### Theory:

The 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 $$47.52$$.