Statistical charts and diagrams

Bar charts

A bar chart shows how often each category or value occurs. It suits qualitative data (favourite fruit, type of vehicle) and discrete data (number of goals, number of children).

When you draw a bar chart:

• give it a title and label both axes;
• make every bar the same width, with equal gaps between the bars;
• start the frequency axis at 0, with an even scale (every square worth the same amount);
• draw each bar up to its frequency exactly, using a ruler.

The bars can be vertical or horizontal. The gaps show that the categories are separate; the order of categories does not matter for qualitative data, but numerical values go in order.

When you read a bar chart, work out the scale first. If the gridlines go up in 2s, a bar ending halfway between 10 and 12 shows 11. Read across from the top of each bar to the axis.

Why the axis must start at 0: the height of a bar is meant to show the frequency. If the axis starts at 40, a bar of 50 looks twice as tall as a bar of 45, which is misleading (see 9.2).

Dual and composite bar charts

Two related sets of data can share one chart.

• A dual bar chart (side-by-side bar chart) draws two bars for each category, one for each set of data, next to each other. It makes comparing the two sets in each category easy. A key says which bar is which.
• A composite bar chart (stacked bar chart) puts the two parts on top of each other in a single bar. The top of the whole bar shows the total, so it is good for comparing totals and how each total splits up.

Reading a composite bar chart needs one extra step. The bottom section is read straight from the axis. The top section is not where its top edge is: it is the length of that section, so

$$\text{top section}=\text{reading at the top of the bar}-\text{reading at the join}$$

In this subject the charts use shading as the key: the shaded bars or sections show one set of data and the unshaded ones the other.

Pie charts

A pie chart shows how a whole is split into parts. Each sector's angle is in proportion to its frequency, and the angles add up to ${360}^{°}$.

Drawing a pie chart.

1. Find the total frequency.
2. Work out the angle for each category: $\text{angle}=\frac{\text{frequency}}{\text{total}}\times {360}^{°}$. It is often quicker to find the angle for one item first, $\frac{{360}^{°}}{\text{total}}$, and multiply.
3. Check that the angles add up to ${360}^{°}$.
4. Draw a circle and a radius. Measure each angle with a protractor, starting from the previous line, and label each sector (or use a key).

Reading a pie chart. Work backwards: $\text{frequency}=\frac{\text{angle}}{{360}^{°}}\times \text{total}$. If you know the frequency for one sector, its angle tells you the number of degrees per item.

A pie chart shows proportions, not amounts. The biggest sector is the biggest share of that group, but two pie charts for groups of different sizes cannot be compared by sector size alone.

Pictograms

A pictogram uses a symbol to stand for a number of items. It needs a key, such as "one square represents 8 cakes", and every symbol must be the same size, lined up in columns so the rows can be compared by their length.

Parts of a symbol stand for parts of the number. With one square representing 8 cakes, half a square is 4 cakes and a quarter of a square is 2 cakes. So choose a key that divides easily into the numbers you need to show.

Reading: count the whole symbols, multiply by the key, then add the value of any part symbol.
Drawing: divide the frequency by the key. 20 cakes with one square for 8 cakes is $20÷8=2.5$ squares: two whole squares and a half.

When the key is not given, find it from a row you know: if 3.5 symbols stand for 14 cars, one symbol stands for $14÷3.5=4$ cars.

Stem-and-leaf diagrams

A stem-and-leaf diagram groups numerical data while keeping every value. For two-digit numbers, the stem is the tens digit and the leaf is the units digit: 47 is written as stem 4, leaf 7.

To draw one:

1. Write the stems in order down the left, with a vertical line after them. Include any stem with no leaves.
2. Go through the data and write each leaf in the row of its stem.
3. Redraw it with the leaves in each row in order, smallest nearest the stem. Cambridge expects an ordered diagram.
4. Add a key, such as "2 | 3 means 23 minutes". Without it, 2 | 3 could mean 23, 2.3 or 230.

Keep the leaves evenly spaced and in neat columns: then the diagram doubles as a bar chart turned on its side, showing the shape of the data.

Because every value is still there, and in order, you can read off averages directly:

• the mode is the leaf that repeats most in one row;
• the median is found by counting along the leaves to the middle position, $\frac{n+1}{2}$;
• the range is the last value minus the first.

A back-to-back stem-and-leaf diagram shares one stem between two sets of data, with leaves going out to the left for one set and to the right for the other. The left-hand leaves are read from the stem outwards, so in "7 | 3 |" the value is 37.

Frequency diagrams for simple frequency distributions

A frequency distribution is a frequency table: the values or classes and how often each occurs. Drawing it as a diagram shows its shape at a glance: where the data is concentrated and whether it is spread out evenly.

• For discrete data in a frequency table, draw a bar chart with the values in order along the horizontal axis.
• For continuous data grouped into classes of equal width, draw a frequency diagram: a bar for each class, with no gaps between the bars, because the classes join on to each other ($0<t⩽5$ ends exactly where $5<t⩽10$ begins). The horizontal axis is a continuous scale, marked with the class boundaries.

In both, the vertical axis is frequency, starting at 0. Read the diagram the other way to recover the table: each bar's height is its frequency.

Once drawn, describe the shape: "most of the times are between 10 and 15 minutes, with few very short or very long times".