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Mean formula

Friday 10 February 2023

                                                                    

                             

      MEAN 


         The most common representative value of a group of a  data is the Arithmetic 

Mean. It is defined as sum of all observations of a data by no. of observations 

of the data. In statistics ,it is one of the measures of central tendency . To 

understand this in a better way ,let us look some examples
         
Two vessels contain 50 and 80 litres of milk respectively. What is the amount that 

each vessel would have , if both share the milk equally? When we ask this 

question we are seeking Arithmetic mean
       
 In this case ,
                         
             Arithmetic mean = Total quantity of milk ÷  Number of vessels
                                               
                                        = (50+80) ÷ 2
                                                   
                                        = 130/ 2 = 65
                      
           thus each vessel would have 65 litres of milk

              The Average or Arithmetic mean is defined as

                               Arithmetic mean or Mean =  Sum of all observation  
                                                                              No .of observation


Example ;

Smith studies for 3 hours,4 hours and 5 hours respectively on three consecutive 

days. How many hours does he study daily  on an average ?
 
 solution :          
                   
 The Average study time of Smith  =     Total no. of study hours             
                                                          No. of days for which he studied

                                                       =   3 +4 + 5 
                                                                 3
                 
                                                       = 12/3 = 4 hours

Example ; 

A batsman scored the following number of runs in six innings ; 

36,40,42,34,46,50,68 . Calculate the mean runs scored by him in an innings?
 
 solution:
               
           The average score of the batsman =  Total runs scored 
                                                                            No .of innings

                                                                  = 36 + 40  +44 +34 +46 +50 +68 
                                                                                             6

                                                                  = 53


 Formula of Arithmetic Mean for ungrouped data


If any data set consisting of the values a1, a2, a3, …., an  then the arithmetic 

mean  is defined as

                                        Mean =  Sum of all observation  

                                                    Total no .of observation
=1=1=1+2+3+.+

If these n observations have corresponding frequencies, the arithmetic mean is 

computed using the formula

=11+22++
using Sigma notation==11ni=1nai

1ni=nai =  a1+ a2 + a3+ ….+ an 
                                                      
                                 n

     
when the observations  like x1, x2, x3, …., xn  some frequencies 
f1, f2, f3, …., fn


Then mean is given by:
                    
         Mean =[ f1x1, f2x2, f3x3, …., fnxn] / N
 









           

Example:

Find the mean of the following data:

                10, 15, 12, 16, 15, 10, 14, 15, 12, 10.

Solution:


   Here data are  repeated in the collection, so we take note of their frequencies.

Variate

(x)

10

12

14

15

16

Total

Frequency

(f)

3

2

1

3

1

10

Therefore, mean  A = 

                                        = 10*3 + 12*2 +14*1 +15*3 +16*1   

                                                                   10

                                        = 129/10

                                        =  12.9

Formula of Arithmetic mean for grouped data


    To find the mean of a grouped data there are three methods 
    
   (i) Direct Method

   (ii) Step deviation Method

   (iii) Assumed mean Method



(i) Direct Method

      In direct method, the datas are arranged  in groups. It is the simplest method to 

find the mean.

    



       For calculating the mean of a grouped data or observation by direct method 

first  we have to find the class mark. The class marks we get is the xi in the data
                                            


                                        Class Marks   =       upper limit+ lower limit
                                                                                                          
                                                                                 2


   Mean formula by direct method is given by:


                                                      
                                   
Example:
      

Calculate the mean of the following distribution: 

Class Interval
Frequency  8               
solution

     class         Mid value xi)                 
0-10                  5          10       50
10-20                 15         20  300
20-30                 25         12  300
30-40                 35        35  1225
40-50                 45        24  1080
50-60                 55        18  990
Total
       1003945

Mean
                
                

(ii) Assumed Mean Method:

      In the assumed mean method we assume that a is an assumed number which the deviation of the observation is 
                                       d = x - a. 

   And substituting this in direct method we will get the formula of assumed mean method. Assumed mean method is given by

               


Mean  =    a + 


          where  ,  a =  assumed number

                

                        d = x - a


                        

                  



 
  
Examples:


Find the mean of the following data, by using the assumed mean method.
Class Interval





Frequency




Solution:

 Class intervalFrequency (  
   
    
    
    
    
    
                                
          x
               f                      

                      
 5 -20 -140
 15  - -80
   0 0
 35  10 130
  20 200
   
Mean =    a + 

    
(iii) Step deviation method:
         
           This method is mainly used when the deviation of the class from the assumed 

mean are large and they all have a common factor. In this method, we use class width  as  ' h ' also . Using the same formula in assumed mean d = x-a  here we use class width u =x-a  
                             h
                                                                                                                                                  
Step deviation formula is given as:
 
                              
                                       Mean =    a + h  ( 
 
                         where , h = class width
                            
                                        a= assumed number                                                                                                                                         
Example:


Find the mean marks of students from the following cumulative frequency distribution:

MarksNumber of students
o and above80
10 and above77
20 and above72
30 and above65
40 and above55
50 and above43
60 and above28
70 and above16
80 and above10
90 and above8
100 and above 0



solution:


Here we have cumulative frequency,

so first we have to change it into an ordinary frequency distribution.

There are 80 student who scored  greater than or equal to 0.

77 students who scored 10 and  more marks .

Therefore the no. of students scored marks between 0 and 10  is 80-77 = 3

similarly, the no. of students who scored  between 10 and 20 is 77- 72= 5

so ,frequency distribution is:

Marks Number of students
0-103
10-205
20-307
30-4010
40-5012
50-6015
60-70 12
70-806
80-902
90-1008


Now, we compute mean by the method of step deviation

Marks
Mid-value Frequency 
0-1053-5-15
10-20155-4-20
20-30257-3-21
30-403510-2-20
40-504512-1-20
50-60551500
60-706512112
70-80756212
80-9085236
90-100958432
Total          
       
= -26
 We have,
   = 


a= 55 and h= 10 


Mean =  a + h( fu / f   )

            =  55 +   10 ( -26 /  80)

             = 55  + 10 ( -0.325)

             = 55  -  3.25

            = 51.75

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