Arithmetic Mean | Definition | Examples - Statistics Part 1

 

Arithmetic  Mean  (A.M )

Let us first take the case of ungrouped raw data. Let x₁, x₂, ............x_n be the values of a variable, x. Then the simple arithmetic mean is defined as the sum of all the values divided by the number of values of the variable. If denotes the simple arithmetic mean, then, by definition, Arithmetic Mean = (x₁ + x₂ + .......x_n)/n .

If x is a discrete variable assuming values x_1,,x₂, ........x_k with frequencies f₁,f₂ .......f_k the arithmetic mean is defined as x̄ = ∑f_ix_i this is so because in the above definition it has been implicitly assumed that all the values of a variable a class interval are equal to the mid-value of that class and thus the, formula only gives an approximate value of the arithmetic mean. Some error will, remain in the calculated value of mean obtained from grouped data. This error is known as grouping error. This error will be negligible if the width of the class interval be compared to the total range of the variable values.

                           

 Example:

 

Weekly wages	Frequency
20-30	10
30-40	8
40-50	6
50-60	4
60-70	2

 

 Solution:

Weekly  wages	Frequency (fi)	Mid Value (xi)
20-30	10	25
30-40	8	35
40-50	6	45
50-60	4	55
60-70	2	65
Total	∑fi = 11

 

Now, by definition, A.M, ∑fixi/∑fi  = 1150/11 = Rs. 38.33

This shows that the average weekly wages is Rs. 38.33