Tables, Charts and Diagrams

Type of data	Best diagram
Categories (colours, sports)	Bar chart or pictogram
Proportions of a whole	Pie chart
Ungrouped discrete numbers	Vertical line chart
Data over time	Line graph (time series)
Comparing two groups by category	Dual or composite bar chart
Grouped continuous data	Histogram (see the next page)

Bars must have gaps for categorical data. Touching bars are reserved for histograms, where the horizontal axis is a continuous scale.

Draw and label both axes, with frequency on the vertical axis.
Choose a sensible, even scale that fits the largest frequency.
Draw bars of equal width with equal gaps between them.
Label every bar with its category.
Give the chart a title.

Dual bar chart — two bars side by side for each category, letting you compare two groups directly. Always include a key.
Composite (stacked) bar chart — one bar per category, split into sections. Good for showing totals as well as parts.

Worked Example 1 — Reading a bar chart

From the chart above, find (a) the modal sport, (b) the total number of students, (c) the fraction who chose tennis.

①Reading the bars: Football $6$, Rugby $8$, Tennis $4$, Netball $10$.

②(a) The tallest bar is Netball, so that is the mode.

③(b) $6 + 8 + 4 + 10 = 28$ students.

④(c) $\dfrac{4}{28} = \dfrac{1}{7}$

Drawing a pie chart

$\text{angle} = \dfrac{\text{frequency}}{\text{total frequency}} \times 360^\circ$

Reading a pie chart

$\text{frequency} = \dfrac{\text{angle}}{360^\circ} \times \text{total frequency}$

The quick way: work out how many degrees represent one item ($360 \div$ total), then multiply each frequency by that. Far faster than doing a separate fraction for every sector.

Worked Example 2 — Drawing a pie chart

$40$ people were asked their favourite fruit: apple $14$, banana $10$, orange $9$, pear $7$. Find the angle for each sector.

①One person $= 360 \div 40 = 9^\circ$.

②Apple: $14 \times 9 = 126^\circ$

③Banana: $10 \times 9 = 90^\circ$

④Orange: $9 \times 9 = 81^\circ$

⑤Pear: $7 \times 9 = 63^\circ$

Check: $126 + 90 + 81 + 63 = 360^\circ$ ✓

Worked Example 3 — Reading a pie chart

A pie chart represents $216$ people. The sector for "cycle" is $75^\circ$. How many cycle?

①$\dfrac{75}{360} \times 216$

②$= 0.20833 \times 216 = 45$ people

You cannot compare two pie charts by sector size alone. A $90^\circ$ sector of a chart representing $40$ people is $10$ people; the same angle on a chart representing $400$ people is $100$. Always check the totals.

Worked Example 4 — Comparing two pie charts

Pie chart A represents $60$ students; its "walk" sector is $120^\circ$. Pie chart B represents $200$ students; its "walk" sector is $72^\circ$. Which group has more walkers?

①A: $\dfrac{120}{360} \times 60 = 20$ students.

②B: $\dfrac{72}{360} \times 200 = 40$ students.

Group B has more walkers, even though its sector angle is much smaller — because the group is far larger.

A pictogram uses a symbol to represent a fixed number of items. It must always have a key saying what one symbol stands for.

Worked Example 5 — Pictogram

In a pictogram, one circle represents $8$ books. How many books are shown by (a) $4$ circles, (b) $3$ and a half circles?

①(a) $4 \times 8 = 32$ books

②(b) $3.5 \times 8 = 28$ books

Half a symbol means half the value. If one symbol is $8$, a quarter-symbol is $2$.

A vertical line chart is like a bar chart but with thin lines instead of bars. It is the correct diagram for ungrouped discrete data such as "number of siblings".

A time series plots a quantity against time, with the points joined by straight lines. Time always goes on the horizontal axis.

Two things to describe

The trend — the overall direction, ignoring the wobbles
Any seasonality — a pattern that repeats at regular intervals

Worked Example 6 — Describing a time series

Describe the pattern in the graph above.

①Trend: sales are generally increasing over the four years.

②Seasonality: within each year there is a rise then a small dip, repeating annually.

③The highest point is in the final quarter shown.

A good answer separates the long-term trend from the short-term fluctuations. Saying only "it goes up and down" earns little credit.

Do not read between the plotted points on a time series of discrete measurements. If sales are recorded quarterly, the joining line is only a visual guide — there is no actual data halfway between two quarters.

Trick	Why it misleads
Vertical axis not starting at zero	Small differences look enormous
Uneven scale	Distorts the shape of the data
Bars of different widths	The eye compares area, not height
3D effects	Front sectors or bars look bigger than they are
No key or labels	The reader cannot tell what is shown
Pictogram symbols of different sizes	Suggests a bigger difference than the data shows

Worked Example 7 — Spotting a misleading graph

A bar chart shows sales of $102$, $104$ and $107$ units, but the vertical axis runs from $100$ to $108$. Explain why this is misleading.

①Because the axis starts at $100$ rather than $0$, the bar for $107$ appears roughly $3.5$ times the height of the bar for $102$.

②In reality the difference is only $5$ units out of about $100$ — an increase of under $5\%$.

The graph exaggerates the differences. Starting the axis at zero would show that the three values are very similar.

Bar chart

Categories; equal widths, gaps between bars.

Pie chart

$\dfrac{\text{frequency}}{\text{total}} \times 360^\circ$; sectors must total $360^\circ$.

Pie shortcut

Find degrees per item first: $360 \div$ total.

Comparing pies

Always convert angles back to actual numbers first.

Pictogram

Needs a key; part-symbols mean part of the value.

Vertical line chart

For ungrouped discrete data.

Time series

Time on the horizontal axis; describe trend and seasonality.

Misleading

Watch for a truncated axis, uneven scales and 3D effects.

Always

Label both axes, add a title, and include a key if needed.

Question 1

$72$ people were surveyed. Find the pie chart angle for a group of $18$ people.

▶ Show solution

One person $= 360 \div 72 = 5^\circ$

Angle $= 18 \times 5 = 90^\circ$

Question 2

A pie chart represents $240$ people. A sector has angle $135^\circ$. How many people does it represent?

▶ Show solution

$\dfrac{135}{360} \times 240 = 0.375 \times 240 = 90$ people

Question 3

$45$ students chose a language: French $20$, Spanish $15$, German $10$. Find the angle of each sector.

▶ Show solution

One student $= 360 \div 45 = 8^\circ$

French: $20 \times 8 = 160^\circ$

Spanish: $15 \times 8 = 120^\circ$

German: $10 \times 8 = 80^\circ$

Check: $160 + 120 + 80 = 360^\circ$ ✓

Question 4

In a pictogram, one star represents $12$ cars. How many cars are shown by $2$ and a quarter stars?

▶ Show solution

$2.25 \times 12 = 27$ cars

Question 5

Name the most suitable diagram for each: (a) the proportion of a budget spent on each category, (b) the number of goals scored per match, (c) monthly rainfall over three years.

▶ Show solution

(a) A pie chart — it shows proportions of a whole.

(b) A vertical line chart — the data are ungrouped and discrete.

(c) A time series line graph — the data are measured over time.

Question 6

Pie chart P represents $80$ people with a $90^\circ$ "tea" sector. Pie chart Q represents $300$ people with a $36^\circ$ "tea" sector. Which group has more tea drinkers?

▶ Show solution

P: $\dfrac{90}{360} \times 80 = 20$ people

Q: $\dfrac{36}{360} \times 300 = 30$ people

Group Q has more tea drinkers, despite its much smaller sector angle.

Question 7

Give two reasons why a bar chart with a vertical axis starting at $50$ could be misleading.

▶ Show solution

1. The bars no longer show the true relative sizes — a bar twice as tall does not represent twice the value.

2. Small differences between the values are visually exaggerated, making them look far more significant than they are.

Question 8

A time series shows shop takings rising each spring and falling each autumn, while gradually increasing year on year. Describe the trend and the seasonality.

▶ Show solution

Trend: takings are increasing overall from year to year.

Seasonality: within each year there is a repeating pattern — a peak in spring and a trough in autumn.

The two effects are separate: the seasonal rises and falls happen around a steadily rising overall level.

Question 9

A pie chart shows how $180$ students travel to school. The angles are: walk $140^\circ$, bus $x^\circ$, car $90^\circ$, cycle $50^\circ$.

(a) Find $x$. (b) How many travel by bus? (c) What fraction walk?

▶ Show solution

(a) $140 + x + 90 + 50 = 360$, so $x = 360 - 280 = 80^\circ$.

(b) One student $= 360 \div 180 = 2^\circ$, so bus $= 80 \div 2 = 40$ students.

(c) $\dfrac{140}{360} = \dfrac{7}{18}$

Question 10

A company's quarterly profits (in £1000s) over two years are: $18, 24, 21, 30$ then $22, 28, 25, 35$.

(a) Describe the trend. (b) Describe the seasonal pattern. (c) Estimate the profit for the first quarter of year 3, explaining your reasoning. (d) Why should this estimate be treated with caution?

▶ Show solution

(a) Comparing matching quarters, every one has risen: $18 \to 22$, $24 \to 28$, $21 \to 25$, $30 \to 35$. The trend is upward, by roughly £$4000$–£$5000$ per year.

(b) Within each year the pattern is the same: a rise in Q2, a dip in Q3, then the highest value in Q4. This seasonal pattern repeats in both years.

(c) Q1 went from $18$ to $22$, an increase of $4$. Continuing that pattern gives an estimate of about $22 + 4 = £26\,000$ for Q1 of year 3.

(d) The estimate assumes the trend continues unchanged, which may not happen — a change in the economy, competition or company circumstances could break the pattern. Two years is also a short record on which to base a prediction.