Learn to read scales, bar graphs, pictograms and line graphs, draw and interpret pie charts, convert between values, fractions, percentages and angles, and recognise graphs that mislead.
A graph turns a table of numbers into a picture that is easier to compare, describe and understand. A good graph shows differences, patterns, trends and proportions at a glance, but only if you read it carefully. Bar graphs compare separate categories, line graphs show how something changes, and pie charts show how the parts of a whole are shared. This guide teaches you to read scales, read and draw each type of graph, work out percentages and angles in both directions, spot graphs that mislead, and choose the right graph for the job. It has 24 worked examples checked by computer, five diagrams, three tools, a practice drill, exam questions with answers, and a 20-question quiz.
A graph (or chart) is a visual way of showing data. The right type depends on what you want to show.
| Graph type | Best for | Typical question |
|---|---|---|
| Pictogram | Simple counts, using symbols | How many symbols does each category have? |
| Vertical bar graph | Comparing separate categories | Which category is largest? |
| Horizontal bar graph | Comparing categories with long names | How much larger is A than B? |
| Line graph | Change over an ordered sequence, often time | When did the value rise most? |
| Pie chart | Parts of one whole | What fraction or percentage belongs to each part? |
A graph with no clear scale or labels may look attractive but tells the reader very little.
Never assume that one gridline means one unit. Read the numbers on the axis first, then work out what each small division is worth.
An axis is labelled 0, 20, 40, 60 and 80 at equally spaced marks. Each major step is 20 units, so a bar that reaches the third mark above zero has value 60.
The labelled marks are 0, 10, 20 with 5 equal small intervals between each pair of labels. Each small interval is 10 ÷ 5 = 2 units. A point 3 small marks above 10 has value 10 + 3 × 2 = 16.
Some graphs do not start at zero. That is allowed, but the reader must notice it (see Section 15).
In a vertical bar graph, categories are along the horizontal axis and the height of each bar shows its value. Bars have equal widths and equal gaps, because only the height carries information.
From Figure 1, Apples 30, Bananas 45, Oranges 25 and Mangoes 50. Mangoes minus bananas = 50 − 45 = 5. The total sold = 30 + 45 + 25 + 50 = 150. The mean per fruit = 150 ÷ 4 = 37.5. Mangoes are 50 ÷ 150 = one third of all fruit sold.
A horizontal bar graph is the same graph turned on its side. It is useful when the category names are long.
Four towns have populations 12 000, 18 000, 15 000 and 21 000. The longest bar is 21 000. The difference between the largest and smallest is 21 000 − 12 000 = 9 000. The total is 66 000 and the mean is 16 500.
A dual (or compound) bar graph shows two groups side by side so they can be compared, for example boys and girls in each class.
Three classes have these numbers of boys and girls: class 6A has 14 boys and 16 girls, class 6B has 18 boys and 12 girls, class 6C has 15 boys and 15 girls. Every class has 14 + 16 = 18 + 12 = 15 + 15 = 30 pupils. Class 6B is the only class with more boys than girls, by 18 − 12 = 6. Across all three classes there are 47 boys and 43 girls.
A pictogram uses a symbol to stand for a number of items. The key tells you the value of one symbol, and a part-symbol stands for a part of that value.
The key says that one book symbol = 4 books. Monday has 5 symbols, so 5 × 4 = 20 books. Tuesday has 3½ symbols, so 3.5 × 4 = 14 books. A half symbol stands for 4 ÷ 2 = 2 books.
A line graph joins plotted points with straight lines. It is used when the horizontal variable has a natural order, such as time. Rising lines show increases, falling lines show decreases and a flat line shows no change.
From Figure 2: Monday to Tuesday +3°C, Tuesday to Wednesday −1°C, Wednesday to Thursday +4°C, Thursday to Friday −1°C. The greatest increase is Wednesday to Thursday. The mean temperature is (22 + 25 + 24 + 28 + 27) ÷ 5 = 126 ÷ 5 = 25.2°C.
On a line graph, the steepness of a segment shows how fast the quantity changes, and the average rate of change is (change in vertical value) ÷ (change in horizontal value).
A graph shows the water in a tank: 0 litres at 0 minutes, 60 litres at 5 minutes, still 60 litres at 8 minutes, and 20 litres at 12 minutes. Filling rate = 60 ÷ 5 = 12 litres per minute. From 5 to 8 minutes the line is flat, so nothing changes for 3 minutes. Emptying rate = (60 − 20) ÷ (12 − 8) = 40 ÷ 4 = 10 litres per minute.
Do not join points with a line when the categories have no order, such as the names of fruits. Use a bar graph instead.
Graphs make comparison easy, but state the comparison exactly. Use the difference (larger − smaller), the total, a ratio, or a percentage change:
Sales rise from 200 to 250 units. Change = 50. Percentage increase = 50 ÷ 200 × 100 = 25%.
Sales then fall back from 250 to 200. The change is again 50, but now the old value is 250, so we divide by 250: 50 ÷ 250 × 100 = 20%. A rise of 25% is not undone by a fall of 25%. Another example: a fall from 80 to 60 is 20 ÷ 80 × 100 = 25% decrease.
A pie chart is a circle divided into sectors. The whole circle is the whole group (100%, or 360°) and each sector is one part.
| Fraction | Percentage | Angle |
|---|---|---|
| 1/2 | 50% | 180° |
| 1/4 | 25% | 90° |
| 1/5 | 20% | 72° |
| 1/10 | 10% | 36° |
| 1/20 | 5% | 18° |
Because 360 ÷ 100 = 3.6, angle = percentage × 3.6° and percentage = angle ÷ 3.6.
(a) In a survey of 200 people, 50 choose A. Angle = 50 ÷ 200 × 360° = 90°.
(b) In a group of 80 items, 12 are in category B. Angle = 12 ÷ 80 × 360° = 54°.
Cricket 90, Football 50, Hockey 30, Other 30. Total = 200. Cricket: 90 ÷ 200 × 360 = 162° (45%). Football: 50 ÷ 200 × 360 = 90° (25%). Hockey: 54° (15%). Other: 54° (15%). Check: 162 + 90 + 54 + 54 = 360°.
How many degrees are in the whole circle?
The fix: a whole pie chart is 360°, not 100°. So 40% of 360° = 40 × 3.6° = 144°.
The same relationship can be rearranged, so you can move between value, fraction, percentage and angle in either direction.
| Task | Formula |
|---|---|
| Angle from a value | (value ÷ total) × 360° |
| Angle from a percentage | percentage × 3.6° |
| Value from an angle | (angle ÷ 360°) × total |
| Percentage from an angle | (angle ÷ 360°) × 100% |
| Fraction from an angle | angle ÷ 360° |
| Total from a known value and angle | value × 360° ÷ angle |
| Missing angle | 360° − sum of the known angles |
| Missing percentage | 100% − sum of the known percentages |
(a) A pie chart represents 600 people and a sector is 72°. Value = 72 ÷ 360 × 600 = 120 people.
(b) A pie chart represents 900 books and a sector is 40°. Value = 40 ÷ 360 × 900 = 100 books.
Is the angle the same as the number of people?
The fix: an angle is a share of 360°, not a count. The share is 72 ÷ 360 = 1/5, and 1/5 of 600 is 120 people.
(a) A 54° sector is 54 ÷ 360 × 100 = 15%.
(b) A 35% sector has angle 35 × 3.6° = 126°.
(c) A 108° sector is 108 ÷ 360 = 3/10 of the circle, which is 30%.
A 72° sector represents 90 people. Since 72° is one fifth of 360°, the whole chart is 5 × 90 = 450 people. By formula, 90 × 360 ÷ 72 = 450.
(a) Three sectors are 80°, 120° and 65°. The fourth is 360 − (80 + 120 + 65) = 360 − 265 = 95°.
(b) Three sectors are 18%, 27% and 35%. The fourth is 100 − 80 = 20%, with angle 20 × 3.6° = 72°.
A pie chart represents 1500 survey responses and one sector is 48°. Value = 48 ÷ 360 × 1500 = 200 responses.
A survey has 800 responses, and 140 are in category X. Fraction = 140 ÷ 800 = 7/40. Percentage = 7/40 × 100 = 17.5%. Angle = 140 ÷ 800 × 360° = 63°.
A pie chart represents 720 items. Three sectors have 180, 144 and 216 items. The total of these is 540, so the fourth category has 720 − 540 = 180 items. Its angle is 180 ÷ 720 × 360° = 90°.
A pie chart shows shares, not amounts. The size of the circle does not tell you how big the group is unless the total is stated. Two pie charts can only be compared by number if you know both totals.
School X has 400 students and a pie chart shows that 25% walk to school. School Y has 1200 students and 15% walk. School X: 0.25 × 400 = 100 walkers. School Y: 0.15 × 1200 = 180 walkers. Even though X has the larger percentage, Y has more students who walk.
What is missing from this comparison?
The fix: the totals. 25% of 400 is 100, but 15% of 1200 is 180. A larger percentage does not mean a larger number when the groups have different sizes.
When angles are rounded to the nearest degree, they may not add up to exactly 360°. Work out the angles to a few decimal places, round them, and then check the total.
A group of 7 people has 1 person in category P, 2 in Q and 4 in R. The exact angles are 51.43°, 102.86° and 205.71°. Rounded to the nearest degree they are 51°, 103° and 206°, which add to 360°. ✓
If instead 7 equal categories each had one person, each angle would be 360 ÷ 7 ≈ 51.43°, and 7 × 51° = 357°. In that case you adjust one or two sectors by a degree so that the angles total 360°, or you draw the chart with the more accurate angles.
More on gathering and organising data is in Basic Data & Statistics.
| Situation | Good choice | Reason |
|---|---|---|
| Number of books in different genres | Bar graph | Separate categories |
| Temperature every day for a week | Line graph | Change over an ordered sequence |
| How a budget is divided | Pie chart | Parts of one whole |
| Many categories with long names | Horizontal bar graph | Labels are easier to read |
| Monthly rainfall through the year | Line graph (or bar graph) | Shows change over time |
| Comparing boys and girls in several classes | Dual bar graph | Two groups side by side |
A graph can use correct numbers and still give a wrong impression. Watch for these:
B is 55 and A is 52. B is larger by 3, which is 3 ÷ 52 × 100 ≈ 5.8%. On the left graph, with the axis from 0, the bars look nearly equal. On the right, the axis starts at 50, so the bars are only 2 and 5 units high, and B looks 5 ÷ 2 = 2.5 times as tall as A. The data have not changed, but the impression has.
What should you check before comparing bar heights?
The fix: check where the axis starts. Bar heights are only proportional to the values when the axis begins at 0. Read the numbers: B = 55 and A = 52, so B is only about 5.8% larger.
A shop graph shows that sales were higher in March than in February. A careful reader asks whether both months have the same number of days, whether the units are the same and whether the scale was changed. A rise in sales may simply reflect a longer month.
(a) A bar graph shows four weekly totals of 120, 150, 135 and 175 units. The total is 580 units and the weekly mean is 580 ÷ 4 = 145 units.
(b) A line graph shows 84 units at one time and 57 units at another. The difference is 84 − 57 = 27 units.
Type one category per line as name, value. The tool finds the total, the fraction, the percentage and the angle for each category, and draws the pie chart. Try the sports survey from Figure 3, then your own data.
Move between values, fractions, percentages and angles in either direction. Choose what you know, type the numbers and see the working.
Find the percentage increase or decrease between an old and a new value, as read from a graph. Notice that the change needed to return to the old value is a different percentage.
A pie chart represents 1200 customers. A sector measures 81°. Find the number of customers it represents.
The whole circle is 360°, so the fraction is 81 ÷ 360 = 0.225. Customers = 0.225 × 1200 = 270. Check: 270 is 22.5% of 1200, and 22.5% of 360° is 22.5 × 3.6° = 81°. ✓
From Figure 3 you can write: “Cricket is the favourite sport of 90 of the 200 pupils, which is 45%. It is more popular than football and hockey together, which have 80 pupils (40%).” A good answer states the numbers and compares them.
Tap the correct answer for each question. Your best score is saved on this device.
1. X: 30 ÷ 100 × 360 = 108°. Y: 180°. Z: 72°. Check: 108 + 180 + 72 = 360°.
2. 126 ÷ 360 × 800 = 0.35 × 800 = 280 people.
3. The increase is 15, so 15 ÷ 60 × 100 = 25%.
1. (a) Friday (42). (b) 24 + 30 + 18 + 36 + 42 = 150 books. (c) 42 ÷ 150 × 100 = 28%.
2. The changes are +3, +6, +3, −5. (a) The greatest rise is between 9 am and 12 noon, by 6°C. (b) It fell by 5°C. (c) 26 − 14 = 12°C.
3. (a) 360 − (90 + 72 + 108) = 90°. (b) Maths 90 ÷ 360 × 400 = 100. Science 72 ÷ 360 × 400 = 80. English 108 ÷ 360 × 400 = 120. Other 90 ÷ 360 × 400 = 100. Check: 100 + 80 + 120 + 100 = 400.
4. Bus: 45 ÷ 120 × 360 = 135°. On foot: 30 ÷ 120 × 360 = 90°. Bicycle: 20 ÷ 120 × 360 = 60°. Car: 25 ÷ 120 × 360 = 75°. Check: 135 + 90 + 60 + 75 = 360°.
5. (a) 33 ÷ 360 × 480 = 44 people. (b) 140 ÷ 480 × 360 = 105°.
6. Class A: 0.40 × 30 = 12. Class B: 0.55 × 20 = 11. Class A has more, by 1 pupil, even though its percentage is lower.
7. (a) (100 − 80) ÷ 80 × 100 = 25% increase. (b) (100 − 75) ÷ 100 × 100 = 25% decrease. Overall, from 80 to 75 is a decrease of 6.25%, so the two changes do not cancel.
| Task | Formula or rule |
|---|---|
| Percentage of a total | value ÷ total × 100% |
| Pie angle from a value | value ÷ total × 360° |
| Pie angle from a percentage | percentage × 3.6° |
| Value from a pie angle | angle ÷ 360° × total |
| Percentage from a pie angle | angle ÷ 360° × 100% |
| Total from a value and its angle | value × 360° ÷ angle |
| Percentage change | (new − old) ÷ old × 100% |
| Average rate of change on a line graph | change in vertical ÷ change in horizontal |
| Pictogram value | number of symbols × value of one symbol |
Start with data the learners collect themselves, such as favourite fruit, how they travel to school or the daily temperature, and ask them to choose which graph suits it. Draw the same data as a bar graph and a pie chart, and ask what each shows best. For pie charts, a paper circle folded into halves, quarters and eighths helps learners see that angles are shares of 360°. Show the two “same data, different axis” graphs from this page and ask learners what is misleading before explaining. The Pie Chart Builder, Pie Chart Converter and Percentage Change Calculator on this page let learners check their own working, but ask them to explain each step in words. Encourage them to write one sentence that states what a graph shows, using numbers.
This article is original EDUSAMBAM writing. Examination requirements differ between boards, so compare your practice with the specification of your own board.
20 questions covering bar graphs, vertical and horizontal graphs, line graphs, scales, comparisons and pie-chart calculations. Answer every question, then submit to see your score instantly.