UPSKILL MATH PLUS
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Learn moreThe formula for finding the arithmetic mean using the direct method is given by:
\(\overline X = \frac{\sum f_ix_i}{\sum f_i}\)
Where \(i\) varies from \(1\) to \(n\), \(x_i\) is the midpoint of the class interval and \(f_i\) is the frequency.
Steps:
1. Calculate the midpoint of the class interval and name it as \(x_i\).
2. Multiply the midpoints\(x_i\) with the frequency\(f_i\) of each class interval and name it as \(f_ix_i\).
3. Find the values \(\sum f_ix_i\) and \(\sum f_i\).
4. Divide \(\sum f_ix_i\) by \(\sum f_i\) to determine the mean of the data.
Example:
The following frequency distribution table shows that the number of trees based on the height in metres. Find the average height of the trees.
| Height (in \(m\)) | \(30 - 40\) | \(40 - 50\) | \(50 - 60\) | \(60 - 70\) | \(70 - 80\) |
| Number of trees | \(124\) | \(156\) | \(200\) | \(10\) | \(10\) |
Solution:
Let us form a frequency distribution table.
Height (in \(m\)) | Number of trees (\(f_i\)) | Midpoint (\(x_i\)) | \(f_ix_i\) |
| \(30 - 40\) | \(124\) | \(35\) | \(4340\) |
| \(40 - 50\) | \(156\) | \(45\) | \(7020\) |
| \(50 - 60\) | \(200\) | \(55\) | \(11000\) |
| \(60 - 70\) | \(10\) | \(65\) | \(650\) |
| \(70 - 80\) | \(10\) | \(75\) | \(750\) |
| Total | \(\sum f_i = 500\) | \(\sum f_ix_i = 23760\) |
Mean \(\overline X = \frac{23760}{500}\) \(= 47.52\)
Therefore, the average height of the trees is \(47.52\).
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