Understanding Central Tendency Properties in Statistics

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# Understanding Central Tendency Properties in Statistics - PowerPoint PPT Presentation

In Statistics, Measures of Central Tendency are numerical values that locate, in some sense, the centre of a set of data. The term average is often associated with all measures of central tendency.

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### Understanding Central Tendency Properties in Statistics

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Numerical DataProperties & Measures

Numerical Data

Properties

Central Tendency

RelativeStanding

Variation

Mean

Range

Percentiles

Interquartile Range

Median

Z–scores

Variance

Mode

Standard Deviation

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n

X

i

X

X

X

1

2

n

1

i

X

n

n

Mean
• Measure of central tendency
• Most common measure
• Acts as ‘balance point’
• Affected by extreme values (‘outliers’)
• Formula (sample mean)
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Raw Data: 10.3 4.9 8.9 11.7 6.3 7.7

X

X

X

X

X

X

X

1

2

3

4

5

6

1

X

n

6

10

.

.

8

9

11

.

6

3

.

6

.

30

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Numerical DataProperties & Measures

Numerical Data

Properties

RelativeStanding

Central

Variation

Tendency

Percentiles

Mean

Range

Median

Interquartile Range

Z–scores

Mode

Variance

Standard Deviation

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Median
• Measure of central tendency
• Middle value in ordered sequence
• If n is odd, middle value of sequence
• If n is even, average of 2 middle values
• Position of median in sequence
• Not affected by extreme values

n

1

Positioning

Point

2

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Median Example Odd-Sized Sample
• Raw Data: 24.1 22.6 21.5 23.7 22.6
• Ordered: 21.5 22.6 22.6 23.7 24.1
• Position: 1 2 3 4 5

n

1

5

1

Positioning

Point

3

.

0

2

2

Median

22

.

6

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Median Example Even-Sized Sample

Raw Data: 10.3 4.9 8.9 11.7 6.3 7.7

Ordered: 4.9 6.3 7.78.9 10.3 11.7

Position: 1 2 34 5 6

n

1

6

1

Positioning

Point

3

.

5

2

2

7

.

7

8

.

9

Median

8

.

30

2

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Numerical DataProperties & Measures

Numerical Data

Properties

RelativeStanding

Central

Variation

Tendency

Range

Mean

Percentiles

Interquartile Range

Median

Z–scores

Mode

Variance

Standard Deviation

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Mode
• Measure of central tendency
• Value that occurs most often
• Not affected by extreme values
• May be no mode or several modes
• May be used for quantitative or qualitative data
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Mode Example
• No ModeRaw Data: 10.3 4.9 8.9 11.7 6.3 7.7
• One ModeRaw Data: 6.3 4.9 8.9 6.3 4.9 4.9
• More Than 1 ModeRaw Data: 21 28 28 41 4343
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Thinking Challenge

You’re a financial analyst for Prudential-Bache Securities. You have collected the following closing stock prices of new stock issues: 17, 16, 21, 18, 13, 16, 12, 11.

Describe the stock pricesin terms of central tendency.

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Central Tendency Solution

Mean

n

X

i

X

X

X

1

2

8

i

1

X

n

8

17

16

21

18

13

16

12

11

8

15

.

5

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Central Tendency Solution

Median

• Raw Data: 17 16 21 18 13 16 12 11
• Ordered: 11 12 13 16 16 17 18 21
• Position: 1 2 3 4 5 6 7 8

n

1

8

1

4

.

5

Positioning Point

2

2

16

16

Median

16

2

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Central Tendency Solution

Mode

Raw Data: 17 16 21 18 13 16 12 11

Mode = 16

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Summary of Central Tendency Measures

Measure

Formula

Description

Mean

Balance Point



X

/

n

i

Median

(

n

+1)

Middle Value

Position

2

When Ordered

Mode

none

Most Frequent

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