1. Finding mean Direct and asssumed

2. Arithmetic mean for ungrouped data
The arithmetic mean is the ratio of the sum of all the observations to the total
number of observations.
Arithmetic mean for ungrouped data
To calculate arithmetic mean for ungrouped data we use the following formula.
In Mathematics, the symbol in Greek letter Σ , is called Sigma
This notation is used to represent the summation Σx
With this symbol, the sum of x1, x2, x3, …… xn or simply as
Then we have
Note: Arithmetic mean is also known as Average or Mean.

3. More about notation: Σ

4. Arithmetic mean for grouped data
Arithmetic mean for grouped data can be obtained in two methods
which are
Direct Method and
Assumed Mean Method

5. To calculate Arithmetic Mean (Direct Method)
Suppose we have the following frequency distribution
Then this table is to be interpreted in the following manner:
The value of x1 occurs f1 times
x2 occurs f2 times
x3 occurs f3 times
xn occurs fn times
Here x1, x2, x3, …… xn are the distinct values of the variable x.

6. In this case, the total number of observations(frequency) is usually denoted by N.
Then the total values observed
Usually, it is written as

7. Example 1:-Calculate the Arithmetic mean of the following data by direct method5 10 15 20 25 30 f 4 7 3 2
Solution:- x f
f x x = fx 5 4
20 (4x5) 10
50 (5x10) 15 7
105 20 80 25 3 75 30 2 60
Total
N=25
Σ fx =390 

8. To calculate Arithmetic Mean (Assumed Mean Method)
In the above example multiplication looks very simple, since the numbers are
small . When the numbers are huge, their multiplications are always tedious or
uninteresting and leads to errors. To overcome this difficulty another simpler
method is devised which is as follows
Step1:- we assume one of the values as mean (A). This assumed value A is
known as assumed mean.
Step2:- Then we calculate the deviation d1, d2, d3, …… dn
∴ d1 = x1 –A , d2 = x2 –A , d3 = x3 –A , …. , dn = xn –A

9. Step3:- Now, multiply d1, d2, d3, …… dn respectively by f1, f2, f3 and add all
these values to get

10. Example 2:-Calculate the Arithmetic mean of the following data by
assumed mean method x 5 10 15 20 25 30 f 4 7 3 2
Solution:-
Step1:- Take the assumed mean A = 15
Step2:- Then we calculate the deviation (d)
You can take any value as
assumed mean.
Take a value which will give
you smaller d value x f
d = x - A
f x d = fd 5 4
-10 (5 - 15)
-40 (4 x -10) 10
-5 ( )
-25 ( 5 x -5) 15 7 20 25 3 30 2
Total
N=25
Σ fd =15 

11. Step3:- Calculate mean value using the deviation
∵ A=15, ∑fd=15 and N=25
= (ans)

12. Weighted Arithmetic Mean (W.A.M.)
Sometimes the variables are associated with various weights and in those
cases the A.M. can be calculated, such an arithmetic mean is known as
For example, let us assume that the variable x1 is associated with the
weight w1, x2 is associated with the weight w2 etc. and finally, xn is
associated with the weight wn then

13. Example 3:-Find the weighted A. M of the price for the following data:
Food stuff
Quantity (in kg) wi
Price (per kg) in Rs. xi
Rice 25 30
Sugar 12
Oil 8 70
Solution
Here the x-values are the price of the given food stuff and the weights
associated are the quantities (in Kg)

14. Try These 1.Calculate the Arithmetic mean of the following data:
2.Calculate the Weighted Arithmetic mean of the following data: