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Slide2Median, Quartiles, Inter-Quartile Range and Box Plots.
Measures of Spread
Remember: The range is the measure of spread that goes with the mean.
Mean
= 7 + 5 + 2 + 7 + 6 + 12 + 10 + 4 + 8 + 9
10
=
70
10
= 7
Range = 12 – 2 = 10
Example
1. Two dice were thrown 10 times and their scores were added together and recorded. Find the mean and range for this data.7, 5, 2, 7, 6, 12, 10, 4, 8, 9
Slide3Median, Quartiles, Inter-Quartile Range and Box Plots.
Measures of Spread
The range is not a good measure of spread because one extreme, (very high or very low value) can have a big affect. The measure of spread that goes with the median is called the inter-quartile range and is generally a better measure of spread because it is not affected by extreme values.
A reminder about the median
Slide4Single middle value
Averages (The Median)
The
median
is the middle value of a set of data once the data has been
ordered
.
Example 1.
Robert hit 11 balls at Grimsby driving range. The recorded distances of his drives, measured in yards, are given below. Find the median distance for his drives.85, 125, 130, 65, 100, 70, 75, 50, 140, 95, 70
Median drive =
85 yards
50, 65, 70, 70, 75, 85, 95, 100, 125, 130, 140
Ordered data
Slide5Two middle values so take the mean.
Averages (The Median)
The
median
is the middle value of a set of data once the data has been
ordered
.
Example 1.
Robert hit 12 balls at Grimsby driving range. The recorded distances of his drives, measured in yards, are given below. Find the median distance for his drives.85, 125, 130, 65, 100, 70, 75, 50, 140, 135, 95, 70
Median drive =
90 yards
50, 65, 70, 70, 75, 85, 95, 100, 125, 130, 135, 140
Ordered data
Slide6Finding the median, quartiles and inter-quartile range.
12, 6, 4, 9, 8, 4, 9, 8, 5, 9, 8, 10
4, 4, 5, 6, 8, 8, 8, 9, 9, 9, 10, 12
Order the data
Inter-Quartile Range = 9 - 5½ =
3½
Example 1: Find the median and quartiles for the data below.
Lower Quartile = 5½
Q
1
Upper Quartile = 9
Q
3
Median = 8
Q
2
Slide7Upper Quartile = 10
Q
3
Lower Quartile = 4
Q
1
Median = 8
Q
2
3, 4, 4, 6, 8, 8, 8, 9, 10, 10, 15,
Finding the median, quartiles and inter-quartile range.
6, 3, 9, 8, 4, 10, 8, 4, 15, 8, 10
Order the data
Inter-Quartile Range = 10 - 4 =
6
Example 2
: Find the median and quartiles for the data below.
Slide82, 5, 6, 6, 7, 8, 8, 8, 9, 9, 10, 15
Median = 8 hours and the inter-quartile range = 9 – 6 = 3 hours.
Battery Life
: The life of 12 batteries recorded in hours is:
2, 5, 6, 6, 7, 8, 8, 8, 9, 9, 10, 15Mean = 93/12 = 7.75 hours and the range = 15 – 2 = 13 hours.
Discuss the calculations below.
The
averages are similar but the measures of spread are significantly different since the extreme values of 2 and 15 are not included in the inter-quartile range.
Slide94
5
6
7
8
9
10
11
12
Median
Lower Quartile
Upper Quartile
Lowest Value
Highest Value
Box
Whisker
Whisker
130
140
150
160
170
180
190
Boys
Girls
cm
Box and Whisker Diagrams.
Box plots are useful for
comparing
two or more sets
of data like that shown below for heights of boys and girls in a class.
Anatomy of a Box and Whisker Diagram.
Box Plots
Slide10Lower Quartile = 5½
Q
1
Upper Quartile = 9
Q
3
Median = 8
Q
2
4
5
6
7
8
9
10
11
12
4, 4, 5, 6, 8, 8, 8, 9, 9, 9, 10, 12
Example 1
: Draw a Box plot for the data below
Drawing a Box Plot.
Slide11Upper Quartile = 10
Q
3
Lower Quartile = 4
Q
1
Median = 8
Q
2
3, 4, 4, 6, 8, 8, 8, 9, 10, 10, 15,
Example 2
: Draw a Box plot for the data below
Drawing a Box Plot.
3
4
5
6
7
8
9
10
11
12
13
14
15
Slide12Upper Quartile = 180
Q
u
Lower Quartile = 158
Q
L
Median = 171
Q
2
Question
: Stuart recorded the heights in cm of boys in his class as shown below. Draw a box plot for this data.
Drawing a Box Plot.
137, 148, 155, 158, 165, 166, 166, 171, 171, 173, 175, 180, 184, 186, 186
130
140
150
160
170
180
190
cm
Slide132.
The boys are taller on average.
Question: Gemma recorded the heights in cm of girls in the same class and constructed a box plot from the data. The box plots for both boys and girls are shown below. Use the box plots to choose some correct statements comparing heights of boys and girls in the class. Justify your answers.
Drawing a Box Plot.
130
140
150
160
170
180
190
Boys
Girls
cm
1.
The girls are taller on average.
3.
The girls show less variability in height.
4.
The boys show less variability in height.
5.
The smallest person is a girl.
6.
The tallest person is a boy.
Slide1410
20
30
40
50
60
70
0
Cumulative Frequency
10
20
30
40
50
60
70
Minutes Late
Median =
27
LQ =
21
UQ =
38
IQR = 38 – 21 =
17 mins
½
¼
¾
0
10
20
30
40
50
60
Box Plot from Cumulative Frequency Curve
CFC
?
?
We can now construct a partial box plot from our earlier work on cumulative frequency curves.
Slide15Finding the median, quartiles and inter-quartile range.
12, 6, 4, 9, 8, 4, 9, 8, 5, 9, 8, 10
Example 1: Find the median and quartiles for the data below.
Finding the median, quartiles and inter-quartile range.
6, 3, 9, 8, 4, 10, 8, 4, 15, 8, 10
Example 2: Find the median and quartiles for the data below.
Worksheet 1
worksheet
Slide16Worksheet 2
3
130
140
150
160
170
180
190
cm
4
5
6
7
8
9
10
11
12
3
4
5
6
7
8
9
10
11
12
13
14
15
0
10
20
30
40
50
60
1
2
4
Box Plots
worksheet
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