# Mean ( Direct method)

Covid-19 has led the world to go through a phenomenal transition .

E-learning is the future today.

Stay Home , Stay Safe and keep learning!!!

Mean (or average) of observations, as we know, is the sum of the values of all the observations divided by the total number of observations.
If x
1 , x 2 ,… x n , are observations with respective frequencies f 1 , f 2 ,, . . ., f n then this means observation x 1 , occurs f 1 times, x 2 , occurs f 2 , times, and so on. Now, the sum of the values of all the observations = f 1 x 1 + f 2 x 2 + . . . + f n x n , and the number of observations = f 1 + f 2 + . . . + f n .

 f1 x1 + f2 x2 + . . . + fn xn x = ---------------------------------            f1 + f2 + . . . + fn

This can be written in short way using Greek letter Σ
 Σfi xix = ------ Σ fi

Example 1 :
 Weight (in Kgs) 67 70 72 73 75 Number of students 4 3 2 2 1

Find the average weight.
Solution :
 Weight (in kg)xi Frequency (fi) fixi 67 4 268 70 3 210 72 2 144 73 2 146 75 1 75 N=Σ fi = 12 Σ fixi = 843

 Σfi xi 843x = ------- = ------- = 70.25 kg Σ fi 12

________________________________________________________________
Example 2 :
If the average of the following data is 20.2, find the value of p:
 x 10 15 20 25 30 f 6 8 P 10 6
Solution :
 x f f x 10 6 60 15 8 120 20 P 20P 25 10 250 30 6 180 30 + P 610 + 20P

 Σfi xix = ------- Σ fi

20.2 = ( 610 + 20P) / ( 30 + P)
20.2 ( 30 + P ) = 610 + 20P
606 + 20.2P = 610 + 20P
20.2P – 20P = 610 – 606
0.2 P = 4
∴ P = 4 / 0.2
P = 20
________________________________________________________________
Example 3: Find the average of the following data :
 Marks 0 - 10 10 - 20 20 - 30 30 - 40 40 - 50 No.of students 4 10 18 12 6

Solution: First find the class mark,
Class Mark = ( upper class limit + Lower class limit ) /2
Example : Class Mark = ( 10 + 0 ) / 2 = 10 / 2 = 5
Prepare the frequency table :
 Class interval fi Class Mark (xi) fi xi 0 - 10 4 5 20 10 - 20 10 15 150 20 -30 18 25 450 30 - 40 12 35 420 40 - 50 6 45 270 Σ fi = 50 Σ fi xi = 1310

 Σfi xix = -------        Σ fi
Mean = (1310 / 50)
= 26.2

• Direct method.
Short cut method.
Step - Deviation method.

From median to measures of central tendency

Statistics