Statistical Charts and Diagrams

Five diagrams, five different jobs

Cambridge names five diagrams in this section: bar charts, pie charts, pictograms, stem-and-leaf diagrams and simple frequency distributions. All five show how often each value or category occurs, but they are good at different things, and a question that asks you to state one advantage is asking whether you know which job each one does.

The same set of data can be drawn as any of them, so the choice is never about which is correct - it is about which one shows the thing the question is asking about. If the question is about a share of a whole, you want a pie chart. If it is about comparing two groups, you want a dual bar chart. If it is about the spread of the individual values, you want a stem-and-leaf diagram.

Bar charts: simple, composite and dual

A bar chart shows the frequency of each category as a bar. The bars all have the same width and there are equal gaps between them - the gaps are what separate a bar chart from a histogram, and Cambridge uses the gap. The vertical axis is labelled frequency, or 'number of ...', and it must start at 0, because the height of the bar is the whole point of the diagram.

Cambridge also uses two extended forms. A composite (stacked) bar chart puts several categories inside one bar, stacked on top of one another, so the top of the bar is the total for that group and each segment is one part of it. A dual (side-by-side) bar chart puts two bars next to each other for every category, so you can compare two groups - boys and girls, or this year and last year.

Reading a composite bar chart has one trap. A question that asks for the total wants the top of the whole bar; a question that asks for one category wants the height of one segment. Measure a segment from the line below it, not from the axis.

• Simple bar chart: one bar per category, read the frequency straight off the vertical axis
• Composite bar chart: segments stacked, so the top of the bar is the total
• Dual bar chart: two bars per category, one for each group, with a key to say which is which

Pie charts and the angle of a sector

A pie chart divides a circle into sectors, one for each category. The whole circle is the total frequency and the whole circle is 360°, so the angle of each sector is the fraction of the total that its category represents.

The formula is the whole topic in one line: $\text{angle}={\displaystyle \frac{\text{frequency}}{\text{total}}}\times 360$°. If 9 out of 36 students choose netball, the sector is $\frac{9}{36}\times 360=90$°. Cancel the fraction first - $\frac{9}{36}=\frac{1}{4}$, and a quarter of 360 is 90 - which is quicker and much safer than multiplying large numbers.

The calculation runs the other way just as often. If a sector is 80° out of a total of 72 people, the frequency is $\frac{80}{360}\times 72=16$ people. Learn it as one calculation with two directions rather than two separate formulas.

• Angle of a sector $={\displaystyle \frac{\text{frequency}}{\text{total}}}\times 360$°
• Frequency $={\displaystyle \frac{\text{angle}}{360}}\times \text{total}$
• All the sector angles must add to 360° - add them as a check
• 72 people, a sector of 80°: $\frac{80}{360}\times 72=16$ people

Pictograms and the key

A pictogram uses a picture or symbol to stand for a number of items, and the key tells you how many. One symbol might stand for 8 cars, so three symbols are 24 cars. Half a symbol stands for half the key value, which is 4 cars.

The key is not decoration: without it a pictogram carries no information at all. When you draw one, put the key beside the first row or at the top of the page.

Counting symbols and forgetting to multiply is the commonest pictogram error. A row of four and a half symbols is not 4.5 of anything - it is $4.5\times 8=36$ cars.

• Key: one symbol represents 8 cars
• Three symbols: $3\times 8=24$ cars
• Half a symbol: 4 cars
• Four and a half symbols: $4.5\times 8=36$ cars

Stem-and-leaf diagrams

A stem-and-leaf diagram splits every value into a stem (the leading digit or digits) and a leaf (the last digit), and it keeps every individual value visible. The key says how to put them back together: '2 | 3 means 23'.

The leaves must be written in order from smallest to largest within each stem. That ordering is what makes the diagram useful: the smallest value is the first leaf of the first stem, the largest is the last leaf of the last stem, and the middle value of the ordered list is the median. Cambridge states both the ordering and the key as requirements, so an unordered diagram loses a mark even if every value is present.

The diagram has one real advantage over a bar chart of grouped data: it shows the shape of the distribution and every original value at the same time, so you can read the median and the range without losing the data. Its disadvantage is that it only works well for a modest number of two- or three-digit values.

• Stems down the left, leaves on the right, in order
• Key: 2 | 3 means 23
• Because the values are ordered, the smallest and largest sit at the two ends
• Every value is still visible - which a bar chart of grouped data cannot manage

Simple frequency distributions

A simple frequency distribution is a table with a column of values or classes and a column of frequencies, and it is the raw material for every diagram in this topic. Tally marks are the quickest way to build one from a list of data: one mark per item, a fifth mark struck through each group of four, then count the groups.

Classes must not overlap and must cover every value. The classes 0-9, 10-19 and 20-29 leave nothing out and put each value in exactly one class; 0-10 and 10-20 do not work, because 10 belongs to both.

Check a frequency table before you use it. The frequencies must add to the total number of items, and that total is the number of marks, the number of people, or whatever the question is counting.

• Tally first, count the tallies, then write the frequency
• Classes must not overlap and must cover every value
• The frequencies must add to the total number of items

Comparing two distributions, and what a chart cannot tell you

Extended questions often give two sets of data - two classes, two years, two machines - and ask you to compare them. Compare two things: an average, which says where the data sits, and the range or spread, which says how consistent it is. A comparison needs both, and it needs numbers.

A good answer sounds like this: 'Class A and Class B have the same mean score of 62, but the range for Class A is 30 marks and for Class B it is 8 marks, so Class B's scores are more consistent.' Every claim carries a number from the data. 'Class B is better' scores nothing at all.

Finally, remember what a diagram cannot show. A chart of a sample tells you about that sample, not about everybody; a chart built from a handful of items can look dramatic by accident; and two charts of the same data drawn with different vertical scales can make the same change look large in one and small in the other. Read the scale before you draw a conclusion.

• Compare an average (where the data sits) and the range (how spread out it is)
• Put a number from the data into every sentence of the comparison
• Read the vertical scale before drawing any conclusion from a chart
• A sample is not the whole population