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Circle graphs

Statistics and data · level 4, lesson 3 of 6

A question to start

A class of 40 students is asked how they get to school. Without counting anything, look at this picture:

Walk45% Bus25% Car20% Bike10% How 40 studentsget to school Walk 18Bus 10Car 8Bike 4

You can see at once that almost half walk, and that bus riders make up a quarter, a slice like the corner of a square. That is what a circle graph (also called a pie chart) is good at: showing how one whole is split into parts. But why is the bus slice a perfect right angle? And when is a circle graph the wrong choice? That is this lesson.

The idea

Parts of a whole

A circle graph shows data where:

  • there is one whole (all 40 students, a whole day, a whole budget);
  • the whole is split into parts that do not overlap (each student is in exactly one group);
  • the parts add up to the whole.

The whole circle stands for the whole group, 100%. Each slice, called a sector, stands for one part. A bigger part gets a bigger slice.

Percents: parts of 100

A percent means "out of 100" (per cent: "for each hundred"). 45% means 45 out of every 100. To turn a part into a percent, write it as a fraction of the whole and then scale it to "out of 100":

\[\text{percent} = \frac{\text{part}}{\text{whole}} \times 100\]

For the walkers: \(\frac{18}{40} \times 100 = 0.45 \times 100 = 45\%\). For the bus: \(\frac{10}{40} = \frac{1}{4} = 25\%\). The four percents add to 100%: \(45 + 25 + 20 + 10 = 100\). They must, because the parts make up the whole.

From percents to angles

The whole circle is a full turn, 360° (degrees). Each part gets the same fraction of 360° as it is of the whole group.

\[\text{angle} = \frac{\text{part}}{\text{whole}} \times 360°\]

Why is this right? A sector's size is set by its angle at the centre. If one part is a quarter of the group, it should get a quarter of the circle, and a quarter of a full turn is \(360 \div 4 = 90°\): a right angle. That is why the bus slice (25%, a quarter) looks like the corner of a square.

50% = 180°25% = 90° 10% = 36°75% = 270°

A handy shortcut: since 100% is 360°, 1% is 3.6°, and an angle is the percent times 3.6.

Way to school Students Fraction Percent Angle
Walk 18 \(\frac{18}{40}\) 45% \(0.45 \times 360 = 162°\)
Bus 10 \(\frac{10}{40}\) 25% \(0.25 \times 360 = 90°\)
Car 8 \(\frac{8}{40}\) 20% \(0.20 \times 360 = 72°\)
Bike 4 \(\frac{4}{40}\) 10% \(0.10 \times 360 = 36°\)
Total 40 1 100% 360°

Always check that the angles add to 360°. \(162 + 90 + 72 + 36 = 360\) ✓. If rounding makes them add to 359° or 361°, adjust the biggest slice by a degree.

Drawing a circle graph

  1. Make the table: count, fraction, percent, angle. Check the total angle is 360°.
  2. Draw a circle with compasses and mark the centre. Draw one radius straight up from the centre (the "12 o'clock" line).
  3. With a protractor centred on the centre, measure the first angle (162°) clockwise from that line, and draw the next radius.
  4. Measure the next angle (90°) from the line you just drew, and so on. The last slice should close up exactly; if it does not, check your measuring.
  5. Label every slice with its name and its percent (or count), and give the graph a title.

Reading a circle graph backwards

If you know the angle, the fraction of the whole is \(\text{angle} \div 360\). If you also know the size of the whole, multiply to get the count. A slice of 72° out of 40 students: \(\frac{72}{360} = \frac{1}{5}\), and \(\frac{1}{5}\) of 40 is 8 students.

When a circle graph fits, and when it does not

A circle graph fits when you want to show shares of one whole, with a few parts (about 2 to 6).

It does not fit when:

  • The parts do not make a whole. Temperatures for each day of a week are not parts of anything; nobody can say what "Tuesday's share of the week's temperature" means. Use a line graph.
  • People could choose more than one answer. In "which sports do you play? (tick all that apply)", one student can be in three groups. The percents add up to more than 100%, and slices would overlap. Use a bar graph.
  • There are many small parts. Twelve slices of about 8% each look nearly identical, and thin slivers cannot be labelled. A bar graph is clearer.
  • You want exact comparisons. People judge angles and areas less accurately than lengths: 23% and 27% look almost the same as slices, but as bars side by side the difference is obvious. Bars also make it easier to compare two groups; comparing two pies means hunting for each slice in each pie.

Circle graphs show shares, not sizes. A pie of 40 students and a pie of 4000 students look the same size. If the size of the whole matters, write it in the title.

Worked examples

1. A day in hours. A student's school day (24 hours): sleep 9, school 7, play 3, eating 2, other 3.

  1. Check the whole: \(9 + 7 + 3 + 2 + 3 = 24\) ✓.
  2. Angles: each hour is \(360 \div 24 = 15°\). Sleep \(9 \times 15 = 135°\), school \(7 \times 15 = 105°\), play \(3 \times 15 = 45°\), eating \(2 \times 15 = 30°\), other \(3 \times 15 = 45°\).
  3. Check: \(135 + 105 + 45 + 30 + 45 = 360\) ✓.
  4. Percents: sleep \(\frac{9}{24} = 37.5\%\), play \(\frac{3}{24} = 12.5\%\).

2. From percents. A survey of favourite fruit gives apples 30%, bananas 15%, grapes 35%, oranges 20%.

  1. Percents add to \(30 + 15 + 35 + 20 = 100\) ✓.
  2. Angles (percent × 3.6): apples \(30 \times 3.6 = 108°\), bananas \(54°\), grapes \(126°\), oranges \(72°\).
  3. Check: \(108 + 54 + 126 + 72 = 360\) ✓.

3. Reading backwards. In a circle graph of 120 library books, the "science" slice has an angle of 48°. How many science books?

\(\frac{48}{360} = \frac{2}{15}\), and \(\frac{2}{15} \times 120 = 16\) books. (Or: each book is \(360 \div 120 = 3°\), and \(48 \div 3 = 16\).)

4. Is a circle graph right? (a) How a club spends its yearly budget on equipment, trips, food, prizes: yes, parts of one whole. (b) The number of goals scored in each month of a season: no, these are values over time; use a bar or line graph. (c) "Which pets have you ever owned?" with answers adding to 160% of the class: no, the answers overlap.

Try it

  1. What angle stands for: (a) 50%; (b) 10%; (c) 5%; (d) 1%?
  2. Of 60 students, 15 chose red. What percent is that, and what angle?
  3. A slice has an angle of 120°. What fraction and roughly what percent of the whole is it?
  4. Make a table for 30 people's favourite season: winter 6, spring 9, summer 12, autumn 3. Find each percent and angle and check the total.
  5. A circle graph of 200 votes has a 54° slice for "maybe". How many people voted "maybe"?
  6. A newspaper's circle graph has slices of 40%, 35%, 20% and 15%. What is wrong?
  7. Why would a circle graph be a poor way to show the heights of the 24 students from the histograms lesson?
  8. Explain in your own words why 1% of a circle graph is 3.6°.
Answers
  1. (a) 180°. (b) 36°. (c) 18°. (d) 3.6°.
  2. \(\frac{15}{60} = \frac{1}{4} = 25\%\), angle 90°.
  3. \(\frac{120}{360} = \frac{1}{3}\), about 33.3%.
  4. Winter \(\frac{6}{30} = 20\%\), 72°. Spring \(\frac{9}{30} = 30\%\), 108°. Summer \(\frac{12}{30} = 40\%\), 144°. Autumn \(\frac{3}{30} = 10\%\), 36°. Total \(72 + 108 + 144 + 36 = 360°\) ✓ and 100% ✓.
  5. \(\frac{54}{360} = 0.15\), and \(0.15 \times 200 = 30\) people.
  6. The percents add to \(40 + 35 + 20 + 15 = 110\%\). Parts of one whole must add to 100%: either there is an error, or people could pick more than one answer, and then a circle graph is the wrong graph.
  7. Heights are not parts of a whole: nothing is "shared out" among the students. You want the spread of the values, so a histogram or stem-and-leaf plot is right.
  8. 100% is the whole circle, 360°. Dividing both by 100, 1% is \(360 \div 100 = 3.6°\).

Common mistake

Using the percent as the angle. A 25% slice drawn at 25° is far too small; the angle is \(25 \times 3.6 = 90°\). Check that your angles add to 360°, not 100.

Forgetting the size of the whole. "60% prefer our cereal" means something very different if 5 people were asked or 5000. Say how many are in the whole.

Where it leads