Distribution and Measure of Central Tendency | Zoology Optional Notes for UPSC PDF Download

 Page 1


M e as u r e of C e n t r al T e n d e n c y: Usually when two or more different data sets are
to be compared it is necessary to condense the data, but for comparison the
condensation of data set into a frequency distribution and visual presentation are
not enough. It is then necessary to summarize the data set in a single value. Such a
value usually somewhere in the center and represent the entire data set and hence it
is called measure of central tendency or averages. Since a measure of central
tendency (i.e. an average) indicates the location or the general position of the
distribution on the X-axis therefore it is also known as a measure of location or
position.
T y p e s o f M e a s u r e o f C e n t r a l T e n d e n c y
1. Arithmetic Mean
2. Geometric Mean
3. Harmonic Mean
4. Mode
5. Median
A r i t h m e t i c M e a n or S i m p l y M e a n : “ A v a l u e o b t a i n e d b y d i v i d i n g t h e s u m
o f a l l t h e o b s e r v a t i o n s b y t h e n u m b e r o f o b s e r v a t i o n i s c a l l e d
a r i t h m e t i c M e a n ”
   
  
Sum of A l l obs e r v at i on
M e an
N um be r of obs e r v at i on
?
N u m e r i c al E xam p l e :
Page 2


M e as u r e of C e n t r al T e n d e n c y: Usually when two or more different data sets are
to be compared it is necessary to condense the data, but for comparison the
condensation of data set into a frequency distribution and visual presentation are
not enough. It is then necessary to summarize the data set in a single value. Such a
value usually somewhere in the center and represent the entire data set and hence it
is called measure of central tendency or averages. Since a measure of central
tendency (i.e. an average) indicates the location or the general position of the
distribution on the X-axis therefore it is also known as a measure of location or
position.
T y p e s o f M e a s u r e o f C e n t r a l T e n d e n c y
1. Arithmetic Mean
2. Geometric Mean
3. Harmonic Mean
4. Mode
5. Median
A r i t h m e t i c M e a n or S i m p l y M e a n : “ A v a l u e o b t a i n e d b y d i v i d i n g t h e s u m
o f a l l t h e o b s e r v a t i o n s b y t h e n u m b e r o f o b s e r v a t i o n i s c a l l e d
a r i t h m e t i c M e a n ”
   
  
Sum of A l l obs e r v at i on
M e an
N um be r of obs e r v at i on
?
N u m e r i c al E xam p l e :
C al c u l at e t h e ar i t h m e t i c m e an f or t h e f ol l ow i n g t h e m ar k s ob t ai n e d b y 9
s t u d e n t s ar e gi ve n b e l ow :
Using formula of arithmetic mean for ungrouped data:
1
n
i
i
x
x
n
?
?
?
9 n ?
360
9
x ? 40 m ar k s ?
? N u m e r i c al E xam p l e :
? C al c u l at e t h e ar i t h m e t i c m e an f or t h e f ol l ow i n g d at a gi ve n b e l ow :
? Using formula of d i r e c t m e t h od of ar i t h m e t i c m e an for gr ou p e d d at a:
1
1
n
i i
i
n
i
i
f x
x
f
?
?
?
?
?
,
1
n
i
i
n f
?
?
?
The weight recorded to the nearest grams of 60 apples picked out at random from a
consignment are given below:
i
x
4 5
3 2
3 7
4 6
3 9
3 6
4 1
4 8
3 6
1
360
n
i
i
x
?
?
?
106 107 76 82 109 107 115 93 187 95 123 125
111 92 86 70 126 68 130 129 139 119 115 128
100 186 84 99 113 204 111 141 136 123 90 115
98 110 78 185 162 178 140 152 173 146 158 194
148 90 107 181 131 75 184 104 110 80 118 82
Page 3


M e as u r e of C e n t r al T e n d e n c y: Usually when two or more different data sets are
to be compared it is necessary to condense the data, but for comparison the
condensation of data set into a frequency distribution and visual presentation are
not enough. It is then necessary to summarize the data set in a single value. Such a
value usually somewhere in the center and represent the entire data set and hence it
is called measure of central tendency or averages. Since a measure of central
tendency (i.e. an average) indicates the location or the general position of the
distribution on the X-axis therefore it is also known as a measure of location or
position.
T y p e s o f M e a s u r e o f C e n t r a l T e n d e n c y
1. Arithmetic Mean
2. Geometric Mean
3. Harmonic Mean
4. Mode
5. Median
A r i t h m e t i c M e a n or S i m p l y M e a n : “ A v a l u e o b t a i n e d b y d i v i d i n g t h e s u m
o f a l l t h e o b s e r v a t i o n s b y t h e n u m b e r o f o b s e r v a t i o n i s c a l l e d
a r i t h m e t i c M e a n ”
   
  
Sum of A l l obs e r v at i on
M e an
N um be r of obs e r v at i on
?
N u m e r i c al E xam p l e :
C al c u l at e t h e ar i t h m e t i c m e an f or t h e f ol l ow i n g t h e m ar k s ob t ai n e d b y 9
s t u d e n t s ar e gi ve n b e l ow :
Using formula of arithmetic mean for ungrouped data:
1
n
i
i
x
x
n
?
?
?
9 n ?
360
9
x ? 40 m ar k s ?
? N u m e r i c al E xam p l e :
? C al c u l at e t h e ar i t h m e t i c m e an f or t h e f ol l ow i n g d at a gi ve n b e l ow :
? Using formula of d i r e c t m e t h od of ar i t h m e t i c m e an for gr ou p e d d at a:
1
1
n
i i
i
n
i
i
f x
x
f
?
?
?
?
?
,
1
n
i
i
n f
?
?
?
The weight recorded to the nearest grams of 60 apples picked out at random from a
consignment are given below:
i
x
4 5
3 2
3 7
4 6
3 9
3 6
4 1
4 8
3 6
1
360
n
i
i
x
?
?
?
106 107 76 82 109 107 115 93 187 95 123 125
111 92 86 70 126 68 130 129 139 119 115 128
100 186 84 99 113 204 111 141 136 123 90 115
98 110 78 185 162 178 140 152 173 146 158 194
148 90 107 181 131 75 184 104 110 80 118 82
S ol u t i on :
Weight (grams) Midpoints (
i
x ) Frequency
(
i
f )
i i
f x
65----84
85----104
105----124
125----144
145----164
165----184
185----204
(65 84) 2 74.5 ? ?
94.5
114.5
134.5
154.5
174.5
194.5
09
10
17
10
05
04
05
9 ?74.5=670.5
945.0
1946.5
1345.0
772.5
698.0
972.5
1
60
n
i
i
f
?
?
?
1
n
i i
i
f x
?
?
?
7350.0
1
1
n
i i
i
n
i
i
f x
x
f
?
?
?
?
?
7350.0
60
? 122.5 ? grams
? Using formula of s h or t c u t m e t h od of ar i t h m e t i c m e an for gr ou p e d d at a:
1
1
n
i i
i
n
i
i
f D
x A
f
?
?
? ?
?
?
,
1
n
i
i
n f
?
?
?
Where
i i
D X A ? ? and A is the provisional or assumed mean
Weight (grams) Frequency
65----84
85----104
105----124
125----144
145----164
165----184
185----204
09
10
17
10
05
04
05
Page 4


M e as u r e of C e n t r al T e n d e n c y: Usually when two or more different data sets are
to be compared it is necessary to condense the data, but for comparison the
condensation of data set into a frequency distribution and visual presentation are
not enough. It is then necessary to summarize the data set in a single value. Such a
value usually somewhere in the center and represent the entire data set and hence it
is called measure of central tendency or averages. Since a measure of central
tendency (i.e. an average) indicates the location or the general position of the
distribution on the X-axis therefore it is also known as a measure of location or
position.
T y p e s o f M e a s u r e o f C e n t r a l T e n d e n c y
1. Arithmetic Mean
2. Geometric Mean
3. Harmonic Mean
4. Mode
5. Median
A r i t h m e t i c M e a n or S i m p l y M e a n : “ A v a l u e o b t a i n e d b y d i v i d i n g t h e s u m
o f a l l t h e o b s e r v a t i o n s b y t h e n u m b e r o f o b s e r v a t i o n i s c a l l e d
a r i t h m e t i c M e a n ”
   
  
Sum of A l l obs e r v at i on
M e an
N um be r of obs e r v at i on
?
N u m e r i c al E xam p l e :
C al c u l at e t h e ar i t h m e t i c m e an f or t h e f ol l ow i n g t h e m ar k s ob t ai n e d b y 9
s t u d e n t s ar e gi ve n b e l ow :
Using formula of arithmetic mean for ungrouped data:
1
n
i
i
x
x
n
?
?
?
9 n ?
360
9
x ? 40 m ar k s ?
? N u m e r i c al E xam p l e :
? C al c u l at e t h e ar i t h m e t i c m e an f or t h e f ol l ow i n g d at a gi ve n b e l ow :
? Using formula of d i r e c t m e t h od of ar i t h m e t i c m e an for gr ou p e d d at a:
1
1
n
i i
i
n
i
i
f x
x
f
?
?
?
?
?
,
1
n
i
i
n f
?
?
?
The weight recorded to the nearest grams of 60 apples picked out at random from a
consignment are given below:
i
x
4 5
3 2
3 7
4 6
3 9
3 6
4 1
4 8
3 6
1
360
n
i
i
x
?
?
?
106 107 76 82 109 107 115 93 187 95 123 125
111 92 86 70 126 68 130 129 139 119 115 128
100 186 84 99 113 204 111 141 136 123 90 115
98 110 78 185 162 178 140 152 173 146 158 194
148 90 107 181 131 75 184 104 110 80 118 82
S ol u t i on :
Weight (grams) Midpoints (
i
x ) Frequency
(
i
f )
i i
f x
65----84
85----104
105----124
125----144
145----164
165----184
185----204
(65 84) 2 74.5 ? ?
94.5
114.5
134.5
154.5
174.5
194.5
09
10
17
10
05
04
05
9 ?74.5=670.5
945.0
1946.5
1345.0
772.5
698.0
972.5
1
60
n
i
i
f
?
?
?
1
n
i i
i
f x
?
?
?
7350.0
1
1
n
i i
i
n
i
i
f x
x
f
?
?
?
?
?
7350.0
60
? 122.5 ? grams
? Using formula of s h or t c u t m e t h od of ar i t h m e t i c m e an for gr ou p e d d at a:
1
1
n
i i
i
n
i
i
f D
x A
f
?
?
? ?
?
?
,
1
n
i
i
n f
?
?
?
Where
i i
D X A ? ? and A is the provisional or assumed mean
Weight (grams) Frequency
65----84
85----104
105----124
125----144
145----164
165----184
185----204
09
10
17
10
05
04
05
Weight (grams) Midpoints (
i
x ) Frequency
(
i
f )
i i
D X A ? ?
114.5 A ? i
f
i
D
65----84
85----104
105----124
125----144
145----164
165----184
185----204
(65 84) 2 74.5 ? ?
94.5
114.5
134.5
154.5
174.5
194.5
09
10
17
10
05
04
05
-40
-20
0
20
40
60
80
-360
-200
0
200
200
240
400
1
60
n
i
i
f
?
?
?
1
n
i i
i
f D
?
?
=480
1
1
n
i i
i
n
i
i
f D
x A
f
?
?
? ?
?
?
480
114.5
60
? ? 122.5 ? grams
? Using formula of s t e p d e vi at i on m e t h od of ar i t h m e t i c m e an for gr ou p e d d at a:
1
1
n
i i
i
n
i
i
f u
x A h
f
?
?
? ? ?
?
?
,
i
i
x A
u
h
?
? , where h is the width of the class interval:
Weight (grams) Midpoints (
i
x ) Frequency
(
i
f )
i
i
X A
u
h
?
?
114.5 A ? , h=20
i
f
i
u
65----84
85----104
105----124
125----144
145----164
165----184
185----204
(65 84) 2 74.5 ? ?
94.5
114.5
134.5
154.5
174.5
194.5
09
10
17
10
05
04
05
-2
-1
0
1
2
3
4
-18
-10
0
10
10
12
20
1
60
n
i
i
f
?
?
?
1
n
i i
i
f u
?
?
=24
1
1
n
i i
i
n
i
i
f u
x A h
f
?
?
? ? ?
?
?
24
114.5 20
60
? ? ? 114.5 08 ? ? 122.5 ? grams (Answer).
Page 5


M e as u r e of C e n t r al T e n d e n c y: Usually when two or more different data sets are
to be compared it is necessary to condense the data, but for comparison the
condensation of data set into a frequency distribution and visual presentation are
not enough. It is then necessary to summarize the data set in a single value. Such a
value usually somewhere in the center and represent the entire data set and hence it
is called measure of central tendency or averages. Since a measure of central
tendency (i.e. an average) indicates the location or the general position of the
distribution on the X-axis therefore it is also known as a measure of location or
position.
T y p e s o f M e a s u r e o f C e n t r a l T e n d e n c y
1. Arithmetic Mean
2. Geometric Mean
3. Harmonic Mean
4. Mode
5. Median
A r i t h m e t i c M e a n or S i m p l y M e a n : “ A v a l u e o b t a i n e d b y d i v i d i n g t h e s u m
o f a l l t h e o b s e r v a t i o n s b y t h e n u m b e r o f o b s e r v a t i o n i s c a l l e d
a r i t h m e t i c M e a n ”
   
  
Sum of A l l obs e r v at i on
M e an
N um be r of obs e r v at i on
?
N u m e r i c al E xam p l e :
C al c u l at e t h e ar i t h m e t i c m e an f or t h e f ol l ow i n g t h e m ar k s ob t ai n e d b y 9
s t u d e n t s ar e gi ve n b e l ow :
Using formula of arithmetic mean for ungrouped data:
1
n
i
i
x
x
n
?
?
?
9 n ?
360
9
x ? 40 m ar k s ?
? N u m e r i c al E xam p l e :
? C al c u l at e t h e ar i t h m e t i c m e an f or t h e f ol l ow i n g d at a gi ve n b e l ow :
? Using formula of d i r e c t m e t h od of ar i t h m e t i c m e an for gr ou p e d d at a:
1
1
n
i i
i
n
i
i
f x
x
f
?
?
?
?
?
,
1
n
i
i
n f
?
?
?
The weight recorded to the nearest grams of 60 apples picked out at random from a
consignment are given below:
i
x
4 5
3 2
3 7
4 6
3 9
3 6
4 1
4 8
3 6
1
360
n
i
i
x
?
?
?
106 107 76 82 109 107 115 93 187 95 123 125
111 92 86 70 126 68 130 129 139 119 115 128
100 186 84 99 113 204 111 141 136 123 90 115
98 110 78 185 162 178 140 152 173 146 158 194
148 90 107 181 131 75 184 104 110 80 118 82
S ol u t i on :
Weight (grams) Midpoints (
i
x ) Frequency
(
i
f )
i i
f x
65----84
85----104
105----124
125----144
145----164
165----184
185----204
(65 84) 2 74.5 ? ?
94.5
114.5
134.5
154.5
174.5
194.5
09
10
17
10
05
04
05
9 ?74.5=670.5
945.0
1946.5
1345.0
772.5
698.0
972.5
1
60
n
i
i
f
?
?
?
1
n
i i
i
f x
?
?
?
7350.0
1
1
n
i i
i
n
i
i
f x
x
f
?
?
?
?
?
7350.0
60
? 122.5 ? grams
? Using formula of s h or t c u t m e t h od of ar i t h m e t i c m e an for gr ou p e d d at a:
1
1
n
i i
i
n
i
i
f D
x A
f
?
?
? ?
?
?
,
1
n
i
i
n f
?
?
?
Where
i i
D X A ? ? and A is the provisional or assumed mean
Weight (grams) Frequency
65----84
85----104
105----124
125----144
145----164
165----184
185----204
09
10
17
10
05
04
05
Weight (grams) Midpoints (
i
x ) Frequency
(
i
f )
i i
D X A ? ?
114.5 A ? i
f
i
D
65----84
85----104
105----124
125----144
145----164
165----184
185----204
(65 84) 2 74.5 ? ?
94.5
114.5
134.5
154.5
174.5
194.5
09
10
17
10
05
04
05
-40
-20
0
20
40
60
80
-360
-200
0
200
200
240
400
1
60
n
i
i
f
?
?
?
1
n
i i
i
f D
?
?
=480
1
1
n
i i
i
n
i
i
f D
x A
f
?
?
? ?
?
?
480
114.5
60
? ? 122.5 ? grams
? Using formula of s t e p d e vi at i on m e t h od of ar i t h m e t i c m e an for gr ou p e d d at a:
1
1
n
i i
i
n
i
i
f u
x A h
f
?
?
? ? ?
?
?
,
i
i
x A
u
h
?
? , where h is the width of the class interval:
Weight (grams) Midpoints (
i
x ) Frequency
(
i
f )
i
i
X A
u
h
?
?
114.5 A ? , h=20
i
f
i
u
65----84
85----104
105----124
125----144
145----164
165----184
185----204
(65 84) 2 74.5 ? ?
94.5
114.5
134.5
154.5
174.5
194.5
09
10
17
10
05
04
05
-2
-1
0
1
2
3
4
-18
-10
0
10
10
12
20
1
60
n
i
i
f
?
?
?
1
n
i i
i
f u
?
?
=24
1
1
n
i i
i
n
i
i
f u
x A h
f
?
?
? ? ?
?
?
24
114.5 20
60
? ? ? 114.5 08 ? ? 122.5 ? grams (Answer).
C h a p t e r 0 3 M e a s u r e s o f C e n t r a l T e n d e n c y