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Inequalities
Recap
Inequalities are:
< Less Than
 Equal to or Less Than
> Greater Than
 Equal to or Greater Than
Inequalities
Notation on the numbers line
Note: the open circle denotes
x<2
-10 -9
-8 -7
-6 -5
-4 -3 -2
x can be very close to, but not equal to 2
-1
0
1
2
3
4
5
-8 -7
-6 -5
-4 -3 -2
7
8
9
Note: the closed circle denotes
x 2
-10 -9
6
x can be very close to, AND equal to 2
-1
0
1
2
3
4
5
6
7
8
9
Means
-10 -9
-10 -9
-8 -7
-8 -7
-6 -5
-6 -5
-4 -3 -2
-4 -3 -2
-1
-1
0
0
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
-6
< x 4
Means
x  -3 or x > 3
Inequalities
List the integer values for -3 < x  1
-10 -9
-8 -7
-6 -5
-4 -3 -2
-1
0
-2, -1,
1
2
3
4
5
6
7
8
0, 1
9
List the integer values for 7  x > -1
0, 1,2,3,4,5,6,7
-10 -9
-8 -7
-6 -5
-4 -3 -2
-1
0
1
2
3
4
5
6
7
8
9
List the integer values for 6  3x > -4
 3 2 x >
-10 -9
-8 -7
-6 -5
-4 -3 -2
-1
0
1
2

3
4
3
-1,
4
5
6
7
8
9
0, 1, 2
Inequalities
Now try these:
List all the integer values that satisfy these inequalities:
-2, -1, 0, 1, 2, 3
1. -2  x  3
-3, -2, -1
2. -4 < x < 0
-2, -1, 0, 1, 2, 3
3. -8  4n  15
-1, 0, 1, 2, 3, 4, 5, 6
4. -3 < 2n  12
-2, -1, 0, 1, 2, 3
5. -5  2n-1 < 6
Types of Data
Discrete Data
Data that can only have a specific value (often whole numbers)
For example
Number of people
Shoe size
You cannot have ½ or ¼ of
a person.
You might have a 6½ or a 7
but not a size 6.23456
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
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.
Data is
grouped
into 6 class
intervals of
width 10.
minutes late (x)
frequency
0 < x ≤ 10
27
10 < x ≤ 20
10
20 < x ≤ 30
7
30 < x ≤ 40
5
40 < x ≤ 50
4
50 < x ≤ 60
2
Averages from Grouped Data
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.
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.
minutes Late
frequency
0 < x ≤ 10
27
5
10 < x ≤ 20
10
15
20 < x ≤ 30
7
30 < x ≤ 40
5
25
35
40 < x ≤ 50
4
45
180
55
110
50 < x ≤ 60
2
f  55
midpoint(x)
mp x f
135
150
175
175
f x
 925
Mean estimate = 925/55 = 16.8 minutes
Averages from Grouped Data
The Modal Class
The modal class is simply the class interval of highest frequency.
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.
minutes late
frequency
0 < x ≤ 10
27
10 < x ≤ 20
10
20 < x ≤ 30
7
30 < x ≤ 40
5
40 < x ≤ 50
4
50 < x ≤ 60
2
Modal class = 0 - 10
Averages from Grouped Data
The Median Class Interval
The Median Class Interval is the class interval containing the median.
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.
minutes late
frequency
0 < x ≤ 10
27
10 < x ≤ 20
10
20 < x ≤ 30
7
30 < x ≤ 40
5
40 < x ≤ 50
4
50 < x ≤ 60
2
(55+1)/2
= 28
The 28th data value is in the 10 - 20 class
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.
Data is
grouped
into 8 class
intervals of
width 4.
number of laps
frequency (x)
1-5
2
6 – 10
9
11 – 15
15
16 – 20
20
21 – 25
17
26 – 30
25
31 – 35
2
36 - 40
1
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.
number of laps
frequency
1-5
2
6 – 10
9
11 – 15
15
16 – 20
20
21 – 25
17
26 – 30
25
31 – 35
2
36 - 40
1
f
 91
midpoint(x)
3
8
13
18
23
28
33
38
mp x f
6
72
195
360
391
700
66
 fx
38
 1828
Mean estimate = 1828/91 = 20.1 laps
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.
number of laps
frequency (x)
1-5
2
6 – 10
9
11 – 15
15
16 – 20
20
21 – 25
17
26 – 30
25
31 – 35
2
36 - 40
1
Modal Class 26 - 30
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. 
number of laps
frequency (x)
1-5
2
6 – 10
9
11 – 15
15
16 – 20
20
21 – 25
17
26 – 30
25
31 – 35
2
36 - 40
1
f
 91
(91+1)/2 =
46
The 46th data value is in the 16 – 20
class
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.
minutes Late
frequency
0 - 10
27
10 - 20
10
20 - 30
7
30 - 40
5
40 - 50
4
50 - 60
2
midpoint(x)
mp x f
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.
number of laps
frequency
1-5
2
6 – 10
9
11 – 15
15
16 – 20
20
21 – 25
17
26 – 30
25
31 – 35
2
36 - 40
1
midpoint(x)
mp x f