UPSKILL MATH PLUS

Learn Mathematics through our AI based learning portal with the support of our Academic Experts!

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 $$47.52$$.