Statistics - Continuous Series Arithmetic Mean

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

Items	0-5	5-10	10-20	20-30	30-40
Frequency	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	0-10	10-20	20-30	30-40
Frequency	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.

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