# Statistics - Continuous Series Arithmetic Mean

When data is given based on ranges alongwith their frequencies. Following is an example of continous series:

 Items Frequency 0-5 5-10 10-20 20-30 30-40 2 5 1 3 12

In case of continous series, a mid point is computed as $\frac{lower-limit + upper-limit}{2}$ and Arithmetic Mean is computed using following formula.

## Formula

$\bar{x} = \frac{f_1m_1 + f_2m_2 + f_3m_3........+ f_nm_n}{N}$

Where −

• ${N}$ = Number of observations.

• ${f_1,f_2,f_3,...,f_n}$ = Different values of frequency f.

• ${m_1,m_2,m_3,...,m_n}$ = Different values of mid points for ranges.

## Example

Problem Statement

Let's calculate Arithmetic Mean for the following continous data −

 Items Frequency 0-10 10-20 20-30 30-40 2 5 1 3

Solution

Based on the given data, we have −

Items Mid-pt
m
Frequency
f
${fm}$
0-10 5 2 10
10-20 15 5 75
20-30 25 1 25
30-40 35 3 105
${N=11}$ ${\sum fm=215}$

Based on the above mentioned formula, Arithmetic Mean $\bar{x}$ will be −

$\bar{x} = \frac{215}{11} \\[7pt] \, = {19.54}$

The Arithmetic Mean of the given numbers is 19.54.