Arithmetic Average or Mean

To calculate the mean of raw data, all the observations of the data are added and their sum is divided by the number of observations.

Mean of Raw Data

The mean of n observations is:

$$ \overline{x} = {\frac{x_{1}+ x_{2} + \cdots + x_{n}}{n}} $$

$$ \overline{x} = \frac {1}{n} \sum_{i=1}^{n} x_{i} $$

Example: The weight of four bags of wheat (in kg) are 103, 105, 102, 104.

Mean weight = (103 +105 +102 +104)/4

= 103.5 kg

Mean of Ungrouped Data

$$ \overline{x} = \frac{\sum f_{i} x_{i}}{\sum f_{i}} $$

To find mean of ungrouped data, first find f_i x_i, by multiplying each x_i with its corresponding frequency f_i and append a column of f_ix_i in the frequency table.

Short-cut Method

Sometimes when the numerical values of x_i and f_i are large, finding the product f_i and x_i becomes time consuming. Here, we choose an arbitrary constant a, also called the assumed mean and subtract it from each of the values x_i. The reduced value, d_i = x_i - a is called the deviation of x_i from a.

$$ \overline{x} = a + \frac {1}{N} \sum f_{i} d_{i} $$

This method of calculation of mean is known as Assumed Mean Method.

Mean of Grouped Data

To find mean of the grouped frequency distribution, we make the following assumption:

• Frequency in any class is centered at its class mark or mid point.

1. Direct Method

$$ \overline{x} = \frac{\sum f_{i} x_{i}}{\sum f_{i}} $$

Class marks (x_i) are the mid-points of the class intervals.

2. Assumed Mean Method

When a is the assumed mean, and d_i = x_i - a. 

$$ \overline{x} = a + \frac {1}{N} \sum f_{i} d_{i} $$

3. Step Deviation Method

When a is the assumed mean, u_i = (x_i - a)/h and h is the class size.

$$ \overline{x} = a + \frac{\sum f_{i} u_{i}}{\sum f_{i}} \times h $$

The mean comes out to be the same by using Direct Method, Assumed Method or Step Deviation Method.