MDM 4U1
LESSON NOTES

DATE: ____________________

2.1

DATA ANALYSIS WITH GRAPHS

STATISTICS is a branch of mathematics that deals with the collection, analysis, interpretation, and presentation of data. The unprocessed information collected for a study is called raw data. The quantity being measured is the variable. A continuous variable can have any value within a given range, such as the height of students in a class. A discrete variable can have only certain separate values, often integers, such as the number of students in a class.
This section introduces a variety of techniques for displaying data in graphs to help us visualize the characteristics of the data.
EXAMPLE 1 – FREQUENCY TABLES AND DIAGRAMS
The following table shows the age distribution of students at a certain secondary school. Complete the column for the cumulative frequency.
Age

Frequency

12

15

13

25

14

112

15

126

16

102

17

86

18

74

19

12

Cumulative
Frequency

We can show the frequency of the ages using a bar graph or a frequency polygon.
A frequency polygon joins the midpoints of the bars in a bar graph.
120

Where is the greatest proportion of students grouped?

Frequency
Frequency

100

80
60

40
20

12 13 14 15

16 17 18 19 20

Age

We can also graph the cumulative frequency using a polygon.

Cumulative Frequency

500
400

300

200

100

12 13 14 15 16

17 18 19 20

Age

EXAMPLE 2 – RELATIVE FREQUENCY DISTRIBUTION
The following table represents the heights of 50 players on the junior basketball team.
Heights of the Players (cm)
174
161
170
190
152

163
174
182
153
161

175
165
163
172
191

170
159
166
179
170

180
177
187
184
173

167
183
179
182
168

182
185
171
157
181

168
188
191
173
181

153
193
166
178
184

171
177
160
177
182

Complete the following table. Adjust the endpoints for your intervals for a continuous range of values.
Height
Interval
150 – 159
160 – 169
170 – 179
180 – 189
190 – 199

Midpoint

Tallies

Frequency

Relative
Frequency

Relative
Cumulative
Cumulative
Frequency
Frequency

We can show the frequency of the heights using a histogram or a frequency polygon. A histogram is a special form of bar graph in which the bars are connected and represent a continuous range of values.
The frequency polygon joins the midpoints of the histogram.
Are any trends or patterns apparent from the table?

20

Frequency

16

12

What proportion of the players was between 170 and 179 cm?

8

4

150

160

170

180

190

200

Height (cm)
Similarly, the relative and cumulative frequencies can be illustrated.
EXAMPLE 3
Below are a class’ scores on a