Week 8 · Organizing data: statistical questions, frequency tables, dot plots, histograms, stem-and-leaf and box plots · 3 lessons
2
Frequency tables with equal intervals and histograms
Mathematics Prep 1 Term 1, Unit 3, Lesson 1
GoalGroup data into equal intervals, make a frequency table and draw and read a histogram.
New words
range · the greatest value minus the smallest valueinterval · a group of values, such as 10 to under 20histogram · a graph of columns that touch, one for each intervalinterval length · the width of each interval
Explanation
When data has many different values, we group it into equal intervals. First find the range: the greatest value minus the smallest. Then choose a convenient interval length, such as 10, so that there are about four to eight intervals. Intervals must not overlap, so we write “10 to under 20”: 10 belongs to this interval and 20 belongs to the next. Count the data in each interval to make the frequency table. A histogram draws one column per interval. The columns touch because the intervals fill the number line with no gaps, and they all have the same width. A histogram shows intervals, not the exact values.
★Remember
Equal intervals, no overlaps, columns touch. A histogram shows intervals, not exact values.
Worked examples
Twenty pupils took these minutes to walk to school: 7, 12, 25, 18, 9, 14, 31, 22, 16, 5, 27, 11, 19, 8, 35, 21, 13, 29, 17, 24. The smallest is 5 and the greatest is 35, so the range is 35 − 5 = 30. An interval length of 10 gives four intervals, which is convenient.
Time (minutes)
Frequency
0 to under 10
4
10 to under 20
8
20 to under 30
6
30 to under 40
2
Count the walks in each interval. Under 10: 7, 9, 5, 8 gives 4. From 10 to under 20: 12, 18, 14, 16, 11, 19, 13, 17 gives 8. From 20 to under 30: 25, 22, 27, 21, 29, 24 gives 6. From 30 to under 40: 31, 35 gives 2. Check: 4 + 8 + 6 + 2 = 20.
The histogram of the same data. The columns touch and have the same width. The tallest column is 10 to under 20 with 8 pupils. Pupils who walk 20 minutes or more: 6 + 2 = 8.
Class activity
Make your own histogram. With a grown-up, measure the length of 12 pencils or leaves in cm. Find the range, choose intervals, make a frequency table and draw the touching columns on squared paper.
Practice
1
The data are 5, 9, 14, 20, 35. What is the range?
35 − 5 = 30.
2
Masses in kg: 41, 52, 47, 63, 58, 45, 70, 66. Make a frequency table with intervals of 10 starting at 40.
40 to under 50: 41, 47, 45 gives 3. 50 to under 60: 52, 58 gives 2. 60 to under 70: 63, 66 gives 2. 70 to under 80: 70 gives 1. Total 8.
3
In which interval does a walk of exactly 30 minutes go?
In “30 to under 40”, because each interval includes its starting value.
4
How many pupils walk less than 20 minutes in the histogram?
4 + 8 = 12 pupils.
5
Mina draws a histogram with gaps between the columns. What is wrong?
The intervals follow each other without gaps, so the columns must touch. Gaps belong to a bar graph of separate values.
