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MEASURES OF MEASURES OF CENTRAL TENDENCY CENTRAL TENDENCY

AND DISPERSION AND DISPERSION AROUND THE AROUND THE

MEDIANMEDIAN

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MMEASURES OF CENTRAL EASURES OF CENTRAL TENDENCYTENDENCY

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 ModeMean, 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 1Activity 1

1) 50,4%2)

3) Maths - 57%,4) 63% - English &

Geography

Test no. Hist Biol Tech Math Eng Geog Zulu

Mark 25% 31% 37% 57% 63% 63% 77%

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Organising Data Using a Organising Data Using a Stem-And-Leaf DiagramStem-And-Leaf Diagram

32 ; 56 ;

stestemm

leavesleaves

33 22

44

55 66

66

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The first number is 32:The stem is 3 and the leaf is 2

The second number is 56:The stem is 5 and the leaf is 6

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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 2Activity 2Stem Leaves

0 2 3 3 5 6 6 6 7 8 8 9 9

1 2 2 2 2 3 4 5 5 8 8

2 0 0 0 0 2 4 5

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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RangeRange

Range = highest value – lowest value200 cm

150 cm

150 cm 100

cm

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QuartilesQuartilesQuartiles divide the distribution into four equal parts.

Set of data items divided into 4 equal parts:

Lower quartile Median Upper quartile

(Q1) (M) (Q3)

1111

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-Finding quartiles on Stem-and-LeafLeaf

Example:Eighteen numbers were listed on a stem and leaf

plot as follows (n = 18)

Stem Leaves

1 1 2

2 0 5

3 0 0 0 2 5|9

4 0 0 0 2 5 8

5 0

6

7 0

KEY: 3/5=35

Median lies between 9th

and 10th data item.

Q1 lies in 5th position.

Q3 lies in the 14th position.

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Activity 3Activity 3

1.1. a) M=7a) M=7 QQ11=5=5 QQ33=9=9

b) M=28,5b) M=28,5 QQ11=22=22 QQ33=35=35

c) M=16,5c) M=16,5 QQ11=13=13 QQ33=19=19

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Leaves for Set 1 Stem Leaves for set 2

1 9

2 7

3 7 8

9 8 5 4 0 2 8 9

7 7 5 2 2 1 1 5 0 9

7 6 4 3 0 6 8

3 2 7 3 6 9

0 8 5 7

9 0 5

KEY: 8/0=80

Set 1: Set 2:M = 57 M = 54,5Q1= 51 Q1= 40

Q3= 66 Q3= 79

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Five-Number SummariesFive-Number Summaries

The five-number summary for a set of

data values consists ofThe Minimum valueThe Lower quartile (Q1)The Median (M)The Upper quartile (Q3)The Maximum value

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Activity 4Activity 4

Min = 1 yearQ1 = 8 years

M = 12 yearsQ3 = 15 years

Maximum = 41 years

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Box and Whisker Box and Whisker DiagramsDiagrams

It is a diagram of the five-number summary.

For example, consider the following data: 1,5,7,8,8,14,17.

Median = 8Q1 = 5Q3 = 14Minimum = 1Maximum = 17

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The box shows the middle 50% or The box shows the middle 50% or half of the data.half of the data.

There is the same number of data There is the same number of data items in each of the four groups.items in each of the four groups.

The varying lengths are The varying lengths are influenced by the value of the influenced by the value of the data itemsdata items

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The Interquartile RangeThe Interquartile Range

• The IQR shows the spread of the The IQR shows the spread of the middle section of data.middle section of data.

• IQR = QIQR = Q33 – Q – Q11

• Semi-interquartile range = IQR Semi-interquartile range = IQR ÷ 2÷ 2