### Theory:

The mean of the ungrouped frequency distribution can be determined using the formula:
$$\overline X = \frac{f_1 x_1 + f_2 x_2 + ... + f_n x_n}{f_1 + f_2 + ... + f_n}$$ $$= \frac{\sum_{i=1}^{n} f_i x_i}{\sum_{i=1}^{n} f_i}$$ $$= \frac{\sum fx}{\sum f}$$
Example:
The marks scored by $$20$$ students in mathematics are $$100$$, $$65$$, $$89$$, $$55$$, $$40$$, $$96$$, $$65$$, $$35$$, $$35$$, $$100$$, $$100$$, $$100$$, $$55$$, $$65$$, $$96$$, $$89$$, $$77$$, $$63$$, $$77$$, $$30$$.

Solution:

In the previous sessions, we have learnt how to find the average using the arithmetic mean method and assumed mean method.

Now, we shall learn the $$3^{rd}$$ method, which is an ungrouped frequency distribution.

Here, in the data, we can find $$100$$ has occured $$4$$ times (i.e. frequency is $$4$$), $$65$$ has frequency of $$3$$ and so on. Then, the frequency distribution table looks like this:

 Marks$$x$$ Number of students$$f$$ $$30$$ $$1$$ $$35$$ $$2$$ $$40$$ $$1$$ $$55$$ $$2$$ $$63$$ $$1$$ $$65$$ $$3$$ $$77$$ $$2$$ $$89$$ $$2$$ $$96$$ $$2$$ $$100$$ $$4$$
We know the formula to find the mean of ungrouped frequency distribution is $$\overline X = \frac{\sum fx}{\sum f}$$
To find the value of $$fx$$, multiply the value of $$x$$ and $$f$$ of each entry.

Consider for the mark $$30$$. That is, $$30 \times 1 = 30$$

Similarly, for the mark $$35$$, we have $$35 \times 2 = 70$$ and so on.

Tabulating these values, we get:

 Marks$$x$$ Frequency$$f$$ $$fx$$ $$30$$ $$1$$ $$30$$ $$35$$ $$2$$ $$70$$ $$40$$ $$1$$ $$40$$ $$55$$ $$2$$ $$110$$ $$63$$ $$1$$ $$63$$ $$65$$ $$3$$ $$195$$ $$77$$ $$2$$ $$154$$ $$89$$ $$2$$ $$178$$ $$96$$ $$2$$ $$192$$ $$100$$ $$4$$ $$400$$ Total $$\sum f = 20$$ $$\sum fx = 1432$$

Substituting the known values in the above formula, we get:

Mean $$\overline X = \frac{1432}{20}$$ $$= 71.6$$

Therefore, the mean of the given data is $$71.6$$.
We can also find the mean of the ungrouped frequency distribution using the assumed mean method.
Example:
Consider the above example. Let the assumed mean be $$A = 65$$.

Then the new frequency distribution table is given by:

 Marks$$x$$ Deviation $$d = x - A$$ Frequency$$f$$ $$fd$$ $$30$$ $$30 - 65 = - 35$$ $$1$$ $$-35$$ $$35$$ $$35 - 65 = - 30$$ $$2$$ $$-60$$ $$40$$ $$40 - 65 = - 25$$ $$1$$ $$-25$$ $$55$$ $$55 - 65 = - 10$$ $$2$$ $$-20$$ $$63$$ $$63 - 65 = -2$$ $$1$$ $$-2$$ $$65$$ $$65 - 65 = 0$$ $$3$$ $$0$$ $$77$$ $$77 - 65 = 12$$ $$2$$ $$24$$ $$89$$ $$89 - 65 = 24$$ $$2$$ $$48$$ $$96$$ $$96 - 65 = 31$$ $$2$$ $$62$$ $$100$$ $$100 - 65 = 35$$ $$4$$ $$140$$ Total $$\sum f = 20$$ $$\sum fd = 132$$

Arithmetic mean $$=$$ Assumed mean $$+$$ Average of sum of deviations

Arithmetic mean $$= 65 + \frac{132}{20}$$ $$= 65 + 6.6$$ $$= 71.6$$

Therefore, the mean is $$71.6$$.