Bar Charts, Pictograms and Pie Charts
A statistical chart should make a comparison easy to understand without distorting the data. Choose a display that suits the variable, then show a clear scale, labels and any key the reader needs.
Foundation & Higher. Use Frequency Tables and Stem-and-Leaf Diagrams first. Pie charts apply fractions and percentages to the 360° in a full turn.
Construct a bar chart for categories
A bar chart compares frequencies for separate categories. Use bars of equal width, gaps between them, a labelled frequency axis starting at zero, and an even scale. The order of unordered categories can be chosen for clarity.
| Method | f |
|---|---|
| A: walk | 15 |
| B: bus | 20 |
| C: cycle | 10 |
| D: car | 15 |
Worked example — build and interpret the bars
There are 60 choices altogether. Mark 0, 5, 10, 15 and 20 at equally spaced positions on the vertical axis. Draw A to 15, B to 20, C to 10 and D to 15. Bus is the modal category; its frequency is 20. Twice as many choose bus as cycle.
Bar height represents frequency because every bar has the same width. Do not use wider bars to make a category look more important.
Use vertical lines for ungrouped discrete values
For a discrete numerical variable, a vertical line chart can show the frequency at each numerical value. Keep positions in numerical order and space the values according to their size; leave missing values on the scale.
Here the horizontal labels are numbers of books, not category codes. The line above 2 reaches frequency 5: five students borrowed two books. Adding the frequencies gives 12 students. Do not join the tops as if they were successive times.
Compare groups and parts with bar charts
A comparative bar chart places a bar for each group beside the corresponding category. A composite or stacked bar divides a total into parts. Use a key and consistent colours.
Key: blue = Week 1; orange = Week 2. R = reading, M = music, S = sport. Sport increases from 10 to 20; music falls from 18 to 14. Week totals are 40 and 50, so compare percentages when the question concerns proportions.
Worked example — read a stacked section
Blue means walk and orange means bus. The top of B is 30, but its orange section has height 30 − 18 = 12. Although B has more walkers, both groups have the same walking proportion: 12/20 = 18/30 = 60%.
Use a pictogram key consistently
A pictogram uses repeated identical symbols to represent a fixed number. The key matters: one symbol need not mean one item. A half-symbol represents half the key amount; it should visibly be half of the same symbol.
Worked example — read and extend the pictogram
The key is one circle = 4 books. Tuesday has 2½ circles, so it represents 2.5 × 4 = 10 books. To show 14 books on Thursday, draw 3½ circles. The three displayed days total 12 + 10 + 16 = 38 books.
Enlarging a picture in both width and height exaggerates the difference in area. Use equal-size repeated symbols rather than different-size pictures.
Construct a pie chart from proportions
A pie chart divides one whole into sectors. Use it when each observation belongs to exactly one category. The angles must add to 360°.
Sector angle = frequency ÷ total × 360°. For the 60 travel choices, one choice represents 360 ÷ 60 = 6°.
| Code | f | Angle |
|---|---|---|
| A | 15 | 90° |
| B | 20 | 120° |
| C | 10 | 60° |
| D | 15 | 90° |
| All | 60 | 360° |
Worked example — draw the sectors
Draw a circle and one radius. With a protractor centred correctly, measure 90° from that radius for A. From the new radius, measure 120° for B, then 60° for C; the remaining 90° is D. Label the sectors or give a key. Check the full turn.
Sector size shows a proportion. Its angle does not directly show a count until you know the total. This pie chart could also describe 120 choices with every frequency doubled.
Recover counts and compare pie charts
Worked example — work backwards
A 72° sector represents 72/360 = 1/5 of the total. If there are 150 responses, its count is 150 ÷ 5 = 30. If instead that sector represents 18 responses, the total is 18 × 360 ÷ 72 = 90.
Worked example — different totals change the comparison
In one survey, a 90° sector represents 1/4 of 80 people: 20. In another, a 72° sector represents 1/5 of 150: 30. The smaller angle represents more people because its survey is larger.
State whether you are comparing numbers or proportions. Do not compare sector areas from circles of different sizes as if they were frequencies.
Spot misleading scales and unsuitable displays
For bar heights, a zero baseline makes height ratios meaningful. If bars for 48 and 52 start at 45, the visible heights are 3 and 7: the second looks more than twice as tall even though the actual values differ by only 4. A clearly marked truncated axis can help inspect variation on some line graphs, but it changes the visual impression and must be interpreted carefully.
Check equal numerical intervals occupy equal distances, axes are labelled, and categories do not overlap. Use a pie chart for parts of one whole; use a time-series graph for values recorded over time. Use scatter graphs for pairs of measurements, not a bar chart of unrelated values.
Think Like a Mathematician
The same data can support several displays. A bar chart makes counts easy to compare; a pie chart emphasises shares of the whole. The best choice depends on the question you want the reader to answer.
Common Mistakes
Stacked-bar reading: subtract the lower boundary from the upper one.
Pie-chart arithmetic: divide by the total before multiplying by 360°.
Different totals: a larger sector proportion need not mean more people.
Missing or distorted scales: equal numerical steps need equal distances.
Practice
1. A pictogram uses one symbol for 6 people. What do 3½ symbols represent?
3.5 × 6 = 21 people.
2. How many symbols represent 27 people with that key?
27 ÷ 6 = 4½ symbols.
3. A category has 18 of 72 responses. Find its pie-chart angle.
18/72 × 360° = 90°.
4. A sector is 120° in a survey of 90 people. Find the frequency.
120/360 × 90 = 30 people.
5. A 45° sector represents 15 people. Find the total.
45° is 1/8 of a circle, so the total is 15 × 8 = 120 people.
6. A stacked section goes from frequency 14 to 32. What is its frequency?
32 − 14 = 18, not 32.
7. A 90° sector represents a group of 80; a 60° sector represents a group of 150. Which contains more people?
First: 90/360 × 80 = 20. Second: 60/360 × 150 = 25. The 60° sector contains more people.
8. Draw a bar chart for A:6, B:10, C:4. State three necessary features.
Bars must have heights 6, 10 and 4, equal widths and gaps; use labelled categories and a uniformly scaled frequency axis starting at zero. Total frequency is 20.
9. Why can bars for 101 and 102 look misleading if the axis starts at 100?
Their visible heights become 1 and 2, suggesting a doubling. The actual increase is only 1, about 0.99% of 101.
Frequently Asked Questions
Are bar charts and histograms interchangeable?
No. Bar charts compare separate categories or ungrouped values. Histograms use numerical class widths and make area represent frequency; that is a Higher lesson.
Should every graph start at zero?
Bar-height comparisons usually need zero. Some other graphs use a clearly labelled restricted range to show detail; always read the scale before judging the size of a change.