Arithmetic mean is the average measure of data distributions which is generally known as the mean in statistics. Arithmetic mean is well defined and easy to compute. It includes all the data values in its computation and can be algebraically treated. One demerit of arithmetic mean is that it is affected by outliers or extreme values in the data set.

Arithmetic mean of two or more quantities is the sum of all the quantities divided by the number of values added in the sum.The general symbol used to denoted arithmetic mean is x. The same notation is used to represent the sample mean while the population mean is denoted by the Greek letter μ.
The arithmetic mean of n observations x1, x2,......xn is given by the formula
x = $\frac{x_{1}+x_{2}+....+x_{n}}{n}$ and written using Sigma notation as $\frac{\sum_{i=1}^{n}x_{i}}{n}$

If these n observations have corresponding frequencies, the arithmetic mean is computed using the formula
x = $\frac{x_{1}f_{1}+x_{2}f_{2}+......+x_{n}f_{n}}{N}$ and using Sigma notation = $\frac{\sum_{i=1}^{n}x_{i}f_{i}}{N}$ where N = f1 + f2 + ..........+ fn.

The above formula can also be used to find the weighted arithmetic mean by taking f1, f2.....fn as the weights of x1, x2...... xn. When the frequencies divided by N are replaced by probabilities p1, p2, ......pn we get the formula for the expected value of a discrete random variable.
X = x1p1 + x2p2 +.......+ xnpn. or using Sigma notation = $\sum_{i=1}^{n}x_{i}p_{i}$
1. When each item in the data set is increased by k, then the arithmetic mean is also increased by k.

2. The algebraic sum of deviations of all the items from the arithmetic mean is zero. That means if x is the arithmetic mean of n observations x1, x2,......xn then (x1 - x) + (x2 - x) +.......(xn - x) = 0. This property can be expressed using Sigma Notation as $\sum_{i=1}^{n}(x_{i}-\overline{x})$ = 0

3. If each of the observations is multiplied by a constant k, then the AM also gets multiplied by k.

Solved Examples

Question 1: Find the arithmetic mean of the following set of observations: 25, 32, 24, 38, 22, 25, 30, 34, 26, 30
Solution:
 
There are 10 observations given here. n =10

    Arithmetic mean of the 10 observations given = $\frac{\sum x}{n}$

                                                                            = $\frac{25+32+24+38+22+25+30+34+26+30}{10}$

                                                                            = $\frac{286}{10}$ = 28.6
 

Question 2: The frequency distribution of the number of children participated under each age in a local Sports event is given below. Find the arithmetic mean of the distribution. What does AM represent here?
Solution:
 
Age in years
         x

  Number of
Participants f

        8
      15
        9
      20
       10
      25
        11
      22
        12
      18
       Total      100

To calculate the arithmetic mean of data given in frequency distribution we add one more column to the table given for fx.
    
Age in years
         x

  Number of
Participants f

       fx   
        8
      15
     120
        9
      20
     180
       10
      25
     250
       11
      22
     242
       12
      18
     216
       Total     N = 100
 ∑fx = 1008

Arithmetic Mean = $\frac{\sum fx}{N}$ = $\frac{1008}{100}$ = 10.08
The Arithmetic mean represents here the average age of participants in the sports event.