Theory:

The formula for finding the arithmetic mean using the direct method is given by:
 
\(\overline X = \frac{\sum fx}{\sum f}\)
 
Where \(x\) is the midpoint of the class interval and \(f\) is the frequency.
 
Steps:
 
1. Calculate the midpoint of the class interval and name it as \(x\).
 
2. Multiply the midpoints\(x\) with the frequency\(f\) of each class interval and name it as \(fx\).
 
3. Find the values \(\sum fx\) and \(\sum f\).
 
4. Divide \(\sum fx\) by \(\sum f\) 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
(\(f\))
Midpoint
(\(x\))
\(fx\)
\(30 - 40\)\(124\)\(35\)\(4340\)
\(40 - 50\)\(156\)\(45\)\(7020\)
\(50 - 60\)\(200\)\(55\)\(11000\)
\(60 - 70\)\(10\)\(65\)\(650\)
\(70 - 80\)\(10\)\(75\)\(750\)
Total\(\sum f = 500\) \(\sum fx = 23760\)
 
Mean \(\overline X = \frac{23760}{500}\) \(= 47.52\)
 
Therefore, the average height of the trees is \(47.52\).