### Theory:

In most situations, we usually consider a very large amount of data(like population census) for a purposeful study. In such cases, we may find it difficult to write the data in ungrouped data.

Hence, to simplify our work, we need to convert the given ungrouped data into grouped data.

Let us convert the given height(in $$cm$$) of the students in the classroom into grouped frequency data.

$$100$$, $$165$$, $$189$$, $$155$$, $$140$$, $$196$$, $$165$$, $$135$$, $$135$$, $$100$$, $$100$$, $$100$$, $$155$$, $$165$$, $$196$$, $$189$$, $$177$$, $$163$$, $$177$$, $$130$$.

Consider the frequency distribution table.

 Height (in cm) $$100 - 120$$ $$120 - 140$$ $$140 - 160$$ $$160 - 180$$ $$180 - 200$$ Students $$4$$ $$3$$ $$3$$ $$6$$ $$4$$

The above frequency table shows that the data are grouped in class intervals.

Consider the interval $$140 - 160$$. There are $$3$$ students in the heights between $$140 - 160$$ metres. In grouped frequency, 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 $$140 - 160$$. Let us find the midpoint of this interval.

Here, $$UCL = 140$$ and $$LCL = 160$$

Midpoint of $$140 - 160$$ is $$\frac{140 + 160}{2} =$$ $$\frac{300}{2}$$ $$= 150$$

Therefore, the midpoint of the interval $$140 - 160$$ is $$150$$.
The mean of a grouped frequency distribution can be determined using any one of the following methods.
• Direct method
• Assumed mean method
• Step deviation method