1. Types of Data
Discrete Data
Data that can only have a specific value (often whole numbers)
For example
Number of people
You cannot have ½ or ¼ of
a person.
Shoe size
You might have a 6½ or a 7
but not a size
Continuous Data
Data that can have any value within a range
For example
Time
A person running a 100m race could finish
at any time between10 seconds and
30 seconds with no restrictions
Height
As you grow from a baby to an adult you
will at some point every height on the way

2. Averages from Grouped Data
Large quantities of data can be much more easily viewed and managed if placed in groups in a frequency table. Grouped data does not enable exact values for the mean, median and mode to be calculated. Alternate methods of analyising the data have to be employed.
Example 1.
During 3 hours at Heathrow airport 55 aircraft arrived late. The number of minutes they were late is shown in the grouped frequency table below. 2
50 < x ≤ 60 4
40 < x ≤ 50 5
30 < x ≤ 40 7
20 < x ≤ 30 10
10 < x ≤ 20 27
0 < x ≤ 10
frequency
minutes late (x)
Data is grouped into 6 class intervals of width 10.

3. Averages from Grouped Data
Example 1.
During 3 hours at Heathrow airport 55 aircraft arrived late. The number of minutes they were late is shown in the grouped frequency table below.
midpoint(x)
mp x f 2
50 < x ≤ 60 4
40 < x ≤ 50 5
30 < x ≤ 40 7
20 < x ≤ 30 10
10 < x ≤ 20 27
0 < x ≤ 10
frequency
minutes Late
Estimating the Mean: An estimate for the mean can be obtained by assuming that each of the raw data values takes the midpoint value of the interval in which it has been placed. 5
135 15
150 25
175 35
175 45
180 55
110
Mean estimate = 925/55 = 16.8 minutes

4. Averages from Grouped Data
Example 1.
During 3 hours at Heathrow airport 55 aircraft arrived late. The number of minutes they were late is shown in the grouped frequency table below.
The Modal Class 2
50 < x ≤ 60 4
40 < x ≤ 50 5
30 < x ≤ 40 7
20 < x ≤ 30 10
10 < x ≤ 20 27
0 < x ≤ 10
frequency
minutes late
The modal class is simply the class interval of highest frequency.
Modal class =

5. Averages from Grouped Data
Example 1.
During 3 hours at Heathrow airport 55 aircraft arrived late. The number of minutes they were late is shown in the grouped frequency table below.
The Median Class Interval
The Median Class Interval is the class interval containing the median. 2
50 < x ≤ 60 4
40 < x ≤ 50 5
30 < x ≤ 40 7
20 < x ≤ 30 10
10 < x ≤ 20 27
0 < x ≤ 10
frequency
minutes late
(55+1)/2
= 28
The 28th data value is in the class

6. Averages from Grouped Data
Example 2.
A group of University students took part in a sponsored race. The number of laps completed is given in the table below. Use the information to:
(a) Calculate an estimate for the mean number of laps.
(b) Determine the modal class.
(c) Determine the class interval containing the median. 1 2
31 – 35 25
26 – 30 17
21 – 25 20
16 – 20 15
11 – 15 9
6 – 10
1 - 5
frequency (x)
number of laps
Data is grouped into 8 class intervals of width 4.

7. Averages from Grouped Data
mp x f
midpoint(x) 1 2
31 – 35 25
26 – 30 17
21 – 25 20
16 – 20 15
11 – 15 9
6 – 10
1 - 5
frequency
number of laps
Example 2.
A group of University students took part in a sponsored race. The number of laps completed is given in the table below. Use the information to:
(a) Calculate an estimate for the mean number of laps.
(b) Determine the modal class.
(c) Determine the class interval containing the median. 3 6 8 72 13
195 18
360 23
391 28
700 33 66 38 38
Mean estimate = 1828/91 = 20.1 laps

8. Averages from Grouped Data
Example 2.
A group of University students took part in a sponsored race. The number of laps completed is given in the table below. Use the information to:
(a) Calculate an estimate for the mean number of laps.
(b) Determine the modal class.
(c) Determine the class interval containing the median. 1 2
31 – 35 25
26 – 30 17
21 – 25 20
16 – 20 15
11 – 15 9
6 – 10
1 - 5
frequency (x)
number of laps
Modal Class

9. Averages from Grouped Data1 2
31 – 35 25
26 – 30 17
21 – 25 20
16 – 20 15
11 – 15 9
6 – 10
1 - 5
frequency (x)
number of laps
Example 2.
A group of University students took part in a sponsored race. The number of laps completed is given in the table below. Use the information to:
(a) Calculate an estimate for the mean number of laps.
(b) Determine the modal class. 
(c) Determine the class interval containing the median.
The 46th data value is in the 16 – 20 class 
(91+1)/2 = 46

10. Averages from Grouped Data
Worksheet 1
Averages from Grouped Data
Example 1.
During 3 hours at Heathrow airport 55 aircraft arrived late. The number of minutes they were late is shown in the grouped frequency table below.
midpoint(x)
mp x f 2 4 5 7 10 27
0 - 10
frequency
minutes Late

11. Averages from Grouped Data
Worksheet 2
Averages from Grouped Data
Example 2.
A group of University students took part in a sponsored race. The number of laps completed is given in the table below. Use the information to:
(a) Calculate an estimate for the mean number of laps.
(b) Determine the modal class.
(c) Determine the class interval containing the median.
mp x f
midpoint(x) 1 2
31 – 35 25
26 – 30 17
21 – 25 20
16 – 20 15
11 – 15 9
6 – 10
1 - 5
frequency
number of laps