Statistics - Arithmetic Mean of Continuous Data Series
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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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