops at $17$ because $17$ students are in the Drama club.
The trap is a section that does not start at $0$ . The top of Drama's girls' section is at $17$ , but $17$ is not the number of girls. A section's frequency is the value at its top minus the value at its bottom.
$17 - 5 = 12 girls$
Choose the chart for the comparison you need. In a dual bar chart both bars start at $0$ , so comparing boys with girls in each club is easy. In a composite bar chart the whole bars show the totals, so seeing which club has the most members is easy: Football, with $22$ .

Part 4 of 7

Pictograms: symbols and a key

A pictogram shows frequencies as rows of identical symbols. The key tells you what one whole symbol stands for, for example: one symbol stands for $4$ books.
To read a row, count the whole symbols and multiply by the key, then add any part symbol. With one symbol for $4$ books, three and a half symbols stand for $3 \times 4 + 2 = 14$ books.
A part symbol stands for the same fraction of the key. Half a symbol is half of $4$ , which is $2$ books. Do not count it as $1$ book just because it is one piece.
To draw a row, divide the frequency by the key. For $9$ books, $9 \div 4 = 2.25$ , so draw $2$ whole symbols and a quarter of a symbol.
Draw every symbol the same size and line the symbols up in columns. Then a longer row really does mean a bigger frequency, and you can compare rows at a glance.
graph diagram — not drawn yet
A pictogram titled 'Books read this term' with four rows labelled Amir, Bea, Chen, Dina. Each symbol is a small book. Amir: 3 whole symbols and the left half of a fourth. Bea: 5 whole symbols. Chen: 2 whole symbols and the left quarter of a third. Dina: 3 whole symbols. Symbols are the same size and lined up in columns. Key below: one book symbol stands for 4 books.
Amir has $14$ , Bea $20$ , Chen $9$ and Dina $12$ books. Each part symbol is a clear fraction of a whole one, and the key is part of the chart.
If you choose the key yourself, choose a number that turns every frequency into whole symbols, halves or quarters. A key of $4$ suits the frequencies $14$ , $20$ , $9$ and $12$ . A key of $7$ would not: $20 \div 7$ is not a whole number, a half or a quarter.

Part 5 of 7

Pie charts: from frequencies to angles

A pie chart shows how one whole group is shared out. The whole group is the full circle, $360 °$ , and each category gets a sector whose angle is its share of $360 °$ .
So the angle of a sector is the category's fraction of the total frequency, multiplied by $360 °$ .
$angle = \frac{frequency}{total frequency} \times 360 °$
For a whole table there is a quicker route: work out how many degrees one item gets, then multiply each frequency by that. $72$ students were asked how they travel to school. $360 \div 72 = 5$ , so each student gets $5 °$ .
$\begin{matrix} Walk & Bus & Car & Cycle & Train & Total \\ Frequency & 26 & 19 & 15 & 8 & 4 & 72 \\ Angle & 130 ° & 95 ° & 75 ° & 40 ° & 20 ° & 360 ° \end{matrix}$
Check that the angles add up to $360 °$ before you draw anything. Here $130 + 95 + 75 + 40 + 20 = 360$ . A total that is not $360 °$ means an angle is wrong.
When an angle is not a whole number, round it to the nearest degree. Rounding can make the total come to slightly more or less than $360 °$ , such as $359 °$ or $361 °$ . A total that is several degrees out means a mistake, so find it before you draw.
To draw the chart, draw a circle with compasses and mark its centre. Draw one radius as a starting line. Put the centre of the protractor on the centre of the circle, with its zero line along the radius, and mark the first angle.
Draw the next radius through your mark. Measure the next angle from this new line, not from the starting line, and carry on round the circle the same way.
You do not need to measure the last sector: it is what is left. Measure it anyway, as a check. If it is not close to the angle in your table, one of the earlier angles was drawn wrongly.
Label every sector with its category, or shade the sectors and give a key. A pie chart without labels shows nothing.
graph diagram — not drawn yet
A pie chart of how 72 students travel to school, starting from a vertical radius and going clockwise: Walk 130°, Bus 95°, Car 75°, Cycle 40°, Train 20°. Each sector is labelled with its category and its angle.
Each angle is measured from the radius drawn just before it. The sectors fill exactly $360 °$ , and every one is labelled.

Part 6 of 7

Reading a pie chart

Reading a pie chart reverses the calculation: a sector's frequency is its angle as a fraction of $360 °$ , multiplied by the total frequency.
$frequency = \frac{angle}{360 °} \times total frequency$
In the travel pie chart, the bus sector is $95 °$ and the total is $72$ students.
$\frac{95}{360} \times 72 = 19 students$
Sometimes the total is not given, but one sector's angle and frequency are. Use that sector to find the degrees for one item, then divide $360 °$ by it. If the cycle sector is $40 °$ and stands for $8$ students, one student is $40 \div 8 = 5 °$ , so the total is $360 \div 5 = 72$ students.
If the angles are not written on the chart, measure them with a protractor, to the nearest degree, before you calculate.
A pie chart shows proportions, not amounts. A big sector means a big share of that group. Two pie charts can be the same size even when one group is much bigger than the other.
Suppose a second school has $180$ students and its walk sector is $60 °$ , smaller than the first school's $130 °$ . Yet $\frac{60}{360} \times 180 = 30$ students walk at the second school, more than the $26$ at the first. To compare numbers from two pie charts, work out the frequencies first.
So use a pie chart to answer 'what fraction?' and a bar chart to answer 'how many?'. A pie chart also needs categories that make up one whole, with each person in exactly one sector.

Part 7 of 7

Stem-and-leaf diagrams: ordered, with a key

A stem-and-leaf diagram shows the shape of a data set, like a bar chart on its side, but it keeps every individual value. Split each value into a stem (the leading digit or digits) and a leaf (the last digit). For $47$ , the stem is $4$ and the leaf is $7$ .
Write the stems in a column, smallest at the top, with a vertical line to their right. Include every stem in the range, even one with no leaves: an empty row shows a gap in the data.
First pass: go through the data in the order given, writing each leaf in its stem's row and crossing the value off. Here are twelve test marks out of $60$ : $34, 52, 40, 38, 45, 29, 43, 56, 33, 47, 41, 30$ .
$\begin{matrix} 2 & 9 \\ 3 & 4 8 3 0 \\ 4 & 0 5 3 7 1 \\ 5 & 2 6 \end{matrix}$
Second pass: rewrite each row with its leaves in order, smallest next to the stem. The syllabus asks for an ordered diagram, so the first pass is rough work, not the answer.
$\begin{matrix} 2 & 9 \\ 3 & 0 3 4 8 \\ 4 & 0 1 3 5 7 \\ 5 & 2 6 \end{matrix} Key: 3 | 4 means 34 marks$
Always add a key, such as 'Key: $3 | 4$ means $34$ marks'. Without it, $3 | 4$ could mean $34$ , $3.4$ or $340$ . The key is part of the diagram.
Write each leaf as a single digit, evenly spaced so the leaves line up in columns. Then the length of each row shows how many values it holds, as a bar would.
Count the leaves to get the number of values: $1 + 4 + 5 + 2 = 12$ . Because the diagram is ordered, you can count along to any position. The median is at position $\frac{12 + 1}{2} = 6.5$ , halfway between the $6$ th value ( $40$ ) and the $7$ th value ( $41$ ), so the median is $40.5$ marks.
The smallest value is the first leaf in the top row, $29$ . The largest is the last leaf in the bottom row, $56$ . The range is $56 - 29 = 27$ marks.
The key handles decimals too. For masses such as $4.7$ kg, the stem is $4$ , the leaf is $7$ , and the key reads 'Key: $4 | 7$ means $4.7$ kg'.
Two data sets can share one column of stems in a back-to-back stem-and-leaf diagram: one set's leaves go to the right, the other set's to the left. On the left, the leaves are ordered outwards from the stem, so they increase from right to left. With set A on the left and set B on the right, a key such as ' $3 | 4 | 1$ means $43$ for set A and $41$ for set B' explains both sides.

Worked examples

Example 1. A café records the drinks it sells in one hour: coffee $14$ , tea $11$ , juice $6$ , water $5$ , hot chocolate $4$ . Calculate the angle of each sector for a pie chart of these data, and describe how to draw the chart.

Show the working

Find the total first, because every angle is a share of it: $14 + 11 + 6 + 5 + 4 = 40$ drinks.
Work out the degrees for one drink.
$\frac{360 °}{40} = 9 ° per drink$
Multiply each frequency by $9 °$ .
$\begin{matrix} Coffee & Tea & Juice & Water & Hot chocolate \\ Frequency & 14 & 11 & 6 & 5 & 4 \\ Angle & 126 ° & 99 ° & 54 ° & 45 ° & 36 ° \end{matrix}$
Check the total: $126 + 99 + 54 + 45 + 36 = 360$ . The angles are right, so you can draw.
Draw a circle and one radius. Measure $126 °$ from the radius for coffee and draw the second radius. From that line measure $99 °$ for tea, then $54 °$ for juice, then $45 °$ for water.
The sector left over is hot chocolate. Measure it as a check: it should be $36 °$ . Then label every sector.

Answer: Coffee $126 °$ , tea $99 °$ , juice $54 °$ , water $45 °$ , hot chocolate $36 °$ (total $360 °$ ). Draw them one after another, each from the radius before it, and label every sector.

Example 2. A pie chart shows the favourite sports of a group of students. The swimming sector has an angle of $84 °$ and represents $21$ students. The tennis sector has an angle of $60 °$ . (a) Find the total number of students. (b) Find the number of students whose favourite sport is tennis. (c) At another school, a pie chart for $150$ students has a tennis sector of $48 °$ . Determine whether more students chose tennis at the second school.

Show the working

(a) The total is not given, so use the sector you know everything about. $84 °$ stands for $21$ students, so one student is $84 \div 21 = 4 °$ .
The whole circle is $360 °$ , so the total is $360 \div 4 = 90$ students.
(b) One student is $4 °$ , so the tennis sector stands for $60 \div 4 = 15$ students. Check with the formula: $\frac{60}{360} \times 90 = 15$ .
(c) Do not compare the angles, because the two schools have different totals. Work out the number of students at the second school.
$\frac{48}{360} \times 150 = 20$
$20 > 15$ , so more students chose tennis at the second school, even though its tennis sector is smaller. The second school is bigger, so a smaller share of it is still more students.

Answer: (a) $90$ students (b) $15$ students (c) Yes: $20$ students chose tennis at the second school, compared with $15$ .

Example 3. The times, in minutes, that $15$ people spent in a gym are $23, 41, 35, 28, 19, 36, 42, 30, 27, 35, 18, 44, 31, 25, 35$ . (a) Draw an ordered stem-and-leaf diagram, with a key, for these times. (b) Find the median, the range and the mode. (c) Find how many people spent more than $30$ minutes in the gym.

Show the working

(a) The times go from the teens to the forties, so the stems are the tens digits $1, 2, 3, 4$ and the leaves are the units digits.
First pass, in the order given, crossing off each time as you write it:
$\begin{matrix} 1 & 9 8 \\ 2 & 3 8 7 5 \\ 3 & 5 6 0 5 1 5 \\ 4 & 1 2 4 \end{matrix}$
Count the leaves: $2 + 4 + 6 + 3 = 15$ . That matches the $15$ people, so no time was missed or written twice.
Second pass: put each row in order and add the key.
$\begin{matrix} 1 & 8 9 \\ 2 & 3 5 7 8 \\ 3 & 0 1 5 5 5 6 \\ 4 & 1 2 4 \end{matrix} Key: 2 | 3 means 23 minutes$
(b) The median is at position $\frac{15 + 1}{2} = 8$ . Count along: the top row holds the $1$ st and $2$ nd values and the second row the $3$ rd to $6$ th, so the $7$ th value is $30$ and the $8$ th is $31$ . The median is $31$ minutes.
The range is the largest minus the smallest: $44 - 18 = 26$ minutes. The leaf $5$ appears three times in the $3$ row, more often than any other value, so the mode is $35$ minutes.
(c) 'More than $30$ ' does not include $30$ . In the $3$ row, the leaves after the $0$ are $1, 5, 5, 5, 6$ , which is $5$ people. The $4$ row adds $3$ more: $5 + 3 = 8$ people.

Answer: (a) $1 | 8 9$ ; $2 | 3 5 7 8$ ; $3 | 0 1 5 5 5 6$ ; $4 | 1 2 4$ , with Key: $2 | 3$ means $23$ minutes. (b) Median $31$ minutes, range $26$ minutes, mode $35$ minutes. (c) $8$ people.

Example 4. A shop sold these numbers of T-shirts at the weekend. Small: $12$ on Saturday, $8$ on Sunday. Medium: $18$ on Saturday, $21$ on Sunday. Large: $10$ on Saturday, $15$ on Sunday. (a) A composite bar chart is drawn with one bar for each size, with Saturday at the bottom. Write down where each section starts and stops on the frequency axis. (b) On the same chart, the extra-large bar has a Saturday section that stops at $9$ and a Sunday section that stops at $16$ . Find the number of extra-large T-shirts sold on Sunday. (c) Explain which kind of bar chart shows more clearly, for each size, whether more were sold on Saturday or on Sunday.

Show the working

(a) Each Saturday section starts at $0$ and goes up to Saturday's frequency. Small: $0$ to $12$ . Medium: $0$ to $18$ . Large: $0$ to $10$ .
Each Sunday section starts where Saturday's stops and adds Sunday's frequency. Small: $12$ to $12 + 8 = 20$ . Medium: $18$ to $18 + 21 = 39$ . Large: $10$ to $10 + 15 = 25$ .
The top of each bar is the weekend total for that size: $20$ , $39$ and $25$ .
(b) The Sunday section sits on top of the Saturday section, so it starts at $9$ , not at $0$ . Its frequency is its top minus its bottom: $16 - 9 = 7$ .
(c) A dual bar chart. There, the Saturday and Sunday bars for a size both start at $0$ and stand side by side, so you compare their heights directly. In the composite chart the Sunday sections start at different heights, so you would have to subtract before you could compare.

Answer: (a) Small: Saturday $0$ – $12$ , Sunday $12$ – $20$ . Medium: Saturday $0$ – $18$ , Sunday $18$ – $39$ . Large: Saturday $0$ – $10$ , Sunday $10$ – $25$ . (b) $7$ T-shirts (c) A dual bar chart: both bars for a size start at $0$ , side by side, so their heights compare directly.

Where people go wrong

Mistake: Reading the top of a section in a composite bar chart as that group's frequency.

Instead: A section's frequency is the value at its top minus the value at its bottom.

Only the bottom section starts at $0$ . Every section above it starts where the one below stops, so the top of the highest section is the total for the whole bar.

Mistake: Deciding which of two groups has more in a category by comparing the angles on their pie charts.

Instead: Work out each frequency as $\frac{angle}{360} \times total$ , then compare the numbers.

An angle shows a share of that group's own total. A small share of a big group can be more people than a big share of a small group.

Mistake: Leaving the leaves of a stem-and-leaf diagram in the order the data came, or leaving out the key.

Instead: Do a rough first pass, then rewrite every row in order from the stem outwards, and add a key with units.

The syllabus asks for ordered data with a key. Without the order you cannot count along to the median; without the key, $3 | 4$ could be $34$ or $3.4$ .

Mistake: Assuming that every small square on a bar chart's axis is worth $1$ .

Instead: Divide the step between two labels by the number of small squares between them.

With labels $0, 10, 20$ and $5$ small squares between labels, each square is $2$ . Reading it as $1$ makes every bar the wrong value.

Mistake: Giving the highest frequency as the mode, for example saying the modal number of siblings is $8$ .

Instead: The mode is the value with the highest frequency: $1$ sibling.

The frequency counts how many times a value occurs. The mode is the value itself.

Check yourself

In a pictogram, one symbol stands for $6$ cars. A row shows $4$ whole symbols and half a symbol. How many cars does the row stand for?
Answer
$4 \times 6 + 3 = 27$ cars. Half a symbol is half of $6$ , which is $3$ .
Using the same key (one symbol for $6$ cars), how many symbols should you draw for $15$ cars?
Answer
$15 \div 6 = 2.5$ , so $2$ whole symbols and half a symbol.
A dice is thrown $50$ times. The scores $1$ to $6$ have frequencies $7, 9, 8, 11, 6, 9$ . How many throws scored more than $4$ , and what is the modal score?
Answer
More than $4$ means $5$ or $6$ : $6 + 9 = 15$ throws. The modal score is $4$ , because it has the highest frequency ( $11$ ). The mode is $4$ , not $11$ .
The frequency axis of a bar chart is labelled $0, 10, 20, 30$ , with $5$ small squares between two labels. A bar stops $3$ small squares above the $20$ line. What frequency does it show?
Answer
Each small square is $10 \div 5 = 2$ , so the bar shows $20 + 3 \times 2 = 26$ .
In a composite bar chart of boys and girls, the girls' section sits on top of the boys' section. In one bar, the girls' section starts at $11$ and stops at $25$ . How many girls does it show, and what is the total for the bar?
Answer
$25 - 11 = 14$ girls. The top of the bar, $25$ , is the total.
$70$ people are surveyed and $23$ of them choose rugby. Calculate the angle of the rugby sector in a pie chart, to the nearest degree.
Answer
$\frac{23}{70} \times 360 = 118.28 \dots$ , so $118 °$ .
A pie chart shows the results for $120$ people. Find the number of people represented by a sector of $135 °$ .
Answer
$\frac{135}{360} \times 120 = 45$ people.
A stem-and-leaf diagram has the key 'Key: $5 | 3$ means $5.3$ kg'. Write down the values in the row $6 | 0 2 2 7$ .
Answer
$6.0$ kg, $6.2$ kg, $6.2$ kg and $6.7$ kg.

Exam questions

Try each part first. Open the mark scheme after you have written your answer.

Question 1. The table shows the main activity chosen by $90$ members of a sports centre. $\begin{matrix} Activity & Swimming & Gym & Yoga & Running & Cycling \\ Frequency & 27 & 24 & 9 & 18 & 12 \\ Angle \end{matrix}$

(a) Show that the angle for swimming in a pie chart of these data is $108 °$ . 2 marks No calculator

Mark scheme

M1 27 / 90 × 360 oe, or 360 / 90 = 4 seen
A1 108 reached from correct working with no errors (e.g. 27 × 4 = 108)

(b) Complete the angle row of the table. 2 marks No calculator

Mark scheme

B2 Swimming 108, Gym 96, Yoga 36, Running 72, Cycling 48. B1 for two or three of 96, 36, 72, 48 correct

(c) Draw a pie chart to show the data. Label each sector. 2 marks No calculator

Mark scheme

B2 all five sectors drawn within 2° of 108, 96, 36, 72, 48 and labelled; FT their angles from (b) if they total 360. B1 for at least three sectors correct, or all five correct but not labelled

(d) Lena looks at a pie chart for $40$ members of a different sports centre. Its swimming sector has an angle of $126 °$ . She says, 'More members chose swimming at the other sports centre, because its swimming sector is bigger.' Comment on Lena's statement. 3 marks No calculator

Mark scheme

M1 126 / 360 × 40 oe
A1 14 (members chose swimming at the other centre)
B1 Lena is wrong: 14 is fewer than 27, because the angle shows a share of a smaller total oe; FT their 14

Question 2. The pulse rates, in beats per minute, of $19$ students are: $72, 65, 88, 59, 70, 81, 76, 64, 92, 68, 75, 70, 83, 61, 77, 70, 58, 86, 73$

(a) Draw an ordered stem-and-leaf diagram to show these pulse rates. Include a key. 3 marks No calculator

Mark scheme

B2 5 | 8 9; 6 | 1 4 5 8; 7 | 0 0 0 2 3 5 6 7; 8 | 1 3 6 8; 9 | 2. B1 for all leaves correct but not in order, or in order with one error or omission
B1 a correct key, e.g. 6 | 4 means 64 (beats per minute)

(b) Find the median pulse rate. 1 mark No calculator

Mark scheme

B1 72; FT the 10th value of their ordered diagram

(c) Find the range of the pulse rates. 1 mark No calculator

Mark scheme

B1 34

Question 3. A composite bar chart shows how the students in Year 10 travel to school. The bar is split into three sections: bus at the bottom, walk in the middle and car at the top. The bus section goes from $0$ to $34$ , the walk section from $34$ to $52$ and the car section from $52$ to $61$ .

(a) Write down the number of Year 10 students who walk to school. 1 mark No calculator

Mark scheme

B1 18

(b) In Year 11, $29$ students travel by bus, $25$ walk and $14$ travel by car. A bar for Year 11 is added to the chart, with its sections in the same order. Write down the value on the frequency axis at the top of each of its three sections. 2 marks No calculator

Mark scheme

B2 bus 29, walk 54, car 68. B1 for two of these correct

(c) Explain why a dual bar chart would show more clearly whether more students walk in Year 10 or in Year 11. 1 mark No calculator

Mark scheme

B1 in a dual bar chart both walk bars start at 0, side by side, so their heights compare directly; in the composite chart the walk sections start at different heights (34 and 29) oe

Question 4. A pictogram shows the number of books a library lent on three days. Key: one symbol stands for $8$ books. Monday: $5$ whole symbols. Tuesday: $3$ whole symbols and a quarter of a symbol. Wednesday: not drawn yet.

(a) Find the number of books the library lent on Tuesday. 1 mark No calculator

Mark scheme

B1 26

(b) The library lent $30$ books on Wednesday. Write down how many symbols should be drawn in Wednesday's row. 1 mark No calculator

Mark scheme

B1 3¾ (3 whole symbols and three-quarters of a symbol) oe

How this is examined

Turning a frequency table into pie-chart angles, then drawing the sectors: a frequency table with an empty angle row: a Show (that) for one angle, a Complete for the other angles, then a Draw inside a circle For the Show (that): the calculation written in full, such as $\frac{27}{90} \times 360 = 108$ , or $360 \div 90 = 4$ and then $27 \times 4 = 108$ . Writing only $108$ shows nothing. For the table: angles that add up to $360 °$ . For the drawing: each sector measured from the radius before it, close to its angle, and every sector labelled.
Building an ordered stem-and-leaf diagram with a key, then reading the median and range from it: an unordered list of two-digit values to Draw as a diagram, followed by one-mark Find parts for the median, the range or the mode Every leaf present and in increasing order from the stem, every stem in the range listed, and a key with units, such as ' $6 | 4$ means $64$ beats per minute'. Count the leaves against the number of values before you move on. The median is at position $\frac{n + 1}{2}$ , counted along the ordered rows; the range is the last leaf of the bottom row minus the first leaf of the top row, with the key applied.
Getting frequencies back out of a pie chart, and why angles cannot compare groups of different sizes: a Find for a frequency or a total from given angles, or a Comment on someone who compares two pie charts by the size of their sectors The method written out: $\frac{angle}{360} \times total$ , or the degrees for one item found from a sector you know. For the Comment: the frequency for each group worked out, your verdict stated, and the reason: a pie chart shows a share of its own total, not a number.
Reading and building composite and dual bar charts: a Write down that reads one group from a stacked bar, or asks where each section of a new stacked bar starts and stops, which may be followed by an Explain about which chart is clearer A section's frequency as top minus bottom ( $52 - 34 = 18$ ), never the top value alone. When you stack, each section starts where the one below stops, and the top of the bar is the total. For 'which chart is clearer for comparing groups': a dual bar chart, because both bars start at $0$ side by side.
Reading scales, keys and part symbols correctly: a one-mark Find or Write down from a bar chart or a pictogram, including a row that ends in part of a symbol The value of one small square worked out from the axis labels before any bar is read. For a pictogram: whole symbols times the key, plus the part symbol as the same fraction of the key. With $8$ books per symbol, a quarter symbol is $2$ books. To draw a row, divide the frequency by the key: $30 \div 8 = 3 \frac{3}{4}$ symbols.
Making and reading a simple frequency table: a raw list to tally into a table, then a Find for how many values are 'at least', 'more than' or 'fewer than' a given value Frequencies that add up to the number of values. The boundary handled correctly: 'more than $2$ ' leaves out $2$ , while 'at least $2$ ' includes it. The mode given as the value with the highest frequency, not as the frequency.

Words to know

Frequency: The number of times a value or category occurs in a data set.
Frequency distribution: A table that lists each value, category or class with its frequency. Also called a frequency table.
Tally: A mark made for each data value as you count, written in groups of five.
Bar chart: A chart that shows each frequency as the height or length of a bar. The bars have equal widths and equal gaps.
Dual bar chart: A bar chart with two or more bars side by side for each category, one for each group. Also called a side-by-side bar chart.
Composite bar chart: A bar chart with one bar for each category, split into stacked sections for the groups. Also called a stacked bar chart. The top of the bar is the category's total.
Key: The part of a chart that says what a symbol, a shading or a stem and leaf stands for.
Pictogram: A chart that shows frequencies as rows of identical symbols, with a key giving the value of one symbol.
Pie chart: A circle divided into sectors, with each sector's angle showing that category's share of $360 °$ .
Sector: A slice of a pie chart, between two radii.
Stem-and-leaf diagram: A diagram that splits each value into a stem (leading digits) and a leaf (last digit). It should be ordered and have a key.
Back-to-back stem-and-leaf diagram: Two stem-and-leaf diagrams that share one column of stems, with one data set's leaves on each side.

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