What you'll learn
Representing data means taking a set of facts you have collected and showing them in a form that can be read at a glance — a tally chart, a frequency table, a pictograph, a bar graph or a line graph. The information itself does not change; only the way it is displayed changes, and a good display answers questions far faster than a list of numbers ever could. On the SEA Mathematics paper Statistics is worth 6 of the 40 items, and questions on this topic usually give you one representation and ask you to read it, complete it, or turn it into another. This guide covers how each display is built, how the key and the scale work, and how to move between a tally, a table and a graph without losing a single count.
Key terms and definitions
Data — the facts you have collected, such as the shoe sizes of everyone in a class.
Survey — the process of collecting data, usually by asking a question or counting something.
Tally — a mark made for each item counted, grouped in fives so that the total is easy to add up.
Frequency — how many times something happened, or how many items are in a group.
Frequency table — a table with one row per category and a column giving the frequency of each.
Category — one of the groups the data is sorted into, such as "mango" or "Tuesday".
Pictograph (picture graph) — a display that uses a repeated symbol to stand for a number of items.
Key — the note on a pictograph or graph telling you what one symbol or one square is worth.
Bar graph — a display using bars of equal width, where the height (or length) of each bar shows the frequency.
Scale — the counting pattern along the numbered axis of a graph, such as going up in 2s or in 5s.
Axis — one of the two lines along the edge of a graph. The horizontal one usually names the categories; the vertical one usually carries the numbers.
Line graph — a display in which points are plotted and joined, used for data that changes over time.
Core concepts
From raw data to a tally chart
Raw data arrives in a messy order. If Anisa asks 30 classmates for their favourite local fruit, she writes answers as they come: mango, orange, mango, sapodilla, banana, and so on. Going back through that list once for every category is slow and easy to get wrong.
A tally chart fixes this. You list the categories once, then make one mark for each answer as you hear it. Every fifth mark is drawn across the previous four to make a gate of five. Gates are counted quickly, so three gates and two single marks is 5 + 5 + 5 + 2 = 17. The advantage is that you pass through the data only once, which is why the tally is the first step in almost every SEA statistics question that begins with a list.
Frequency tables
Once the tallying is finished, the marks are counted and written as numbers. That gives a frequency table, which is the tidiest form of the data.
Here is Anisa's survey of 30 pupils written out fully:
| Fruit | Tally | Frequency |
|---|---|---|
| Mango | one gate of 5, then 4 single marks | 9 |
| Orange | one gate of 5, then 1 single mark | 6 |
| Banana | one gate of 5 | 5 |
| Pommecythere | 4 single marks | 4 |
| Sapodilla | one gate of 5, then 1 single mark | 6 |
The frequency column adds to 9 + 6 + 5 + 4 + 6 = 30, which matches the number of pupils asked. Checking that total is the single most useful habit in this topic: if it does not match, a mark has been lost or double-counted somewhere.
Pictographs and the key
A pictograph shows each frequency as a row of repeated symbols. The key tells you what one symbol is worth. If one mango picture stands for 2 pupils, then Anisa's data is drawn as:
| Fruit | Symbols (1 symbol = 2 pupils) |
|---|---|
| Mango | 4 whole symbols and a half |
| Orange | 3 whole symbols |
| Banana | 2 whole symbols and a half |
| Pommecythere | 2 whole symbols |
| Sapodilla | 3 whole symbols |
Multiply back to check: 4½ × 2 = 9, 3 × 2 = 6, 2½ × 2 = 5, 2 × 2 = 4, 3 × 2 = 6. The symbols total 15, and 15 × 2 = 30 pupils.
Half symbols are how a pictograph shows an odd number when the key is 2. If the key were 5, then a quarter symbol would be worth 1 and a half symbol worth 2½ — which is why keys of 2, 5 and 10 are chosen and keys of 3 or 7 almost never are.
Bar graphs and their scale
A bar graph draws a bar for each category. Every bar is the same width and the bars are separated by equal gaps; only the height carries information.
The scale is the part that questions attack. If the vertical axis is numbered 0, 5, 10, 15, 20 with five small squares between each pair of numbers, one square is worth 1. If there are only two squares between them, one square is worth 2½. Before reading any bar, work out what one square is worth: take the gap between two printed numbers and divide by the squares between them.
If Anisa's fruit data were drawn with the fruits along the bottom and frequency up the side going up in 2s, the mango bar would be tallest at 9, half a square above the 8 line. Orange and sapodilla would match at 6, banana would reach 5, and pommecythere would be shortest at 4.
Line graphs
A line graph is used when the data changes over time — temperature through a day, rainfall through a year, a plant's height week by week. Points are plotted and joined by straight lines, and the slope between two points tells you whether the quantity rose or fell, and how steeply.
Here is the rainfall Devon recorded at his school in Sangre Grande:
| Month | Rainfall (mm) |
|---|---|
| June | 180 |
| July | 220 |
| August | 240 |
| September | 190 |
| October | 210 |
Drawn as a line graph, the line would climb from June to August, drop sharply to September, then rise again. The steepest rise is June to July (up 40 mm) and the steepest fall is August to September (down 50 mm). A line graph makes those changes visible in a way a table does not.
Bar graphs are for comparing separate categories; line graphs are for following one thing as it changes. Using a line graph for favourite fruits would suggest mango turns gradually into orange, which is nonsense.
Choosing the right representation
- Counting answers as they come in: tally chart.
- Tidy final record of the counts: frequency table.
- A display for young readers, with a small set of round numbers: pictograph.
- Comparing the sizes of separate categories: bar graph.
- Showing change over time: line graph.
Worked examples
Example 1: Building a frequency table from a tally
Kern stood by the school gate for twenty minutes and tallied the vehicles that passed. He recorded cars as four gates and four extra marks, maxi taxis as one gate and three extra marks, trucks as five single marks, and bicycles as three single marks. Make a frequency table and find the total.
Cars: four gates is 4 × 5 = 20, plus 4 extra marks, so 24. Maxi taxis: one gate is 5, plus 3, so 8. Trucks: 5. Bicycles: 3.
| Vehicle | Frequency |
|---|---|
| Car | 24 |
| Maxi taxi | 8 |
| Truck | 5 |
| Bicycle | 3 |
Total: 24 + 8 + 5 + 3 = 40 vehicles.
Example 2: Reading a pictograph with a key
A pictograph shows the books borrowed from the school library each day. The key says one book symbol = 10 books. Monday shows 3 symbols, Tuesday 4 symbols, Wednesday 2 symbols and a half, Thursday 6 symbols. How many books were borrowed altogether, and on which day were the fewest borrowed?
Multiply each row by the key.
Monday: 3 × 10 = 30 books. Tuesday: 4 × 10 = 40 books. Wednesday: 2½ × 10 = 25 books, because the half symbol is worth half of 10, which is 5. Thursday: 6 × 10 = 60 books.
Total: 30 + 40 + 25 + 60 = 155 books.
The fewest were borrowed on Wednesday, with 25. The half symbol is the whole point of the question: a pupil who glances past it and reads Wednesday as 2 symbols gets 20 books and a total of 150, and both of those answers are wrong.
Example 3: Turning a table into a described bar graph
Riya recorded how many pupils in her class walk, take a maxi taxi, come by car or ride a bicycle to school: walk 12, maxi taxi 9, car 6, bicycle 3. She draws a bar graph with a scale going up in 3s. How many squares tall is each bar if one square is worth 3, and how many pupils are in the class?
Walk: 12 ÷ 3 = 4 squares. Maxi taxi: 9 ÷ 3 = 3 squares. Car: 6 ÷ 3 = 2 squares. Bicycle: 3 ÷ 3 = 1 square.
Class total: 12 + 9 + 6 + 3 = 30 pupils.
The bars step down neatly by one square each time, which is a useful check — if one of your heights broke the pattern you would go back and look at it.
Common mistakes and how to avoid them
Ignoring the key on a pictograph. Counting 4 symbols as 4 when the key says each is worth 10. Fix: read the key aloud before counting anything, and write it at the top of your working.
Assuming one square equals one on a bar graph. Fix: find two printed numbers on the axis, subtract, and divide by the number of squares between them.
Miscounting a tally. Fix: count the gates in fives first, then add the loose marks at the end. Never count the marks one by one.
Forgetting to check the total. Fix: add the frequency column and compare it with the number of people or items surveyed. They must match.
Using a line graph for categories. Fix: ask whether the horizontal axis shows time. If it does not, use bars.
Leaving the graph unlabelled. Fix: every graph needs a title, both axes named, and a key if symbols are used. Marks are given for these.
How parents can help at home
You do not need to remember any statistics to help with this topic. Every skill in it is counting, sorting and reading, and your kitchen table is a good enough laboratory.
Collect something together. Ask your child to tally the colours of cars passing the house for ten minutes, or the different brands in the fridge, or how many pages they read each evening for a week. Real data collected by the child is remembered far better than data printed on a worksheet.
Insist on the total check. After any tally, ask "how many did you count altogether, and does that match?" This one question catches most of the errors in the topic.
Point at graphs in the newspaper or on the news. Ask two questions: "what is one square worth?" and "which is biggest?" That is exactly what the exam asks.
When your child is stuck, say "make the table first." Almost every difficulty in this topic disappears once the data is in a neat frequency table with a checked total.
Exam technique for representing data
Read the title, the key and the scale before you look at a single bar. The information you need to answer correctly is almost always in those three places, and the wrong answers offered in multiple-choice items are usually built from ignoring one of them.
When a question asks you to complete a table or a graph, find the total first. If the total is given and one category is missing, subtract to get it rather than trying to read a partly drawn bar.
Write the frequency next to each bar or row as you read it. Two seconds of writing prevents you from re-reading the same graph three times.
Statistics is 6 of the 40 items on the Mathematics paper, so it is worth a little over an eighth of the marks. These are usually among the quickest marks on the paper, because the working is short — do not rush them and lose them to a misread key.
Quick revision summary
- Tally in gates of five, then count the gates in fives and add the loose marks.
- A frequency table has one row per category; its total must match the number surveyed.
- On a pictograph, always read the key first: one symbol may be worth 2, 5 or 10.
- A half symbol is worth half of whatever the key says.
- On a bar graph, work out the value of one square by dividing the gap between two printed numbers by the squares between them.
- Bars are equal in width with equal gaps; only the height carries meaning.
- Use bar graphs to compare categories and line graphs to show change over time.
- Every display needs a title, labelled axes and, where symbols are used, a key.
- Check the total at the end of every question in this topic.