Discover Mathematics P6: Term 1, Unit 6, Lessons 1–2
GoalRead a histogram, understand intervals of equal width and tell a histogram from a bar graph.
New words
histogram · bars that show how many numerical data values fall in each intervalinterval · a range of values such as 10–19frequency · how many data values fall in an intervalbar graph · bars for categories, with gaps between them
Explanation
When the numbers are many and different, a dot plot gets crowded. A histogram groups them into intervals of the same width, for example 0–9, 10–19, 20–29. Every value belongs to exactly one interval, so the intervals must not overlap. The height of each bar is the frequency, the number of values in that interval. In a histogram the bars touch because the intervals follow each other on the number line. In a bar graph the bars are for categories and there are gaps between them. Add the frequencies and you get the number of data values. (Our drawings leave a thin gap between the bars; in your notebook draw them touching.) A histogram does not show exact values: we know 7 pupils read between 10 and 19 minutes, but not how long each one read.
★Remember
Histogram: numerical data, equal intervals, bars touch, height = frequency. Bar graph: categories, gaps between bars.
Worked examples
Twenty pupils told us how many minutes they read last night. The histogram shows 3 pupils in 0–9, 7 in 10–19, 6 in 20–29 and 4 in 30–39. The tallest bar is 10–19. Check: 3 + 7 + 6 + 4 = 20.
Use the same histogram to answer questions. How many pupils read fewer than 20 minutes? 3 + 7 = 10 pupils. How many read at least 20 minutes? 6 + 4 = 10 pupils. Can we say how many pupils read exactly 15 minutes? No: the histogram only says 7 pupils are in 10–19.
Bar graph
Histogram
Data
categories
numbers in intervals
Bars
gaps between
touch
Order
any order
number-line order
Bar graph or histogram? Favourite sports are categories, so we draw a bar graph with gaps. Ages are numbers that can be grouped, so we draw a histogram. Unequal intervals such as 0–9 and then 10–29 are not allowed, because the wide bar would look big only because it is wide.
Class activity
Histogram on squared paper. Measure the length of the right foot of 12 people in your family and neighbourhood, in cm, rounded to whole numbers. Use the intervals 10–14, 15–19, 20–24, 25–29, count the values in each, and draw the bars on squared paper so that they touch.
Practice
1
The histogram shows the heights of 24 pupils. How many pupils are in 140–149? How many pupils are there altogether?
9 pupils are in 140–149. Altogether 4 + 9 + 8 + 3 = 24 pupils.
2
In the same histogram, which interval has the fewest pupils, and how many pupils are shorter than 150 cm?
160–169 has the fewest (3 pupils). Shorter than 150 cm: 4 + 9 = 13 pupils.
3
Why are the intervals 0–10, 10–20, 20–30 a problem for whole-number data?
The value 10 would belong to two intervals. Use 0–9, 10–19, 20–29 so that each value has exactly one place.
4
Which display fits: (a) the favourite colour of 30 pupils, (b) the masses of 30 school bags?
(a) a bar graph, because colours are categories. (b) a histogram, because masses are numbers that can be grouped.
5
Sort these 12 test scores into the intervals 0–4, 5–9, 10–14, 15–19: 3, 8, 12, 17, 9, 14, 5, 19, 11, 6, 13, 10.
