ARITHMETIC MEAN, MODE, MEDIANARITHMETIC MEAN - CAT PDF Download

 Page 1


 
 
ARITHMETIC MEAN, MODE, MEDIAN
ARITHMETIC MEAN 
We have studied that data can be represented in different 
forms, like tabular form, graphical from. Some times, we 
represent data arithmetically. That means, we represent the 
data by some arithmetic value such that some information 
about data can be can be derived from that arithmetic value. 
 
Arithmetic Descriptors 
The arithmetic values to represent certain features of data are 
called Arithmetic Descriptors. 
 
Central Tendency Or Central Value 
Average or central value of a data is the value, which 
represents the entire data. Because this value represents the 
entire data, therefore such value lies somewhere in between 
the data. In other words, such value lies between two 
extremes of data, i.e. between largest value in the data and 
smallest value on the data. 
 
Note 
Central tendency is an arithmetic descriptor. 
 
Measure Of Central Tendency Or Measure Of 
Location 
The methods used to obtain or calculate central tendency are 
called measure of central tendency or measure of location. 
 
There are different methods to measure central tendency. 
1. Arithmetic Mean 
2. Geometric Mean 
3. Harmonic Mean 
4. Median 
5. Mode 
 
Arithmetic Mean 
Arithmetic mean is simply called Mean. Arithmetic mean is the 
sum of observations divided by the number of observations. 
 
The data may be in different forms like ungrouped data, 
grouped data. So, there are different methods to calculate 
arithmetic mean. Here is a list of various methods to calculate 
arithmetic mean depending upon the form of the available 
data. 
 
1. Raw Data 
 (i) Direct Method 
 
2. Ungrouped Data 
 (i) Direct Method 
 (ii) Shortcut Method 
 (iii) Step Deviation Method 
 
3. Grouped Data 
 (i) Direct Method 
 (ii) Shortcut Method 
 (iii) Step Deviation Method 
 
Now, we will study each of these methods. 
1. Raw Data 
Raw data is unarranged data. 
 
(i) Direct Method 
Sum of the observations is divided by the number of 
observations. 
 
n
x ...... x x x x
X
n 4 3 2 1
+ + + + +
= 
Where  
 X   ?  Arithmetic Mean 
 
n 3 2 1
x ......, , x , x , x  ? Observations 
 n    ? Number of observations 
 
Example 
The daily pocket allowances (In Rs.) of ten college students 
are 26, 27, 20, 29, 21, 23, 25, 30, 28, 21. 
Find the mean daily pocket allowance. 
Mean ( ) x  =  
?
=
n
1 i
i
x
n
1
 
   ] 21 28 30 25 23 21 29 20 27 26 [
10
1
+ + + + + + + + + = 
  ] 250 [
10
1
= 
  25 = 
? The mean daily pocket allowance is Rs 25. 
 
2. Ungrouped Data 
The following methods are used to calculate arithmetic mean 
of ungrouped data: - 
 
(i) Direct Method 
If observation 
n 3 2 1
x ....., , x , x , x have corresponding 
frequencies 
n 3 2 1
f ...., , f , f , f , then 
n 3 2 1
n n 3 3 2 2 1 1
f ....... f f f
x f ........ x f x f x f
X
+ + + +
+ + + +
= 
  Or 
X =  
N
f x
n
1 i
i i
?
=
 
Where  
 X   ? Arithmetic Mean 
 
n 3 2 1
x ....., , x , x , x ? Observations 
 
n 3 2 1
f ...., , f , f , f ? Frequencies 
 
?
  ? Sum 
 i    ? Sequence Number 
 N   ? Sum Of Frequencies 
 
Example 
Find the mean of following observations: - 
x  : 10 30 50 70 89  
f : 7 8 10 15 10 
 
Page 2


 
 
ARITHMETIC MEAN, MODE, MEDIAN
ARITHMETIC MEAN 
We have studied that data can be represented in different 
forms, like tabular form, graphical from. Some times, we 
represent data arithmetically. That means, we represent the 
data by some arithmetic value such that some information 
about data can be can be derived from that arithmetic value. 
 
Arithmetic Descriptors 
The arithmetic values to represent certain features of data are 
called Arithmetic Descriptors. 
 
Central Tendency Or Central Value 
Average or central value of a data is the value, which 
represents the entire data. Because this value represents the 
entire data, therefore such value lies somewhere in between 
the data. In other words, such value lies between two 
extremes of data, i.e. between largest value in the data and 
smallest value on the data. 
 
Note 
Central tendency is an arithmetic descriptor. 
 
Measure Of Central Tendency Or Measure Of 
Location 
The methods used to obtain or calculate central tendency are 
called measure of central tendency or measure of location. 
 
There are different methods to measure central tendency. 
1. Arithmetic Mean 
2. Geometric Mean 
3. Harmonic Mean 
4. Median 
5. Mode 
 
Arithmetic Mean 
Arithmetic mean is simply called Mean. Arithmetic mean is the 
sum of observations divided by the number of observations. 
 
The data may be in different forms like ungrouped data, 
grouped data. So, there are different methods to calculate 
arithmetic mean. Here is a list of various methods to calculate 
arithmetic mean depending upon the form of the available 
data. 
 
1. Raw Data 
 (i) Direct Method 
 
2. Ungrouped Data 
 (i) Direct Method 
 (ii) Shortcut Method 
 (iii) Step Deviation Method 
 
3. Grouped Data 
 (i) Direct Method 
 (ii) Shortcut Method 
 (iii) Step Deviation Method 
 
Now, we will study each of these methods. 
1. Raw Data 
Raw data is unarranged data. 
 
(i) Direct Method 
Sum of the observations is divided by the number of 
observations. 
 
n
x ...... x x x x
X
n 4 3 2 1
+ + + + +
= 
Where  
 X   ?  Arithmetic Mean 
 
n 3 2 1
x ......, , x , x , x  ? Observations 
 n    ? Number of observations 
 
Example 
The daily pocket allowances (In Rs.) of ten college students 
are 26, 27, 20, 29, 21, 23, 25, 30, 28, 21. 
Find the mean daily pocket allowance. 
Mean ( ) x  =  
?
=
n
1 i
i
x
n
1
 
   ] 21 28 30 25 23 21 29 20 27 26 [
10
1
+ + + + + + + + + = 
  ] 250 [
10
1
= 
  25 = 
? The mean daily pocket allowance is Rs 25. 
 
2. Ungrouped Data 
The following methods are used to calculate arithmetic mean 
of ungrouped data: - 
 
(i) Direct Method 
If observation 
n 3 2 1
x ....., , x , x , x have corresponding 
frequencies 
n 3 2 1
f ...., , f , f , f , then 
n 3 2 1
n n 3 3 2 2 1 1
f ....... f f f
x f ........ x f x f x f
X
+ + + +
+ + + +
= 
  Or 
X =  
N
f x
n
1 i
i i
?
=
 
Where  
 X   ? Arithmetic Mean 
 
n 3 2 1
x ....., , x , x , x ? Observations 
 
n 3 2 1
f ...., , f , f , f ? Frequencies 
 
?
  ? Sum 
 i    ? Sequence Number 
 N   ? Sum Of Frequencies 
 
Example 
Find the mean of following observations: - 
x  : 10 30 50 70 89  
f : 7 8 10 15 10 
 
 
Calculation Of Mean 
i
x 
i
f 
i i
f x 
10 7 70 
30 8 240 
50 10 500 
70 15 1050 
89 10 890 
 N = 
i
f ? = 50 
i i
f x ? = 2750 
We have, 
N = 50,  
i i
f x ? = 2750 
Mean X =  
N
f x
n
1 i
i i
?
=
 
 = 
50
2750
 
 = 55 
? Mean = 55 
 
(ii) Shortcut Method 
If the observations are large, then the calculations become 
very tedious and time consuming. In such case, we take an 
assumed mean, say A and subtract this assumed mean A from 
different observations.  
 
The difference between an observation 
1
x and A is called 
deviation of 
1
x from A and is represented by d. So, A is 
chosen in such a way that deviations of different observations 
from A should be small, i.e. d should be small. 
n
f d
A X
n
1 i
i i
?
=
+ = 
Where 
  X   ? Arithmetic Mean 
 A   ? Assumed Mean 
 
n 3 2 1
d ...., , d , d , d ? Deviations 
 
n 3 2 1
f ....., , f , f , f ? Frequencies 
 
?
  ?  Sum 
 i   ? Sequence Number 
 N   ? Sum Of Frequencies 
 
Example 
Find the mean wage from the data given below: - 
 
Wage (in Rs.)    : 800  820   860   900   920   980   1000 
No. Of Workers :  7    14      19     25    20     10      5 
 
Let the assumed mean be A= 900 
Calculation Of Mean 
We have, 
100 N = , 880 d f
i i
- = ? , 900 A = 
 Mean 
n
f d
A X
n
1 i
i i
?
=
+ = 
 
?
?
?
?
?
?
?
? -
+ =
100
880
900 
 8 . 8 900 - = 
 2 . 891 = 
? Mean wage = Rs 891.2 
 
(iii) Step Deviation Method 
If the observations are large, we use shortcut method to 
obtain small deviations. Some times, even the deviations are 
too large and the deviations have a common factor. In such 
case, we divide the deviations by the same common factor and 
reduce the values. 
The common factor is the largest factor of all deviations and is 
represented by h. The value obtained by dividing the deviation 
of observation 
1
x by common factor h is represented by u. 
X  =  
N
f u h
A
n
1 i
i i
?
=
+ 
 Where 
X   ? Arithmetic Mean 
 A   ?  Assumed Mean 
 h   ? Common Factor Of 
     Deviations 
 
n 3 2 1
u ...., , u , u , u ? Quotients Obtained On 
     Dividing  Deviations By  
     Common factor 
 
?
  ?  Sum 
 i   ? Sequence Number 
 N   ? Sum Of Frequencies 
 
Example 
The table below gives the distribution of villages under 
different heights from sea level in a certain region. Compute 
the mean height of the region. 
 
Height (In metrer)  200    600    1000    1400    1800   2200 
No. Of Villages          142    265     560      271       89       16 
 
Let the assumed mean  A = 1400 
Let common factor h = 400  
Calculation Of Mean 
We have, 
A = 1400, h = 400,         N = 1343,     
i i
u f ? = 1395 - 
Height 
(In 
meters) 
i
x 
No. Of 
Villages 
i
f 
i
d = 
1400 x
i
-
 
i
u = 
400
1400 x
i
-
 
i i
u f 
200 142 1200 - -3 426 - 
600 265 800 - -2 530 - 
1000 560 400 - -1 560 - 
1400 ? A 271 0 0 0 
1800 89 400 1 89 
2200 16 800 2 32 
 = N 
i
f ? =1343 
 
 
i i
u f ? =       
1395 - 
Wage 
(In Rs) 
i
x 
No.of workers 
i
f 
A x d
i i
- = 
   = 900 x
i
- 
i i
d f 
800 7 -100 -700 
820 14 -80 -1120 
860 19 -40 -760 
900  ? A 25 0 0 
920 20 20 400 
980 10 80 800 
1000 5 100 500 
 
100 f N
i
= ? =
 
 
880 d f
i i
- = ?
 
 
Page 3


 
 
ARITHMETIC MEAN, MODE, MEDIAN
ARITHMETIC MEAN 
We have studied that data can be represented in different 
forms, like tabular form, graphical from. Some times, we 
represent data arithmetically. That means, we represent the 
data by some arithmetic value such that some information 
about data can be can be derived from that arithmetic value. 
 
Arithmetic Descriptors 
The arithmetic values to represent certain features of data are 
called Arithmetic Descriptors. 
 
Central Tendency Or Central Value 
Average or central value of a data is the value, which 
represents the entire data. Because this value represents the 
entire data, therefore such value lies somewhere in between 
the data. In other words, such value lies between two 
extremes of data, i.e. between largest value in the data and 
smallest value on the data. 
 
Note 
Central tendency is an arithmetic descriptor. 
 
Measure Of Central Tendency Or Measure Of 
Location 
The methods used to obtain or calculate central tendency are 
called measure of central tendency or measure of location. 
 
There are different methods to measure central tendency. 
1. Arithmetic Mean 
2. Geometric Mean 
3. Harmonic Mean 
4. Median 
5. Mode 
 
Arithmetic Mean 
Arithmetic mean is simply called Mean. Arithmetic mean is the 
sum of observations divided by the number of observations. 
 
The data may be in different forms like ungrouped data, 
grouped data. So, there are different methods to calculate 
arithmetic mean. Here is a list of various methods to calculate 
arithmetic mean depending upon the form of the available 
data. 
 
1. Raw Data 
 (i) Direct Method 
 
2. Ungrouped Data 
 (i) Direct Method 
 (ii) Shortcut Method 
 (iii) Step Deviation Method 
 
3. Grouped Data 
 (i) Direct Method 
 (ii) Shortcut Method 
 (iii) Step Deviation Method 
 
Now, we will study each of these methods. 
1. Raw Data 
Raw data is unarranged data. 
 
(i) Direct Method 
Sum of the observations is divided by the number of 
observations. 
 
n
x ...... x x x x
X
n 4 3 2 1
+ + + + +
= 
Where  
 X   ?  Arithmetic Mean 
 
n 3 2 1
x ......, , x , x , x  ? Observations 
 n    ? Number of observations 
 
Example 
The daily pocket allowances (In Rs.) of ten college students 
are 26, 27, 20, 29, 21, 23, 25, 30, 28, 21. 
Find the mean daily pocket allowance. 
Mean ( ) x  =  
?
=
n
1 i
i
x
n
1
 
   ] 21 28 30 25 23 21 29 20 27 26 [
10
1
+ + + + + + + + + = 
  ] 250 [
10
1
= 
  25 = 
? The mean daily pocket allowance is Rs 25. 
 
2. Ungrouped Data 
The following methods are used to calculate arithmetic mean 
of ungrouped data: - 
 
(i) Direct Method 
If observation 
n 3 2 1
x ....., , x , x , x have corresponding 
frequencies 
n 3 2 1
f ...., , f , f , f , then 
n 3 2 1
n n 3 3 2 2 1 1
f ....... f f f
x f ........ x f x f x f
X
+ + + +
+ + + +
= 
  Or 
X =  
N
f x
n
1 i
i i
?
=
 
Where  
 X   ? Arithmetic Mean 
 
n 3 2 1
x ....., , x , x , x ? Observations 
 
n 3 2 1
f ...., , f , f , f ? Frequencies 
 
?
  ? Sum 
 i    ? Sequence Number 
 N   ? Sum Of Frequencies 
 
Example 
Find the mean of following observations: - 
x  : 10 30 50 70 89  
f : 7 8 10 15 10 
 
 
Calculation Of Mean 
i
x 
i
f 
i i
f x 
10 7 70 
30 8 240 
50 10 500 
70 15 1050 
89 10 890 
 N = 
i
f ? = 50 
i i
f x ? = 2750 
We have, 
N = 50,  
i i
f x ? = 2750 
Mean X =  
N
f x
n
1 i
i i
?
=
 
 = 
50
2750
 
 = 55 
? Mean = 55 
 
(ii) Shortcut Method 
If the observations are large, then the calculations become 
very tedious and time consuming. In such case, we take an 
assumed mean, say A and subtract this assumed mean A from 
different observations.  
 
The difference between an observation 
1
x and A is called 
deviation of 
1
x from A and is represented by d. So, A is 
chosen in such a way that deviations of different observations 
from A should be small, i.e. d should be small. 
n
f d
A X
n
1 i
i i
?
=
+ = 
Where 
  X   ? Arithmetic Mean 
 A   ? Assumed Mean 
 
n 3 2 1
d ...., , d , d , d ? Deviations 
 
n 3 2 1
f ....., , f , f , f ? Frequencies 
 
?
  ?  Sum 
 i   ? Sequence Number 
 N   ? Sum Of Frequencies 
 
Example 
Find the mean wage from the data given below: - 
 
Wage (in Rs.)    : 800  820   860   900   920   980   1000 
No. Of Workers :  7    14      19     25    20     10      5 
 
Let the assumed mean be A= 900 
Calculation Of Mean 
We have, 
100 N = , 880 d f
i i
- = ? , 900 A = 
 Mean 
n
f d
A X
n
1 i
i i
?
=
+ = 
 
?
?
?
?
?
?
?
? -
+ =
100
880
900 
 8 . 8 900 - = 
 2 . 891 = 
? Mean wage = Rs 891.2 
 
(iii) Step Deviation Method 
If the observations are large, we use shortcut method to 
obtain small deviations. Some times, even the deviations are 
too large and the deviations have a common factor. In such 
case, we divide the deviations by the same common factor and 
reduce the values. 
The common factor is the largest factor of all deviations and is 
represented by h. The value obtained by dividing the deviation 
of observation 
1
x by common factor h is represented by u. 
X  =  
N
f u h
A
n
1 i
i i
?
=
+ 
 Where 
X   ? Arithmetic Mean 
 A   ?  Assumed Mean 
 h   ? Common Factor Of 
     Deviations 
 
n 3 2 1
u ...., , u , u , u ? Quotients Obtained On 
     Dividing  Deviations By  
     Common factor 
 
?
  ?  Sum 
 i   ? Sequence Number 
 N   ? Sum Of Frequencies 
 
Example 
The table below gives the distribution of villages under 
different heights from sea level in a certain region. Compute 
the mean height of the region. 
 
Height (In metrer)  200    600    1000    1400    1800   2200 
No. Of Villages          142    265     560      271       89       16 
 
Let the assumed mean  A = 1400 
Let common factor h = 400  
Calculation Of Mean 
We have, 
A = 1400, h = 400,         N = 1343,     
i i
u f ? = 1395 - 
Height 
(In 
meters) 
i
x 
No. Of 
Villages 
i
f 
i
d = 
1400 x
i
-
 
i
u = 
400
1400 x
i
-
 
i i
u f 
200 142 1200 - -3 426 - 
600 265 800 - -2 530 - 
1000 560 400 - -1 560 - 
1400 ? A 271 0 0 0 
1800 89 400 1 89 
2200 16 800 2 32 
 = N 
i
f ? =1343 
 
 
i i
u f ? =       
1395 - 
Wage 
(In Rs) 
i
x 
No.of workers 
i
f 
A x d
i i
- = 
   = 900 x
i
- 
i i
d f 
800 7 -100 -700 
820 14 -80 -1120 
860 19 -40 -760 
900  ? A 25 0 0 
920 20 20 400 
980 10 80 800 
1000 5 100 500 
 
100 f N
i
= ? =
 
 
880 d f
i i
- = ?
 
 
 
Mean X =  
N
f u h
A
n
1 i
i i
?
=
+ 
1343
1395 400
1400
- ×
+ = 
49 . 415 1400 - = 
51 . 984 = 
? Mean height of region = 51 . 984 meters 
 
3. Grouped Data 
Grouped data consists of classes and their frequencies. In case 
of classes, all observations lose their individual value. So, all 
observations in a class are represented by the midpoint of that 
class, which is called Class Mark. 
 
The following methods are used to calculate arithmetic mean 
of grouped data: - 
 
(i) Direct Method 
If observations are grouped into classes and each class has its 
own frequency, then 
X =  
N
f x
n
1 i
i i
?
=
 
Where  
 X   ?  Arithmetic Mean 
 
n 3 2 1
x ....., , x , x , x ? Midpoints / Class Marks 
 
n 3 2 1
f ...., , f , f , f ? Frequencies 
 
?
  ? Sum 
 i   ? Sequence Number 
 N   ?  Sum Of Frequencies 
 
Note 
Here 
n 3 2 1
x ....., , x , x , x represent midpoints or class-marks 
of classes and not individual observation. 
 
Example 
Calculate the arithmetic mean of the marks scored by students 
of a class in a class test from the following data: - 
 
Marks 0 - 10 10 - 20 20 - 30 
Number Of 
Students 
 
12 18 27 
 
 
30 – 40 40 - 50 50 - 60 Total 
 
20 
 
17 6 100 
 
Calculation Of Mean 
Marks No. of Students 
i
f 
Mid-point 
i
x 
i i
x f 
0 – 10 12 5 60 
10 – 20 18 15 270 
20 – 30 27 25 675 
30 – 40 20 35 700 
40 – 50 17 45 765 
50 – 60 6 55 330 
Total 100  2800 
Mean X = 
N
x f
n
1 i
i i
?
=
 
2800
100
1
× = 
28 = 
? Mean = 28 marks 
 
(ii) Shortcut Method OR Assumed Mean Method 
If the observations are large, then we take an assumed mean 
A and subtract it from midpoints/class-marks of different 
classes. 
 
The difference between midpoint of a class 
i
x and assumed 
mean A is called deviation of 
1
x from A and is represented by 
d. 
 
N
f d
A X
n
1 i
i i
?
=
+ = 
Where  
X   ? Arithmetic Mean 
 A   ?  Assumed Mean 
 
n 3 2 1
d ...., , d , d , d ? Deviations 
 
n 3 2 1
f ....., , f , f , f ? Frequencies 
 
?
  ?  Sum 
 i   ? Sequence Number 
 N   ? Sum Of Frequencies 
 
Note 
Here 
n 3 2 1
d ...., , d , d , d represent deviations of midpoints or 
class-marks from assumed mean and not deviations of 
individual observations from assumed mean. 
 
Example 
Calculate the arithmetic mean of the marks scored by students 
of a class in a class test from the following data: - 
 
 
Marks 0 – 10 10 - 20 20 - 30 
Number Of 
Students 
 
12 18 27 
 
 
30 - 40 40 - 50 50 - 60 Total 
 
20 
 
17 6 100 
 
 
We take Assumed Mean A = 25 
 
Marks 
No.of 
Students 
Mid-point 
i
x 
Deviation 
25 x d
i i
- = 
i i
d f 
0-10 12 5 -20 -240 
10-20 18 15 -10 -180 
20-30 27 25 ? A 0 0 
30-40 20 35 10 200 
40-50 17 45 20 340 
50-60 6 55 30 180 
Total 100   300 
 
Page 4


 
 
ARITHMETIC MEAN, MODE, MEDIAN
ARITHMETIC MEAN 
We have studied that data can be represented in different 
forms, like tabular form, graphical from. Some times, we 
represent data arithmetically. That means, we represent the 
data by some arithmetic value such that some information 
about data can be can be derived from that arithmetic value. 
 
Arithmetic Descriptors 
The arithmetic values to represent certain features of data are 
called Arithmetic Descriptors. 
 
Central Tendency Or Central Value 
Average or central value of a data is the value, which 
represents the entire data. Because this value represents the 
entire data, therefore such value lies somewhere in between 
the data. In other words, such value lies between two 
extremes of data, i.e. between largest value in the data and 
smallest value on the data. 
 
Note 
Central tendency is an arithmetic descriptor. 
 
Measure Of Central Tendency Or Measure Of 
Location 
The methods used to obtain or calculate central tendency are 
called measure of central tendency or measure of location. 
 
There are different methods to measure central tendency. 
1. Arithmetic Mean 
2. Geometric Mean 
3. Harmonic Mean 
4. Median 
5. Mode 
 
Arithmetic Mean 
Arithmetic mean is simply called Mean. Arithmetic mean is the 
sum of observations divided by the number of observations. 
 
The data may be in different forms like ungrouped data, 
grouped data. So, there are different methods to calculate 
arithmetic mean. Here is a list of various methods to calculate 
arithmetic mean depending upon the form of the available 
data. 
 
1. Raw Data 
 (i) Direct Method 
 
2. Ungrouped Data 
 (i) Direct Method 
 (ii) Shortcut Method 
 (iii) Step Deviation Method 
 
3. Grouped Data 
 (i) Direct Method 
 (ii) Shortcut Method 
 (iii) Step Deviation Method 
 
Now, we will study each of these methods. 
1. Raw Data 
Raw data is unarranged data. 
 
(i) Direct Method 
Sum of the observations is divided by the number of 
observations. 
 
n
x ...... x x x x
X
n 4 3 2 1
+ + + + +
= 
Where  
 X   ?  Arithmetic Mean 
 
n 3 2 1
x ......, , x , x , x  ? Observations 
 n    ? Number of observations 
 
Example 
The daily pocket allowances (In Rs.) of ten college students 
are 26, 27, 20, 29, 21, 23, 25, 30, 28, 21. 
Find the mean daily pocket allowance. 
Mean ( ) x  =  
?
=
n
1 i
i
x
n
1
 
   ] 21 28 30 25 23 21 29 20 27 26 [
10
1
+ + + + + + + + + = 
  ] 250 [
10
1
= 
  25 = 
? The mean daily pocket allowance is Rs 25. 
 
2. Ungrouped Data 
The following methods are used to calculate arithmetic mean 
of ungrouped data: - 
 
(i) Direct Method 
If observation 
n 3 2 1
x ....., , x , x , x have corresponding 
frequencies 
n 3 2 1
f ...., , f , f , f , then 
n 3 2 1
n n 3 3 2 2 1 1
f ....... f f f
x f ........ x f x f x f
X
+ + + +
+ + + +
= 
  Or 
X =  
N
f x
n
1 i
i i
?
=
 
Where  
 X   ? Arithmetic Mean 
 
n 3 2 1
x ....., , x , x , x ? Observations 
 
n 3 2 1
f ...., , f , f , f ? Frequencies 
 
?
  ? Sum 
 i    ? Sequence Number 
 N   ? Sum Of Frequencies 
 
Example 
Find the mean of following observations: - 
x  : 10 30 50 70 89  
f : 7 8 10 15 10 
 
 
Calculation Of Mean 
i
x 
i
f 
i i
f x 
10 7 70 
30 8 240 
50 10 500 
70 15 1050 
89 10 890 
 N = 
i
f ? = 50 
i i
f x ? = 2750 
We have, 
N = 50,  
i i
f x ? = 2750 
Mean X =  
N
f x
n
1 i
i i
?
=
 
 = 
50
2750
 
 = 55 
? Mean = 55 
 
(ii) Shortcut Method 
If the observations are large, then the calculations become 
very tedious and time consuming. In such case, we take an 
assumed mean, say A and subtract this assumed mean A from 
different observations.  
 
The difference between an observation 
1
x and A is called 
deviation of 
1
x from A and is represented by d. So, A is 
chosen in such a way that deviations of different observations 
from A should be small, i.e. d should be small. 
n
f d
A X
n
1 i
i i
?
=
+ = 
Where 
  X   ? Arithmetic Mean 
 A   ? Assumed Mean 
 
n 3 2 1
d ...., , d , d , d ? Deviations 
 
n 3 2 1
f ....., , f , f , f ? Frequencies 
 
?
  ?  Sum 
 i   ? Sequence Number 
 N   ? Sum Of Frequencies 
 
Example 
Find the mean wage from the data given below: - 
 
Wage (in Rs.)    : 800  820   860   900   920   980   1000 
No. Of Workers :  7    14      19     25    20     10      5 
 
Let the assumed mean be A= 900 
Calculation Of Mean 
We have, 
100 N = , 880 d f
i i
- = ? , 900 A = 
 Mean 
n
f d
A X
n
1 i
i i
?
=
+ = 
 
?
?
?
?
?
?
?
? -
+ =
100
880
900 
 8 . 8 900 - = 
 2 . 891 = 
? Mean wage = Rs 891.2 
 
(iii) Step Deviation Method 
If the observations are large, we use shortcut method to 
obtain small deviations. Some times, even the deviations are 
too large and the deviations have a common factor. In such 
case, we divide the deviations by the same common factor and 
reduce the values. 
The common factor is the largest factor of all deviations and is 
represented by h. The value obtained by dividing the deviation 
of observation 
1
x by common factor h is represented by u. 
X  =  
N
f u h
A
n
1 i
i i
?
=
+ 
 Where 
X   ? Arithmetic Mean 
 A   ?  Assumed Mean 
 h   ? Common Factor Of 
     Deviations 
 
n 3 2 1
u ...., , u , u , u ? Quotients Obtained On 
     Dividing  Deviations By  
     Common factor 
 
?
  ?  Sum 
 i   ? Sequence Number 
 N   ? Sum Of Frequencies 
 
Example 
The table below gives the distribution of villages under 
different heights from sea level in a certain region. Compute 
the mean height of the region. 
 
Height (In metrer)  200    600    1000    1400    1800   2200 
No. Of Villages          142    265     560      271       89       16 
 
Let the assumed mean  A = 1400 
Let common factor h = 400  
Calculation Of Mean 
We have, 
A = 1400, h = 400,         N = 1343,     
i i
u f ? = 1395 - 
Height 
(In 
meters) 
i
x 
No. Of 
Villages 
i
f 
i
d = 
1400 x
i
-
 
i
u = 
400
1400 x
i
-
 
i i
u f 
200 142 1200 - -3 426 - 
600 265 800 - -2 530 - 
1000 560 400 - -1 560 - 
1400 ? A 271 0 0 0 
1800 89 400 1 89 
2200 16 800 2 32 
 = N 
i
f ? =1343 
 
 
i i
u f ? =       
1395 - 
Wage 
(In Rs) 
i
x 
No.of workers 
i
f 
A x d
i i
- = 
   = 900 x
i
- 
i i
d f 
800 7 -100 -700 
820 14 -80 -1120 
860 19 -40 -760 
900  ? A 25 0 0 
920 20 20 400 
980 10 80 800 
1000 5 100 500 
 
100 f N
i
= ? =
 
 
880 d f
i i
- = ?
 
 
 
Mean X =  
N
f u h
A
n
1 i
i i
?
=
+ 
1343
1395 400
1400
- ×
+ = 
49 . 415 1400 - = 
51 . 984 = 
? Mean height of region = 51 . 984 meters 
 
3. Grouped Data 
Grouped data consists of classes and their frequencies. In case 
of classes, all observations lose their individual value. So, all 
observations in a class are represented by the midpoint of that 
class, which is called Class Mark. 
 
The following methods are used to calculate arithmetic mean 
of grouped data: - 
 
(i) Direct Method 
If observations are grouped into classes and each class has its 
own frequency, then 
X =  
N
f x
n
1 i
i i
?
=
 
Where  
 X   ?  Arithmetic Mean 
 
n 3 2 1
x ....., , x , x , x ? Midpoints / Class Marks 
 
n 3 2 1
f ...., , f , f , f ? Frequencies 
 
?
  ? Sum 
 i   ? Sequence Number 
 N   ?  Sum Of Frequencies 
 
Note 
Here 
n 3 2 1
x ....., , x , x , x represent midpoints or class-marks 
of classes and not individual observation. 
 
Example 
Calculate the arithmetic mean of the marks scored by students 
of a class in a class test from the following data: - 
 
Marks 0 - 10 10 - 20 20 - 30 
Number Of 
Students 
 
12 18 27 
 
 
30 – 40 40 - 50 50 - 60 Total 
 
20 
 
17 6 100 
 
Calculation Of Mean 
Marks No. of Students 
i
f 
Mid-point 
i
x 
i i
x f 
0 – 10 12 5 60 
10 – 20 18 15 270 
20 – 30 27 25 675 
30 – 40 20 35 700 
40 – 50 17 45 765 
50 – 60 6 55 330 
Total 100  2800 
Mean X = 
N
x f
n
1 i
i i
?
=
 
2800
100
1
× = 
28 = 
? Mean = 28 marks 
 
(ii) Shortcut Method OR Assumed Mean Method 
If the observations are large, then we take an assumed mean 
A and subtract it from midpoints/class-marks of different 
classes. 
 
The difference between midpoint of a class 
i
x and assumed 
mean A is called deviation of 
1
x from A and is represented by 
d. 
 
N
f d
A X
n
1 i
i i
?
=
+ = 
Where  
X   ? Arithmetic Mean 
 A   ?  Assumed Mean 
 
n 3 2 1
d ...., , d , d , d ? Deviations 
 
n 3 2 1
f ....., , f , f , f ? Frequencies 
 
?
  ?  Sum 
 i   ? Sequence Number 
 N   ? Sum Of Frequencies 
 
Note 
Here 
n 3 2 1
d ...., , d , d , d represent deviations of midpoints or 
class-marks from assumed mean and not deviations of 
individual observations from assumed mean. 
 
Example 
Calculate the arithmetic mean of the marks scored by students 
of a class in a class test from the following data: - 
 
 
Marks 0 – 10 10 - 20 20 - 30 
Number Of 
Students 
 
12 18 27 
 
 
30 - 40 40 - 50 50 - 60 Total 
 
20 
 
17 6 100 
 
 
We take Assumed Mean A = 25 
 
Marks 
No.of 
Students 
Mid-point 
i
x 
Deviation 
25 x d
i i
- = 
i i
d f 
0-10 12 5 -20 -240 
10-20 18 15 -10 -180 
20-30 27 25 ? A 0 0 
30-40 20 35 10 200 
40-50 17 45 20 340 
50-60 6 55 30 180 
Total 100   300 
 
 
 
 Mean X = 
N
d f
A
n
1 i
i i
?
=
+ 
   
100
300
25 + = 
   3 25 + = 
=28 
 
? Mean = 28 marks 
 
(iii) Step Deviation Method 
If observations are large, then we take deviations 
n 3 2 1
d ...., , d , d , d of observations from assumed mean A. If 
the deviations are still large, then we divide the deviations 
from the same common factor h. The result of dividing 
deviation by same common factor h is represented by u. 
 
X =  
N
f u h
A
n
1 i
i i
?
=
+ 
 
Where 
X   ? Arithmetic Mean 
 A   ?  Assumed Mean 
 h   ? Common Factor Of 
     Deviations 
 
n 3 2 1
u ...., , u , u , u ? Quotients Obtained On 
     Dividing  Deviations By  
     Common factor 
 
n 3 2 1
f ....., , f , f , f ? Frequencies 
 
?
  ?  Sum 
 i   ? Sequence Number 
 
 
Note 
Here 
n 3 2 1
d ...., , d , d , d are deviations of midpoint or class- 
marks from assumed mean, so h is common factor of deviation 
of midpoint/class marks from assumed mean and not common 
factor of deviation of individual observations. 
 
Example 
The following table gives the distribution of total household 
expenditure (in rupees) of manual workers in a city. 
 
Find the average expenditure (in Rs) per household. 
 
Expenditure 
(In Rs) 100–150 150–200 200–250 250–300 
Freauency 
 
24 40 33 28 
 
        
300–350 350–400 400–450 450–500l 
 
30 
 
22 16 7 
 
 
Let the assumed mean  A = 325  
Let the common factor  h = 50 
 
Calculation Of Mean 
 
Expenditure 
(in Rs.) 
i
x 
Frequency 
 
i
f 
Mid-values  
 
i
x 
A x d
i i
- = 
 
  = 325 x
i
- 
100 - 150 24 125 -200 
150 - 200 40 175 -150 
200 - 250 33 225 -100 
250 - 300 28 275 -50 
300 - 350 30 325 ? A 0 
350 - 400 22 375 50 
400 - 450 16 425 100 
450 - 500 7 475 150 
 
200 f N
i
= ? = 
  
 
 
 
i
u = 
h
A x
i
-
 
    = 
50
325 x
i
-
 
i i
u f 
-4 -96 
-3 -120 
-2 -66 
-1 -28 
0 0 
1 22 
2 32 
3 21 
 
235 u f
i i
- = ?
 
 
 
We have, 
 , 200 N = , 325 A =  , 50 h =  235 u f
i i
- = ? 
Mean X =  
N
f u h
A
n
1 i
i i
?
=
+ 
  
200
235 50
325
- ×
+ = 
  75 . 58 325 - = 
  
4
235
325 - = 
  75 . 58 325 - =  
  25 . 266 = 
 
? Average expenditure is Rs. 266.25 
 
 
MODE  
The mode the value which has the highest frequency in 
the data. In other words, mode of a data distribution is 
the value which is repeated highest times in the data. 
 
In case of a grouped or class frequency distribution 
with equal class intervals, we use the following 
algorithm to compute the mode: - 
 
Algorithm 
(i) Obtain the grouped frequency distribution 
(ii) Determine the class which has the maximum 
frequency. This class is the modal class. 
 
Page 5


 
 
ARITHMETIC MEAN, MODE, MEDIAN
ARITHMETIC MEAN 
We have studied that data can be represented in different 
forms, like tabular form, graphical from. Some times, we 
represent data arithmetically. That means, we represent the 
data by some arithmetic value such that some information 
about data can be can be derived from that arithmetic value. 
 
Arithmetic Descriptors 
The arithmetic values to represent certain features of data are 
called Arithmetic Descriptors. 
 
Central Tendency Or Central Value 
Average or central value of a data is the value, which 
represents the entire data. Because this value represents the 
entire data, therefore such value lies somewhere in between 
the data. In other words, such value lies between two 
extremes of data, i.e. between largest value in the data and 
smallest value on the data. 
 
Note 
Central tendency is an arithmetic descriptor. 
 
Measure Of Central Tendency Or Measure Of 
Location 
The methods used to obtain or calculate central tendency are 
called measure of central tendency or measure of location. 
 
There are different methods to measure central tendency. 
1. Arithmetic Mean 
2. Geometric Mean 
3. Harmonic Mean 
4. Median 
5. Mode 
 
Arithmetic Mean 
Arithmetic mean is simply called Mean. Arithmetic mean is the 
sum of observations divided by the number of observations. 
 
The data may be in different forms like ungrouped data, 
grouped data. So, there are different methods to calculate 
arithmetic mean. Here is a list of various methods to calculate 
arithmetic mean depending upon the form of the available 
data. 
 
1. Raw Data 
 (i) Direct Method 
 
2. Ungrouped Data 
 (i) Direct Method 
 (ii) Shortcut Method 
 (iii) Step Deviation Method 
 
3. Grouped Data 
 (i) Direct Method 
 (ii) Shortcut Method 
 (iii) Step Deviation Method 
 
Now, we will study each of these methods. 
1. Raw Data 
Raw data is unarranged data. 
 
(i) Direct Method 
Sum of the observations is divided by the number of 
observations. 
 
n
x ...... x x x x
X
n 4 3 2 1
+ + + + +
= 
Where  
 X   ?  Arithmetic Mean 
 
n 3 2 1
x ......, , x , x , x  ? Observations 
 n    ? Number of observations 
 
Example 
The daily pocket allowances (In Rs.) of ten college students 
are 26, 27, 20, 29, 21, 23, 25, 30, 28, 21. 
Find the mean daily pocket allowance. 
Mean ( ) x  =  
?
=
n
1 i
i
x
n
1
 
   ] 21 28 30 25 23 21 29 20 27 26 [
10
1
+ + + + + + + + + = 
  ] 250 [
10
1
= 
  25 = 
? The mean daily pocket allowance is Rs 25. 
 
2. Ungrouped Data 
The following methods are used to calculate arithmetic mean 
of ungrouped data: - 
 
(i) Direct Method 
If observation 
n 3 2 1
x ....., , x , x , x have corresponding 
frequencies 
n 3 2 1
f ...., , f , f , f , then 
n 3 2 1
n n 3 3 2 2 1 1
f ....... f f f
x f ........ x f x f x f
X
+ + + +
+ + + +
= 
  Or 
X =  
N
f x
n
1 i
i i
?
=
 
Where  
 X   ? Arithmetic Mean 
 
n 3 2 1
x ....., , x , x , x ? Observations 
 
n 3 2 1
f ...., , f , f , f ? Frequencies 
 
?
  ? Sum 
 i    ? Sequence Number 
 N   ? Sum Of Frequencies 
 
Example 
Find the mean of following observations: - 
x  : 10 30 50 70 89  
f : 7 8 10 15 10 
 
 
Calculation Of Mean 
i
x 
i
f 
i i
f x 
10 7 70 
30 8 240 
50 10 500 
70 15 1050 
89 10 890 
 N = 
i
f ? = 50 
i i
f x ? = 2750 
We have, 
N = 50,  
i i
f x ? = 2750 
Mean X =  
N
f x
n
1 i
i i
?
=
 
 = 
50
2750
 
 = 55 
? Mean = 55 
 
(ii) Shortcut Method 
If the observations are large, then the calculations become 
very tedious and time consuming. In such case, we take an 
assumed mean, say A and subtract this assumed mean A from 
different observations.  
 
The difference between an observation 
1
x and A is called 
deviation of 
1
x from A and is represented by d. So, A is 
chosen in such a way that deviations of different observations 
from A should be small, i.e. d should be small. 
n
f d
A X
n
1 i
i i
?
=
+ = 
Where 
  X   ? Arithmetic Mean 
 A   ? Assumed Mean 
 
n 3 2 1
d ...., , d , d , d ? Deviations 
 
n 3 2 1
f ....., , f , f , f ? Frequencies 
 
?
  ?  Sum 
 i   ? Sequence Number 
 N   ? Sum Of Frequencies 
 
Example 
Find the mean wage from the data given below: - 
 
Wage (in Rs.)    : 800  820   860   900   920   980   1000 
No. Of Workers :  7    14      19     25    20     10      5 
 
Let the assumed mean be A= 900 
Calculation Of Mean 
We have, 
100 N = , 880 d f
i i
- = ? , 900 A = 
 Mean 
n
f d
A X
n
1 i
i i
?
=
+ = 
 
?
?
?
?
?
?
?
? -
+ =
100
880
900 
 8 . 8 900 - = 
 2 . 891 = 
? Mean wage = Rs 891.2 
 
(iii) Step Deviation Method 
If the observations are large, we use shortcut method to 
obtain small deviations. Some times, even the deviations are 
too large and the deviations have a common factor. In such 
case, we divide the deviations by the same common factor and 
reduce the values. 
The common factor is the largest factor of all deviations and is 
represented by h. The value obtained by dividing the deviation 
of observation 
1
x by common factor h is represented by u. 
X  =  
N
f u h
A
n
1 i
i i
?
=
+ 
 Where 
X   ? Arithmetic Mean 
 A   ?  Assumed Mean 
 h   ? Common Factor Of 
     Deviations 
 
n 3 2 1
u ...., , u , u , u ? Quotients Obtained On 
     Dividing  Deviations By  
     Common factor 
 
?
  ?  Sum 
 i   ? Sequence Number 
 N   ? Sum Of Frequencies 
 
Example 
The table below gives the distribution of villages under 
different heights from sea level in a certain region. Compute 
the mean height of the region. 
 
Height (In metrer)  200    600    1000    1400    1800   2200 
No. Of Villages          142    265     560      271       89       16 
 
Let the assumed mean  A = 1400 
Let common factor h = 400  
Calculation Of Mean 
We have, 
A = 1400, h = 400,         N = 1343,     
i i
u f ? = 1395 - 
Height 
(In 
meters) 
i
x 
No. Of 
Villages 
i
f 
i
d = 
1400 x
i
-
 
i
u = 
400
1400 x
i
-
 
i i
u f 
200 142 1200 - -3 426 - 
600 265 800 - -2 530 - 
1000 560 400 - -1 560 - 
1400 ? A 271 0 0 0 
1800 89 400 1 89 
2200 16 800 2 32 
 = N 
i
f ? =1343 
 
 
i i
u f ? =       
1395 - 
Wage 
(In Rs) 
i
x 
No.of workers 
i
f 
A x d
i i
- = 
   = 900 x
i
- 
i i
d f 
800 7 -100 -700 
820 14 -80 -1120 
860 19 -40 -760 
900  ? A 25 0 0 
920 20 20 400 
980 10 80 800 
1000 5 100 500 
 
100 f N
i
= ? =
 
 
880 d f
i i
- = ?
 
 
 
Mean X =  
N
f u h
A
n
1 i
i i
?
=
+ 
1343
1395 400
1400
- ×
+ = 
49 . 415 1400 - = 
51 . 984 = 
? Mean height of region = 51 . 984 meters 
 
3. Grouped Data 
Grouped data consists of classes and their frequencies. In case 
of classes, all observations lose their individual value. So, all 
observations in a class are represented by the midpoint of that 
class, which is called Class Mark. 
 
The following methods are used to calculate arithmetic mean 
of grouped data: - 
 
(i) Direct Method 
If observations are grouped into classes and each class has its 
own frequency, then 
X =  
N
f x
n
1 i
i i
?
=
 
Where  
 X   ?  Arithmetic Mean 
 
n 3 2 1
x ....., , x , x , x ? Midpoints / Class Marks 
 
n 3 2 1
f ...., , f , f , f ? Frequencies 
 
?
  ? Sum 
 i   ? Sequence Number 
 N   ?  Sum Of Frequencies 
 
Note 
Here 
n 3 2 1
x ....., , x , x , x represent midpoints or class-marks 
of classes and not individual observation. 
 
Example 
Calculate the arithmetic mean of the marks scored by students 
of a class in a class test from the following data: - 
 
Marks 0 - 10 10 - 20 20 - 30 
Number Of 
Students 
 
12 18 27 
 
 
30 – 40 40 - 50 50 - 60 Total 
 
20 
 
17 6 100 
 
Calculation Of Mean 
Marks No. of Students 
i
f 
Mid-point 
i
x 
i i
x f 
0 – 10 12 5 60 
10 – 20 18 15 270 
20 – 30 27 25 675 
30 – 40 20 35 700 
40 – 50 17 45 765 
50 – 60 6 55 330 
Total 100  2800 
Mean X = 
N
x f
n
1 i
i i
?
=
 
2800
100
1
× = 
28 = 
? Mean = 28 marks 
 
(ii) Shortcut Method OR Assumed Mean Method 
If the observations are large, then we take an assumed mean 
A and subtract it from midpoints/class-marks of different 
classes. 
 
The difference between midpoint of a class 
i
x and assumed 
mean A is called deviation of 
1
x from A and is represented by 
d. 
 
N
f d
A X
n
1 i
i i
?
=
+ = 
Where  
X   ? Arithmetic Mean 
 A   ?  Assumed Mean 
 
n 3 2 1
d ...., , d , d , d ? Deviations 
 
n 3 2 1
f ....., , f , f , f ? Frequencies 
 
?
  ?  Sum 
 i   ? Sequence Number 
 N   ? Sum Of Frequencies 
 
Note 
Here 
n 3 2 1
d ...., , d , d , d represent deviations of midpoints or 
class-marks from assumed mean and not deviations of 
individual observations from assumed mean. 
 
Example 
Calculate the arithmetic mean of the marks scored by students 
of a class in a class test from the following data: - 
 
 
Marks 0 – 10 10 - 20 20 - 30 
Number Of 
Students 
 
12 18 27 
 
 
30 - 40 40 - 50 50 - 60 Total 
 
20 
 
17 6 100 
 
 
We take Assumed Mean A = 25 
 
Marks 
No.of 
Students 
Mid-point 
i
x 
Deviation 
25 x d
i i
- = 
i i
d f 
0-10 12 5 -20 -240 
10-20 18 15 -10 -180 
20-30 27 25 ? A 0 0 
30-40 20 35 10 200 
40-50 17 45 20 340 
50-60 6 55 30 180 
Total 100   300 
 
 
 
 Mean X = 
N
d f
A
n
1 i
i i
?
=
+ 
   
100
300
25 + = 
   3 25 + = 
=28 
 
? Mean = 28 marks 
 
(iii) Step Deviation Method 
If observations are large, then we take deviations 
n 3 2 1
d ...., , d , d , d of observations from assumed mean A. If 
the deviations are still large, then we divide the deviations 
from the same common factor h. The result of dividing 
deviation by same common factor h is represented by u. 
 
X =  
N
f u h
A
n
1 i
i i
?
=
+ 
 
Where 
X   ? Arithmetic Mean 
 A   ?  Assumed Mean 
 h   ? Common Factor Of 
     Deviations 
 
n 3 2 1
u ...., , u , u , u ? Quotients Obtained On 
     Dividing  Deviations By  
     Common factor 
 
n 3 2 1
f ....., , f , f , f ? Frequencies 
 
?
  ?  Sum 
 i   ? Sequence Number 
 
 
Note 
Here 
n 3 2 1
d ...., , d , d , d are deviations of midpoint or class- 
marks from assumed mean, so h is common factor of deviation 
of midpoint/class marks from assumed mean and not common 
factor of deviation of individual observations. 
 
Example 
The following table gives the distribution of total household 
expenditure (in rupees) of manual workers in a city. 
 
Find the average expenditure (in Rs) per household. 
 
Expenditure 
(In Rs) 100–150 150–200 200–250 250–300 
Freauency 
 
24 40 33 28 
 
        
300–350 350–400 400–450 450–500l 
 
30 
 
22 16 7 
 
 
Let the assumed mean  A = 325  
Let the common factor  h = 50 
 
Calculation Of Mean 
 
Expenditure 
(in Rs.) 
i
x 
Frequency 
 
i
f 
Mid-values  
 
i
x 
A x d
i i
- = 
 
  = 325 x
i
- 
100 - 150 24 125 -200 
150 - 200 40 175 -150 
200 - 250 33 225 -100 
250 - 300 28 275 -50 
300 - 350 30 325 ? A 0 
350 - 400 22 375 50 
400 - 450 16 425 100 
450 - 500 7 475 150 
 
200 f N
i
= ? = 
  
 
 
 
i
u = 
h
A x
i
-
 
    = 
50
325 x
i
-
 
i i
u f 
-4 -96 
-3 -120 
-2 -66 
-1 -28 
0 0 
1 22 
2 32 
3 21 
 
235 u f
i i
- = ?
 
 
 
We have, 
 , 200 N = , 325 A =  , 50 h =  235 u f
i i
- = ? 
Mean X =  
N
f u h
A
n
1 i
i i
?
=
+ 
  
200
235 50
325
- ×
+ = 
  75 . 58 325 - = 
  
4
235
325 - = 
  75 . 58 325 - =  
  25 . 266 = 
 
? Average expenditure is Rs. 266.25 
 
 
MODE  
The mode the value which has the highest frequency in 
the data. In other words, mode of a data distribution is 
the value which is repeated highest times in the data. 
 
In case of a grouped or class frequency distribution 
with equal class intervals, we use the following 
algorithm to compute the mode: - 
 
Algorithm 
(i) Obtain the grouped frequency distribution 
(ii) Determine the class which has the maximum 
frequency. This class is the modal class. 
 
 
(iii) Using the modal class, determine the values of the 
variables used to calculate mode, i.e. l, 
1
f , 
0
f , 
2
f , 
h 
 Where, 
  l Lower limit of the modal class 
 
1
f  Frequency of the modal class 
 
0
f  Frequency of the previous class than  
modal class 
 
2
f  Frequency of the next class than modal  
class 
 h  Class size (Width) of median class 
(iv) Calculate the mode using the formula 
 Mode = 
10
10 2
f f
l h
2f f f
?? -
+ ×
??
--
??
 
Example 
Calculate the mode of the following data: - 
Class 
0–
50 
50–
100 
100–
150 
150–
200 
200–
250 
250–
300 
Frequency 90 150 100 80 70 10 
 
Here, 
Highest frequency is 150, so 50 – 100 is modal class. 
Then, 
l = 50, 
1
f 150 = , 
0
f 90 = , 
2
f 100 = , h = 50 
Then, 
Mode = 
10
10 2
f f
l h
2f f f
?? -
+ ×
??
--
??
 
  = 
( )
150 90
50 50
2 150 90 100
??
-
+ × ??
??
--
??
 
 = 
60
50 50
300 90 100
? ?
+×
? ?
--
? ?
 
 = 
60
50 50
110
??
+×
??
??
 
 = 
30
50
11
+ 
 = 
550 30
11
+
 
 = 
580
11
 
 = 52.72 
 
 
MEDIAN 
The median is the value which is at the mid of data. In 
other words, median of a data distribution is the value 
which divides the data into two equal parts. So, median 
is the value of the variable such that the number of 
observations above it and the number of observations 
below it are equal. 
 
There are different methods to calculate the median 
depending upon the type of distribution of the data. 
1. Median Of Individual Observations 
2. Median Of Discrete Frequency Distribution 
3. Median Of Grouped/Class Frequency Distribution 
 
1. Median Of Individual Observations 
If x
1
, x
2
, x
3
, ……………, x
n
 are the observations such 
that there are n number of observations, then we use 
the following algorithm to find the median: - 
 
Algorithm 
(i) Arrange the observations x
1
, x
2
, x
3
, ……………, x
n
 in 
ascending or descending order of magnitude. 
(ii) Determine the total number of observations. Let 
the total number of observations be n. 
(iii) If number of observations are odd, i.e. if n is odd, 
then median is value of 
Th
n1
2
+ ??
??
??
observation. 
(iv) If number of observations is even, i.e. if n is even, 
then median is 
Th Th
n n
1
2 2
2
?? ? ?
++
?? ? ?
?? ? ?
observation., i.e. 
the Arithmetic-Mean of the observations at the 
position 
n
2
??
??
??
and 
n
1
2
??
+
??
??
. 
 
Example 
1. The following are the marks of 9 students in a 
class: -34, 32, 48, 38, 24, 30, 27, 21, 35 
 Find the median. 
 
 Arranging the data in ascending order 
 21, 24, 27, 30, 32, 34, 35, 38, 48 
  
 Number of observations = n = 9 
 Here, number of observation is odd. Therefore,  
 Median = 
Th
n1
2
+ ??
??
??
observation  
       = 
th
91
2
+ ??
??
??
observation 
       = 5
th
 observation  
       = 32    [5
th
 observation = 32] 
 ? 32 is median 
 
2. Find the median of the daily wages of ten workers 
from the following data: - 
 20, 25, 17, 18, 8, 15, 22, 11, 9, 14 
 
 Arranging the data in ascending order 
 8, 9, 11, 14, 15, 17, 18, 20, 22, 25 
 Number of observations = n = 10 
 Here, number of observation is even. Therefore,  
 Median = 
Th Th
n n
1
2 2
2
?? ? ?
++
?? ? ?
?? ? ?
 observation 
       = 
th th
10 10
1
22
2
?? ? ?
++
?? ? ?
?? ? ?
 observation 
       = 
th th
5 6
2
+
 observation 
       = 
15 17
2
+
 
       = 
32
2
 
       = 16  
 
 ? Median is 16 
 
2. Median Of Discrete Frequency 
Distribution 
In case of a discrete frequency distribution, we 
calculate the median by using the following algorithm: -