Statistical Charts

1 · One set of data, three pictures

Imagine your teacher asks everyone in the class, "What is your favourite fruit?" The answers are your data: the information you collected. Each answer belongs to one category (a group with a name, such as "Apple" or "Banana"). Data like this, where the answers are names rather than measurements, is called categorical data.

You count how many people chose each fruit. That count is the frequency: how often something happens. Putting the frequencies in a table gives a frequency table (there's a whole page on building those, with tallies: see frequency tables).

A table is accurate but dull. A chart lets you see the answers. Below, the same table is drawn three different ways. Press + and − to change the class's answers and watch all three pictures change together.

Notice what each picture is good at. The pictogram is friendly and easy to count. The bar chart makes it easy to read exact numbers and to see which is biggest. The pie chart shows what share (fraction) of the whole class chose each fruit. The rest of this page takes each one in turn.

2 · Pictograms

A pictogram uses little pictures, called symbols, to show frequencies. Each row is one category. The most important part of a pictogram is the key: it tells you how many things one symbol stands for. Without the key a pictogram is meaningless.

If the key says ■ = 4 cars, then two symbols mean 8 cars, and half a symbol means half of 4, which is 2 cars. A quarter of a symbol means 1 car. So:

frequency = number of symbols × key

When you draw one, go the other way: number of symbols = frequency ÷ key. Try changing the key below. The data (cars passing the school gate in a 5-minute survey) stays the same; only the picture changes.

Choosing the key is your job when you draw a pictogram. A good key makes every row a whole number of symbols, or an easy fraction like ½ or ¼. Too small a key and the rows get very long; too big and you end up drawing 0.6 of a symbol, which nobody can read.

Try it: read a pictogram row

3 · Bar charts

A bar chart shows each frequency as the height of a bar. It has two axes (the plural of axis), the two lines the chart is built on:

• The horizontal axis (across the bottom) lists the categories. Give it a label saying what they are, e.g. "Pet".
• The vertical axis (up the side) is the frequency scale. It must start at 0 and go up in equal steps (0, 2, 4, 6…). Label it "Frequency" or "Number of pupils".
• Every bar is the same width, and there are equal gaps between the bars. The gaps show that the categories are separate things: there is nothing "between" Dog and Cat.
• Give the chart a title so a reader knows what it shows.

Reading the scale is the skill people rush. Before reading any bar, work out what one small gridline is worth. If the labels go 0, 2, 4, 6 and there is one line between each, every line is 1.

Try it: draw the bar chart

Drag (or tap) inside each column to set the height of the bar so it matches the table. Every gridline is 1 pet owner. Then press Check.

Comparing two groups: dual and composite bar charts

Sometimes you want to compare two groups, such as Year 7 and Year 8. There are two ways to do it.

• A dual bar chart puts the two groups' bars side by side for each category, with a key (legend) saying which colour is which group. It is best for comparing the groups category by category.
• A composite bar chart (also called a stacked bar chart) draws one tall bar per group, split into sections, one for each category. It is best for comparing the totals and seeing how each total is made up. To read a section, subtract: top of the section − bottom of the section.

4 · Drawing a pie chart

A pie chart is a circle cut into slices, called sectors. The whole circle stands for everybody in the survey; each sector shows one category's share. Think of a pizza shared out: whoever gets more votes gets a bigger slice.

The key fact: a full turn is 360°. So we share the 360° out fairly between everyone in the survey. Let's do it with concrete numbers first. Suppose 36 people were asked.

• 360° shared between 36 people is 360 ÷ 36 = 10° per person.
• If 9 people said "Walk", Walk gets 9 × 10° = 90° (a quarter of the circle, which makes sense: 9 is a quarter of 36).

That's the "degrees per person" method. Writing it as one formula:

angle = frequency ÷ total × 360°

Both methods give the same answer: 9 ÷ 36 × 360° = 90°. Use degrees per person when 360 ÷ total is a nice number, and the formula otherwise. Always check that your angles add up to 360°.

Drawing it: draw a circle with compasses and mark the centre. Draw one radius straight up from the centre. Put your protractor on that line, measure the first angle and draw the next radius. Then measure the second angle from the line you just drew, and so on round the circle. Label each sector.

Try it: draw the pie chart

Work out each angle first. Then drag the round handles round the edge so each sector has the right angle. The marks round the outside act like a protractor, every 10°, with numbers every 30°. You're allowed to be within 2°.

5 · Reading a pie chart, and what it can't tell you

To go backwards, from an angle to a frequency, undo the steps. You need to know the total number of people.

frequency = angle ÷ 360° × total

Example: a pie chart shows 60 people. A sector of 120° is 120/360 = ⅓ of the circle, so it is ⅓ of 60 = 20 people. Or with degrees per person: 360 ÷ 60 = 6° each, and 120 ÷ 6 = 20 people.

If you don't know the total but you know one sector's frequency, you can still find it. If a 90° sector stands for 15 people, then 90° is a quarter of the circle, so the total is 4 × 15 = 60.

Pie charts show proportion, not size

A pie chart shows proportions: what fraction of the group is in each category. It does not show how many people that is. Every pie chart is the same size whether it shows 10 people or 10 000. So you cannot compare two pie charts by slice size unless they have the same total. Try it: change the sizes of the two schools.

Try it: read the chart

A mix of questions on bar charts, pictograms and pie charts. Each one is new.

6 · Scatter graphs

All the charts so far count one thing about each person (their favourite fruit, their pet). Sometimes we measure two things about each person, because we want to know if they are connected. Does revising for longer go with higher test scores? Do taller people have bigger feet? Data in pairs like this is called bivariate data ("bi" means two, "variate" means variable: a quantity that varies).

A scatter graph plots each pair as a single point, exactly like plotting coordinates. Each point is one person (or one thing): how far across shows their first value, how far up shows their second. You don't join the points up.

Choosing the axes: put the variable you think might affect the other on the horizontal x-axis, and the one that might be affected on the vertical y-axis. Revising might affect the score, so hours go across and score goes up. Each axis needs its own sensible scale and a label with units.

Once the points are plotted, look at the overall shape. Here the points drift upwards from left to right: pupils who revised more tended to score more. A pattern like that is called correlation. Describing correlation and drawing a line of best fit are covered on the distributions and correlation page.

7 · Choosing a chart, and spotting misleading ones

Try it: which chart?

Spot the misleading chart

Charts can be drawn to give the wrong impression, sometimes by accident and sometimes on purpose (adverts, news headlines). Three classic tricks are below. For each one, decide what looks true, then press the button to see the honest version.

Worked examples

Quick check

What you've learned

• Categorical data has named categories; the frequency is how many are in each.
• Pictograms need a key: frequency = symbols × key, and symbols = frequency ÷ key. Part-symbols count, and all symbols must be the same size.
• Bar charts have labelled axes, a frequency scale starting at 0 in equal steps, equal-width bars and gaps between categories. Work out what one gridline is worth before reading.
• Dual bar charts compare groups category by category; composite bar charts compare totals, and each section is top − bottom.
• Pie charts share 360°: angle = frequency ÷ total × 360°, or frequency × degrees per person. Angles must add to 360°. Reading back: frequency = angle ÷ 360° × total.
• Pie charts show proportion, not size: don't compare slices of pies with different totals.
• Scatter graphs plot pairs of values; each point is one person or thing. The possible cause goes on the x-axis. Patterns are called correlation (see distributions and correlation).
• Watch for misleading charts: axes not starting at 0, unequal symbols, 3D pies, missing labels.
• To summarise data with a single number instead of a picture, see averages and range.