Measures of Central Tendency
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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