Warm-up
Three quick ones from earlier work. Try each in your head or on paper before you open the answer.
- (Lesson 3 of this unit) In a frequency table the frequencies are 4, 6, 5 and 5. How many values were tallied?
Answer
20. 4 + 6 + 5 + 5 = 20. - (Unit 2, lesson 1) 9 of 20 students walk to school. What percentage is that?
Answer
45%. Each student is 100 ÷ 20 = 5%, and 9 × 5 = 45. - (Unit 1, lesson 7) Round 2.6 to the nearest whole number.
Answer
3. The tenths digit is 6, which is 5 or more, so round up.
Words used in this lesson
- bar chart
- bars of equal width whose heights show frequencies, for categories or counts. For example: favourite sport.
- histogram
- touching bars for grouped measurements on a number line. For example: heights in classes 2 cm wide.
- scale
- the numbers and steps along an axis. For example: 0, 5, 10, 15.
- axis
- a line with a scale and a label. For example: the vertical axis shows frequency.
- gridline
- a line across a chart that helps read a value. For example: a line at 10.
- title
- the words that say what the chart shows. For example: Favourite sport of 40 students.
A table is exact, a bar is quick
A sports teacher asked 40 students for their favourite sport. Football won, with 14 students, and she wrote every count in a table. The table is exact, but a bar for each makes the biggest and smallest jump out. Predict first: next to the Football bar, how tall will the bar for Other, the smallest group, look? Then draw it, and change the scale.
Gridlines every 5, scale to 15. The Football bar has frequency 14, which is 35% of the 40 students.
Show me the steps
- Gridlines every 2: the scale runs 0 to 14. The Football bar fills the whole height.
- Gridlines every 5: the scale runs 0 to 15. The Football bar stops just below the 15 line, at 14.
- Gridlines every 10: the scale runs 0 to 20. The Football bar is 14 out of 20 of the height, 70%.
- Reading the Hockey bar: its top is between the 5 and 10 lines when the step is 5, and its frequency is 9, which is 22.5% of 40.
A bar's height is its frequency. The bars are the same width, with equal gaps between them, so only the height carries the number. The vertical axis is an even scale starting at 0, with the same step between every pair of gridlines.
Why a good chart has a title and both axes labelled: a reader can only trust a bar if they know what was counted and which unit the scale is in.
Back to the sports table: with gridlines every 5, the Football bar stops just below the 15 line, because 14 is one short of 15. And Other, at 4, is a little over a quarter of the Football bar: 4 is more than a quarter of 14.
When the categories are measurements
For sports the bars are separate groups with nothing between them, so there are gaps. Measurements are different: the class “8 to under 10” is followed straight away by “10 to under 12”. Draw lesson 3's plant heights, then try the gaps.
Classes 2 cm wide, bars touching. The tallest bar has frequency 4.
Show me the steps
- Classes 2 cm wide, touching: 8 bars with heights 0, 2, 3, 4, 4, 4, 2 and 1. They add to 20.
- Classes 4 cm wide, touching: 4 bars with heights 2, 7, 8 and 3. The tallest is 8.
- Classes 8 cm wide, touching: 2 bars with heights 9 and 11. The tallest is 11.
- Same classes with gaps: the heights do not change, but the gaps wrongly suggest there are no plants between one class and the next.
A histogram has touching bars. Bars for classes sit on a number line, and each bar starts exactly where the one before ends. For categories and counts, such as sports or numbers of siblings, a bar chart has gaps.
The classes have equal widths, so the height of a bar is the frequency and the bars can be compared by height.
A common mistake: “Bars should have gaps, it looks neater.” For measurements a gap says there are no values there, and that is false. Gaps belong to separate categories only.
Worked example: drawing a bar chart
40 students travel to school: Walk 12, Bus 18, Car 7, Cycle 3. Plan a bar chart with a vertical axis 10 cm tall.
- The largest frequency is 18, so the scale must go up to at least 18. Gridlines every 5 give a top of 20.
- The axis is 10 cm for 20 students, so 1 student is 10 ÷ 20 = 0.5 cm.
- Heights: Walk 12 × 0.5 = 6 cm, Bus 18 × 0.5 = 9 cm, Car 7 × 0.5 = 3.5 cm, Cycle 3 × 0.5 = 1.5 cm. 6 cm, 9 cm, 3.5 cm and 1.5 cm.
Your go. The first step is given. Times of 20 runners, in minutes: 10 to under 15 has 3 runners, 15 to under 20 has 8, 20 to under 25 has 6, and 25 to under 30 has 3. Plan a histogram.
- Times are measured, so the bars touch with no gaps.
- Now choose the top of the frequency scale if gridlines are every 2, and find how many runners took 20 minutes or more, then check.
Check the answer
The tallest bar is 8, so gridlines every 2 give a top of 8. Runners of 20 minutes or more: 6 + 3 = 9. Check the total: 3 + 8 + 6 + 3 = 20.
Check before you go on
1. A chart shows the masses of 30 parcels in classes 0 to under 5 kg, 5 to under 10 kg and 10 to under 15 kg. Which feature is right?
Your turn
Start at Level 1 and go as far as you can. A wrong answer gets a reason, not just a cross: read it, then try again.
Level 1 · Getting started
2. A bar chart has gridlines labelled 0, 10, 20 and 30. The top of the bar for Cycle is exactly halfway between the 10 line and the 20 line. What is its frequency?
3. On a bar chart, the Walk bar has height 9, the Bus bar 14 and the Car bar 6. How much taller is the Bus bar than the Car bar?
Level 2 · Building
4. Which list has everything a good bar chart needs?
5. A bar chart is drawn so that 1 cm of bar height stands for 4 students. The Netball bar is 3.5 cm tall. How many students chose Netball?
6. A histogram of 40 values has four bars. The first three have heights 8, 14 and 10 on the frequency axis. How tall is the fourth bar?
Level 3 · Stretch
7. The largest frequency on a bar chart is 37. The gridlines are every 5. What is the largest gridline value the axis needs so that every bar fits?
8. A histogram shows ages in classes 10 to under 20, 20 to under 30 and 30 to under 40, with bars of heights 6, 12 and 7. Which statement is correct?
Something to think about
Two students draw the same bar chart. One has gridlines every 2 up to 14, the other every 10 up to 20. Whose chart makes a difference of 5 students look bigger? Is either one dishonest?
One way to think about it
The 2-step chart: 5 students takes up 5/14 of the height, about 36%, while on the 10-step chart it is 5/20, which is 25%. The first makes differences look bigger. Neither is dishonest if the scale is labelled and starts at 0, because the numbers can be read. But a reader who only glances at the bars should be told the scale, so label it clearly.
Exit ticket
9. Which data would be shown with a histogram, bars that touch?