Measures of Central Tendency

Measures of Central Tendency

Measures of Central Tendency

•Definition•Measures of Central Tendency (Mean,

Median, Mode)

Central Tendency

•Refers to a characteristic where the frequency of a variable tends to cluster around the ‘center’

Measures of Central Tendency

•Arithmetic Mean•Median•Mode

Arithmetic Mean

•Data (units produced by workers)10, 20, 30

•Mean =

n

x

3

302010

Mean =

•Ungrouped data (1)

=20

Arithmetic Mean

•Data (units produced by workers)10, 20, 20, 25, 25, 25, 25, 30, 30, 50, 50, 50

•Ungrouped data (2)

Units (x) Worker(f)

10 1

20 2

25 4

30 2

50 3

Total

Mean(x)=360

30 units12

fx

f

10

40

100

60

150

12 360

fx

Arithmetic Mean

•Data (units produced by workers)12, 24, 24, 25, 25, 25, 25, 32, 32, 45, 45, 45

•Grouped data

Units Worker(f)

10 – 20 1

20 – 30 6

30 – 40 2

40 – 50 3

Total

Mean(x)=370

30.8 units12

fm

f

Midpoint(m)

15

25

35

45

fm

15

150

70

135

12 370

Arithmetic Mean•Ungrouped data

n

x

fm

f

•Grouped data

fx

f

Features of Arithmetic Mean

•Commonly used •Easily understood

•Greatly affected by extreme values

Median

1. Array2. Median position3. median

Median

• Data (units produced by workers) 20, 10, 30 (odd)

•Ungrouped data (1)

① Array10, 20, 30

② Median position1

2

n

③ Median

20

3 1

2

2

Median

• Data (units produced by workers) 20, 10, 40, 30 (even)

•Ungrouped data (1)

① Array10, 20, 30, 40

② Median position1

2

n

② Median

2

302025

4 1

2

2.5

Median

•Data (units produced by workers)10, 20, 20, 25, 25, 25, 25, 30, 30, 50, 50, 50

1 12 16.5

2 2

n

Median position=

•Ungrouped data (2)

Units (x) Worker(f)

10 1

20 2

25 4

30 2

50 3

Total 12

25 units

√

Median=

c.f.

1

3

7

9

12

Median

•Data (units produced by workers)12, 24, 24, 25, 25, 25, 25, 32, 32, 45, 45, 45

2nC

L if

Median =

•Grouped data (2)

Units Worker(f)

10 – 20 1

20 – 30 6

30 – 40 2

40 – 50 3

Total 12

Median position =1 12 1

2 2

n

6.5

3.28

√

Median Class =20-

30

c.f.1

7

9

12

121

220 106

Median•Ungrouped data

even)(odd, 2

1n

•Grouped data

2nC

L if

1

2

n

Features of Median•Not affected by extreme values•When data is skewed, the median is often

a better indicator of “average” than the mean.

•Time consuming•Unfamiliar to most people

Mode

•Data (units produced by workers)

10, 20, 20, 30 • Mode =

•Ungrouped data (1)

20

Mode

•Data (units produced by workers)10, 20, 20, 25, 25, 25, 25, 30, 30, 50, 50, 50

•Ungrouped data (2)

Units (x) Worker(f)

10 1

20 2

25 4

30 2

50 3

Total 12

√ Mode =

25

Mode

•Data (units produced by workers)12, 24, 24, 25, 25, 25, 25, 32, 32, 45, 45, 45

1

1 2

dL id d

Mode =

•Grouped data (2)

Units Worker(f)

10 – 20 1

20 – 30 6

30 – 40 2

40 – 50 3

Total 12

The highest frequency:6

6.25

√ Modal group=

20-30 units

6 120 10

(6 1) (6 2)

Mode•Ungrouped data

•Grouped data

Data with the highest frequency

1

1 2

dL id d

Features of Mode

•Not affected by extreme values

•May be more than one mode, or no mode•May not give a good indication of central

values

Skewness of Data DistributionNormal

Mode = mean =median

Skewness of Data Distribution

Positively skewed

Mode < median< mean

Skewness of Data DistributionNegatively skewed

Mean < median< mode

Arithmetic Mean•ungrouped data

n

x

fm

f

•grouped data

fx

f

Median•ungrouped data

even)(odd, 2

1n

•grouped data

2nC

L if

1

2

n

Mode•ungrouped data

•grouped data

Data with the highest frequency

1

1 2

dL id d

Measures of Dispersion

Measures of Dispersion

•Definition•Measures of Dispersion(Range, Quartile

Deviation, Mean Deviation, Standard Deviation, Variance, Coefficient of Variation)

Dispersion

•It describes the level of variation and also indicates the level of consistency in the distribution.

Measures of Dispersion

•Range•Quartile Deviation•Mean Deviation•Standard Deviation•Variance•Coefficient of Variation

Range

•It measures the difference between the highest and the lowest piece of data.

Data1: Data2:10, 20, 30 0, 20, 40

Range1 = xmax – xmin = 30 - 10 = 20

Range2 = xmax – xmin = 40 - 0 = 40

Feature

•It is easy to calculate and easy to understand.

•It is distorted by extreme values.

Quartile Deviation

1. Array 2. Quartile position3. Quartile Value4. IQR,QD

Quartile Deviation

•It excludes the first and last quarters of information and in doing so concentrates on the main core of data, ignoring extreme values.

45 46 50 55 60 65 67 69 69 70 71 72 73 74 76 78 78 79 80 82 83 85 90 95

Q1 Q2 Q3

Interquartile Range = Q3 - Q1Quartile Deviation = 2

13 QQ

Quartile Deviation (ungrouped)

Q1 position=

Q3 position=

4

1n

4

)1(3 n4

124 25.6

4

)124(3 75.18

5.13665.79.. 13 QQRQI66

2

6765

5.792

8079

Q1 value=

Q3 value=

Grouped data

if

cn

LQ

411

if

cn

LQ

43

33

2.. 13 QQDQ

Amount Spent ($)

Number of Staff

0-10 2

10-20 3

20-30 4

30-40 3

40-50 1

Total 13

1 positionQ 1

4

n

13 1

4

3.5

c. f.

2

5

9

12

13

√

Amount Spent ($)

Number of Staff

0-10 2

10-20 3

20-30 4

30-40 3

40-50 1

Total 13

c. f.

2

5

9

12

13

√

3 positionQ3( 1)

4

n

3 (13 1)

4

10.5

Amount Spent ($)

Number of Staff

0-10 2

10-20 3

20-30 4

30-40 3

40-50 1

Total 13

c. f.

2

5

9

12

13

√

1 14 value

nc

Q L if

132

410 103

14.17

Amount Spent ($)

Number of Staff

0-10 2

10-20 3

20-30 4

30-40 3

40-50 1

Total 13

c. f.

2

5

9

12

13

√

3 3

34 value

nc

Q L if

3 139

430 103

32.5

2

5

9

12

13

3 1 32.5 14.17 18.33IQR Q value Q value

Amount Spent ($)

Number of Staff

0-10 2

10-20 3

20-30 4

30-40 3

40-50 1

Total 13

c. f.

3 1 32.5 14.17 18.33. . 9.165

2 2 2

Q value Q valueQ D

Feature

•Not effected by extreme values.

•Not widely used or understood.

Quartile Deviation

4

1n

4

)1(3 n

Q1 =

Q3=

• Ungrouped:

I.Q.R= Q3 value- Q1 value

Quartile Deviation =

3 1

2

Q value Q value

Quartile Deviation

1

4

n

3( 1)

4

n

Q1 =

Q3=

• Grouped:

if

cn

LQ

411

if

cn

LQ

43

33

I.Q.R= Q3 value- Q1 value

Quartile Deviation =

3 1

2

Q value Q value

Mean Deviation

•The absolute distance of each score away from the mean.

Mean Deviation

•Ungrouped data

n

xxDM

||

..

Mean Deviation

•Ungrouped data

Team 1: 20 22 23 25 25 26 26 26 28 29

Team 2: 12 14 18 24 28 30 30 30 31 33

2510

292826262625252322201

x

2510

333130303028241814122

x

Mean Deviation

•Ungrouped data

10

|2529||2528||2526||2526|

|2526||2525||2525|

|2523||2522||2520|

..1

DM

Team 1: 20 22 23 25 25 26 26 26 28 29

Team 2: 12 14 18 24 28 30 30 30 31 33

2

Mean Deviation

•Ungrouped data

10

|2533||2531||2530||2530|

|2530||2528||2524|

|2518||2514||2512|

..2

DM

Team 1: 20 22 23 25 25 26 26 26 28 29

Team 2: 12 14 18 24 28 30 30 30 31 33

4.6

Mean Deviation

•Ungrouped data

Team 1: 20 22 23 25 25 26 26 26 28 29

Team 2: 12 14 18 24 28 30 30 30 31 33

M.D. 1 = 2 M.D. 2=6.4

Mean Deviation

•Grouped data

| |. .

f m xM D

n

Units Midpoint(m)

Worker(f)

fm f|m – |

20-30 5

30-40 10

40-50 20

50-60 15

Total

xfm

f

xx

| |. .

f m xM D

n

25

35

45

55

125

350

900

825

44

44

44

44

95

90

20

165

50 2,200 370

2,20044

50

3707.4

50

Mean Deviation

•Grouped data

| |. .

f m xM D

n

•Ungrouped data

n

xxDM

||

..

Standard Deviation/Variance

•Ungrouped data

n

xx

2)(

n

xx

22

)(

Standard Deviation/Variance

•Ungrouped data

Team 1: 20 22 23 25 25 26 26 26 28 29

Team 2: 12 14 18 24 28 30 30 30 31 33

2510

292826262625252322201

x

2510

333130303028241814122

x

Standard Deviation/Variance

•Ungrouped data

57.210

)2520()2528()2526()2526(

)2526()2525()2525(

)2523()2522()2520(

2222

222

222

1

Team 1: 20 22 23 25 25 26 26 26 28 29

Team 2: 12 14 18 24 28 30 30 30 31 33

Standard Deviation/Variance

•Ungrouped data

24.710

)2533()2531()2530()2530(

)2530()2528()2524(

)2518()2514()2512(

2222

222

222

2

Team 1: 20 22 23 25 25 26 26 26 28 29

Team 2: 12 14 18 24 28 30 30 30 31 33

Standard Deviation/Variance

•Ungrouped data

24.7 57.2 21

Team 1: 20 22 23 25 25 26 26 26 28 29

Team 2: 12 14 18 24 28 30 30 30 31 33

4.52 6.6 22

21

Standard Deviation (Variance)

•Grouped data

2( )f m x

n

22 ( )f m x

n

Units(x)

Worker(f)

20-30 5

30-40 10

40-50 20

50-60 15

Total

xfm

f

2( )f m x

n

x

25

35

45

55

Midpoint(m)

fm

125

350

900

825

44

44

44

44

1,805

810

20

1,815

2f m x

50 2,200

2,20044

50

4,450

4,4509.43

50

2,200x 44

50

fm

f

2

2 ( ) 4,45089

50

f m x

n

Units Midpoint

(m)

Worker(f)

fm f(m – )2

20-30 25 5 125 44 1,805

30-40 35 10 350 44 810

40-50 45 20 900 44 20

50-60 55 15 825 44 1,815

Total - 50 2,200 4,450

x x

Standard Deviation/Variance

•Ungrouped data

n

xx

2)(

22 ( )m x

n

Standard Deviation (Variance)

•Grouped data

2( )f m x

n

22 ( )f m x

n

Coefficient of Variation

%100.. x

VC

Coefficient of Variation (100 Students)Height:

Weight:

9cm 168 cmx

5kg 52 kgx

Height C.V.: %36.5%100168

9%100

x

Weight C.V.: %62.9%10052

5%100

x

Weight is more variant than Height.

Population & sample

2( )f m x

n

22 ( )f x x

n

2( )

1

f m x

n

22 ( )

1

f x x

n