Download 2 - Maths Excellence

MEASURES OF
CENTRAL
TENDENCY AND
DISPERSION
AROUND THE
MEDIAN
1
MEASURES OF CENTRAL
TENDENCY
A Measure of Central Tendency is a single
value representing a set of data
Three Measures of Central Tendency are
– Mean (dealt with first in Grade 7)
– Median (dealt with first in Grade 6)
– Mode (dealt with first in Grade 5)
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Mean, Median and Mode
The mean – the equal shares
average;
The median – the middle value;
The mode – the value that occurs
most often.
Their use depends on the sort of
information you need your data to
show.
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Activity 1
1) 50,4%
2)
Test no.
Hist
Biol
Tech
Math
Eng
Geog
Zulu
Mark
25%
31%
37%
57%
63%
63%
77%
3) Maths - 57%,
4) 63% - English &
Geography
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Organising Data Using a
Stem-And-Leaf Diagram
32 ; 56 ;
stem
The first
number is 32:
The stem is 3
and the leaf is 2
3
leaves
2
The second
number is 56:
The stem is 5
and the leaf is 6
4
5
6
6
7
5
The leaf is the ‘units’ digit – i.e.
furthest to the right in the number.
The stem is the ‘tens’ digit – i.e.
furthest to the left in the number.
If the number includes ‘hundreds’
and ‘thousands’ digits then the
stem includes these digits as well.
If the list of numbers includes a
single digit number then the stem
must be 0.
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Redraw the display with the leaves
written in ascending order.
Leaves must be carefully written
underneath each other.
Squared paper!
Find median (or middle value) by
counting the leaves.
Two data sets can be written as
displays on either side of the same
stem.
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Activity 2
Stem
Leaves
0 2
3
3
5
6
6
6
7
8
8
1 2
2
2
2
3
4
5
5
8
8
2 0
0
0
0
2
4
5
9
9
3 0
KEY: 2/5=25
Median lies
between 15th and
16th value.
Median is 12 hrs
Mode = 20 hrs and 12 hrs
We say the data is bimodal
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Range
Range = highest value – lowest value
200 cm
150 cm
150 cm
100 cm
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Quartiles
Quartiles divide the distribution into four
equal parts.
Set of data items divided into 4 equal parts:
Lower quartile
(Q1)
Median
Upper quartile
(M)
(Q3)
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The lower quartile (Q1) is a quarter
of the way through the
distribution,
The middle quartile which is the
same as the median (M) is midway
through the distribution.
The upper quartile (Q3) is three
quarters of the way through the
distribution.
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Finding quartiles on Stem-and-Leaf
Example:
Eighteen numbers were listed on a stem and leaf
plot as follows (n = 18)
Stem
Leaves
1 12
Q1 lies in 5th
position.
2 05
Median lies
between 9th and
10th data item.
3 0 0 0 2 5|9
4 000258
5 0
6
7 0
Q3 lies in the
14th position.
KEY: 3/5=35
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Activity 3
1. a) M=7
b) M=28,5
c) M=16,5
Q1=5
Q1=22
Q1=13
Q3=9
Q3=35
Q3=19
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Leaves for Set 1
7
7
Stem
Leaves for set 2
1
9
2
7
3
7
8
9
8
5
4
0
2
9
5
2
2
1
1
5
0
7
6
4
3
0
6
8
3
2
7
3
6
0
8
5
7
9
0
5
8
9
9
KEY: 8/0=80
Set 1:
M = 57
Q1= 51
Q3= 66
Set 2:
M = 54,5
Q1= 40
Q3= 79
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Five-Number Summaries
The five-number summary for a set of
data values consists of
The Minimum value
The Lower quartile (Q1)
The Median (M)
The Upper quartile (Q3)
The Maximum value
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Activity 4
Min = 1 year
Q1 = 8 years
M = 12 years
Q3 = 15 years
Maximum = 41 years
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Box and Whisker Diagrams
It is a diagram of the five-number
summary.
For example, consider the following data:
1,5,7,8,8,14,17.
Median = 8
Q1 = 5
Q3 = 14
Minimum = 1
Maximum = 17
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The box shows the middle 50% or
half of the data.
There is the same number of data
items in each of the four groups.
The varying lengths are
influenced by the value of the
data items
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The Interquartile Range
• The IQR shows the spread of the
middle section of data.
• IQR = Q3 – Q1
• Semi-interquartile range = IQR ÷ 2
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