### Theory:

Let us consider the steps for finding the mean of grouped data using the step deviation method.

Steps:

1. Calculate the midpoint of the class interval and name it as $$x_i$$.

2. From the data of $$x_i$$, choose any value(preferably in the middle) as the assumed mean($$a$$).

3. Determine the deviation ($$d = x_i - a$$) for each class.

4. Determine the deviation ($$u = \frac{x_i - a}{h}$$ where $$h$$ is the class size) for each class.

5. Multiply the frequency and $$u_i$$ of each class interval and name it as $$f_iu_i$$.

6. Calculate the mean by applying the formula $$\overline X = a + \left[\frac{\sum fd}{\sum f} \times h \right]$$.
Example:
Find the mean of the following frequency distribution:

 Class interval $$30 - 40$$ $$40 - 50$$ $$50 - 60$$ $$60 - 70$$ $$70 - 80$$ Frequency $$124$$ $$156$$ $$200$$ $$10$$ $$10$$

Solution:

Let the assumed mean be $$a = 55$$ and class width $$h = 10$$.

 Class interval Frequency$$f_i$$ Midpoint$$x_i$$ Deviation$$d_i = x_i - 55$$ $$u_i = \frac{x - a}{h}$$ $$f_iu_i$$ $$30 - 40$$ $$124$$ $$35$$ $$-20$$ $$-2$$ $$-248$$ $$40 - 50$$ $$156$$ $$45$$ $$-10$$ $$-1$$ $$-156$$ $$50 - 60$$ $$200$$ $$55$$ $$0$$ $$0$$ $$0$$ $$60 - 70$$ $$10$$ $$65$$ $$10$$ $$1$$ $$10$$ $$70 - 80$$ $$10$$ $$75$$ $$20$$ $$2$$ $$20$$ Total $$\sum f_i = 500$$ $$\sum f_iu_i = -374$$
We know that the mean of the grouped frequency distribution using the step deviation method can be determined using the formula, $$\overline X = a + \left[\frac{\sum f_iu_i}{\sum f_i} \times h \right]$$.
Substituting the known values in the above formula, we have:

$$\overline X = 55 + \left[\frac{-374}{500} \times 10 \right]$$

$$\overline X = 55 - 7.48$$

$$\overline X = 47.52$$

Therefore, the mean of the given data is $$47.52$$.