### Theory:

Consider the frequency distribution table.

 Height (in m) $$30 - 40$$ $$40 - 50$$ $$50 - 60$$ $$60 - 70$$ $$70 - 80$$ Number of trees $$124$$ $$156$$ $$200$$ $$10$$ $$10$$

The above frequency table shows that the data are grouped in class intervals. This table shows that the number of trees with various heights.

Consider the interval $$50 - 60$$. There are $$200$$ trees in the heights between $$50 - 60$$ metres. In grouped frequency, the individual observations are not available. Thus, we need to determine the value that indicates the particular interval. This value is called a midpoint or class mark. The midpoint can be determined using the formula:

Midpoint $$= \frac{UCL + LCL}{2}$$

Where $$UCL$$ is the upper class limit and $$LCL$$ is the lower class limit.
Example:
Consider the interval $$40 - 50$$. Let us find the midpoint of this interval.

Here, $$UCL = 40$$ and $$LCL = 50$$

Midpoint of $$40 - 50$$ is $$\frac{40 + 50}{2} =$$ $$\frac{90}{2}$$ $$= 45$$

Therefore, the midpoint of the interval $$40 - 50$$ is $$45$$.
The arithmetic mean of a grouped frequency distribution can be determined using any one of the following methods.
• Direct method
• Assumed mean method
• Step deviation method