Mean formula

                                                                    

                             

      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

$\begin{array}{l}=\frac{1}{�}\sum _{�=1}^{�}{�}_{�}=\frac{{�}_1+{�}_2+{�}_3+\dots .+{�}_{�}}{�}\end{array}$

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

computed using the formula

$\begin{array}{l}�=\frac{{�}_1{�}_1+{�}_2{�}_2+\dots \dots +{�}_{�}{�}_{�}}{�}\end{array}$

$\begin{array}{l}\text{using Sigma notation}=\frac{\sum _{�=1}^{�}}{}\end{array}$1n∑i=1nai

$\frac{1}{n}\sum _{i=}^n{a}_{\mathrm{i\; = }}$ 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   

$\frac{\mathrm{ \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; 10<br>}}{\mathrm{<br>}}$

                                        = 12910$\frac{\mathrm{129/}}{10}$

                                        =  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	$0-10$	$10-20$	$20-30$	$30-40$	$40-50$	$50-60$
Frequency	8	5	1	3	2	1

solution

class	Mid value x (xi)	$f$	x
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		100	3945

Mean$(\bar{x})=\frac{\sum fx}{\sum f}$
                $=\frac{3945}{100}$
                $=39.45$

(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${}_i$ = x${}_i$ - a. 

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

               

Mean  =    a + $\frac{\Sigma f_i{d}_i}{\Sigma f_i}$

          where  ,  a =  assumed number

                

                        d = x - a

                        Σf = Total frequency

                  

    

Examples:

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

Class Interval	0−10	10−20	20−30	30−40	40−50
Frequency	7	8	1	1	$1$

Solution:

Class interval	Frequency ($f_i$)	$x_i=\frac{\text{lower limit + upper limit }}{2}$	$f_i{x}_i$
$0-10$	$7$	5	35
$10-20$	8	15	120
$20-30$	12	25	300
$30-40$	13	35	455
$40-50$	10	45	450
	$\Sigma f_i=50$		​ ​

x	f                       ​	${d\; =x-\; a}^{}$	$f_{}{d}^{}$
5	$7$	-20	-140
15	8	-10	-80
$25\; =\; a$	12	0	0
35	13	10	130
$45$	10	20	200
	$\Sigma f_i=50$		Σf ​d ​=110

Mean =    a + Σfi​Σfi​di​​=   25  +50110​=27.2

    

(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  ( $\frac{\Sigma f_i{u}_i}{\Sigma f_i}$

 

                         where , h = class width

                            

                                        a= assumed number                                                                                                                                         

Example:

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

Marks Number of students o and above 80 10 and above 77 20 and above 72 30 and above 65 40 and above 55 50 and above 43 60 and above 28 70 and above 16 80 and above 10 90 and above 8 100 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
0-10	3
10-20	5
20-30	7
30-40	10
40-50	12
50-60	15
60-70	12
70-80	6
80-90	2
90-100	8

Now, we compute mean by the method of step deviation

Marks (xi​)	Mid-value (xi​)	Frequency	ui=​10xi​−55​	fi​ui​
0-10	5	3	-5	-15
10-20	15	5	-4	-20
20-30	25	7	-3	-21
30-40	35	10	-2	-20
40-50	45	12	-1	-20
50-60	55	15	0	0
60-70	65	12	1	12
70-80	75	6	2	12
80-90	85	2	3	6
90-100	95	8	4	32
Total	∑fi​=80			∑fi​ui​= -26

 We have,
   $\sum f_i$= 

80,

∑fi​ui​=−26, 

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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