Warm-up
Three quick ones from earlier work. Try each in your head or on paper before you open the answer.
- From Lesson 1: the midpoint of the class 40 to under 60 is ?
Answer
50. (40 + 60) ÷ 2. - From Year 7: the area of a rectangle 6 wide and 4 tall is ?
Answer
24. 6 × 4. - From Year 8: 45 ÷ 9 = ?
Answer
5.
Words used in this lesson
- histogram
- a graph of continuous data in which the area of each bar is the frequency. For example: bars with no gaps and a density axis.
- frequency density
- frequency divided by class width. For example: values per minute of waiting time.
- class width
- upper boundary minus lower boundary. For example: the width of the class 30 to under 45.
- continuous data
- data that can take any value on a scale. For example: heights or times.
- area
- width times height; in a histogram, the frequency. For example: the size of a bar.
- modal class
- the class with the highest frequency density. For example: the tallest bar when the heights show density.
The age bands
A town in Chile publishes its population by age in unequal bands: 0 to under 5, 5 to under 18, 18 to under 65, and 65 to under 100. A reporter draws one bar per band and makes each bar's height equal to the number of people in it.
Predict first: which bar will look the biggest, and is that band really the busiest per year of age? Can a picture be dishonest even when every bar is the right height for its number? Use the first figure to switch what the height shows, then the second to see how width and height trade off.
Class 0 to under 10: 16 values ÷ width 10 = frequency density 1.6, and 1.6 × 10 = 16.
Show me the steps
- Height is frequency: the class 20 to under 40 has the tallest bar (30 values) and the class 40 to under 80 is the widest. The picture suggests the wide classes dominate.
- Press Height is frequency density: the heights become 1.6, 2.1, 1.5 and 0.5. Now the class 10 to under 20 is tallest, and the widest class is the lowest.
- Slide to class 4: 20 values over a width of 40 is a density of 0.5, but it still has an area of 0.5 × 40 = 20. The area, not the height, shows the frequency.
Frequency density makes area show the frequency.
- Frequency density = frequency ÷ class width. It is the number of values per unit of the horizontal scale.
- In a histogram the area of a bar is the frequency: width × height = frequency.
- The vertical axis is labelled “frequency density”, not “frequency”.
- With equal widths, density is proportional to frequency, so the tallest bar is also the modal class. With unequal widths, the modal class is the one with the highest density.
There are no gaps between bars of a histogram, because the horizontal axis is a continuous scale.
Frequency 30 over a width of 20: height 30 ÷ 20 = 1.5, and the area 20 × 1.5 = 30.
Show me the steps
- Frequency 30, width 20: height 30 ÷ 20 = 1.5 and area 20 × 1.5 = 30.
- Slide the width to 10: the same 30 values are squeezed into half the width, so the height doubles to 3 and the area is still 30.
- Slide the frequency to 60 with the width still 10: the height is 6 and the area is 60. Doubling the values doubles the area.
Do not read a bar's height as a frequency. A histogram with unequal widths has a vertical axis in values per unit, and the number of values is found by multiplying the height by the width. And the tallest bar is not always the class with the most values: a wide, low bar can hold more. The tallest bar shows the highest density; to find how many values a class holds, use its area.
Worked example: journey times
A histogram has bars on 0 to under 20 minutes (height 1.5), 20 to under 30 (height 4.2) and 30 to under 60 (height 0.8). Find each frequency and the total.
- 0 to under 20: width 20, so 20 × 1.5 = 30.
- 20 to under 30: width 10, so 10 × 4.2 = 42.
- 30 to under 60: width 30, so 30 × 0.8 = 24.
- The total is 30 + 42 + 24 = 96 journeys.
Your go. The first step is given. To draw a histogram, find the height of each bar for: 15 to under 25 (36 values), 25 to under 40 (45 values) and 40 to under 60 (26 values).
- The widths are 10, 15 and 20.
- Divide each frequency by its width to get the height.
Check the answer
36 ÷ 10 = 3.6, 45 ÷ 15 = 3 and 26 ÷ 20 = 1.3. A common slip is to use the frequencies as heights, which makes the last two bars look far larger than they are per unit.
Check before you go on
1. On a histogram, a bar is 8 units wide and 2.5 units tall on the frequency-density axis. How many values does it represent?
Your turn
Start at Level 1 and go as far as you can. Type negative numbers with a minus sign, for example -7. A wrong answer gets a reason, not just a cross: read it, then try again.
Level 1 · Getting started
2. A class 12 units wide holds 54 values. What is its frequency density?
3. A histogram bar is 5 units wide and 3.6 units tall on the frequency-density axis. What is the frequency of the class?
Level 2 · Building
4. The class 10 to under 25 has frequency 33. What is its frequency density?
5. A histogram has these bars: 0 to under 10 with height 1.8; 10 to under 30 with height 2.4; 30 to under 60 with height 0.6. How many values are there in total?
Level 3 · Stretch
6. On a histogram the class 5 to under 10 has frequency density 4.8. The class 10 to under 30 holds the same number of values. What is its frequency density?
7. A histogram has bars: 0 to under 10 (height 2), 10 to under 15 (height 6), 15 to under 25 (height 4) and 25 to under 45 (height 1). Estimate the median, assuming values are spread evenly within a class.
Something to think about
A histogram has one very wide class at the right with a tiny bar, and the report says that few people are in it. Is that a safe conclusion? What would you check?
One way to think about it
Not safe on its own. A tiny height means a low density, but a very wide class can still hold many values: area is height times width. Check the width and find the area, or read the frequency from the table. A low bar spread across a wide class may hold more values than a tall bar in a narrow class.
Exit ticket
8. On a histogram the bar for 40 to under 60 has height 0.7 on the frequency-density axis. How many values are in that class?