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 \(\overline X = a + \left[\frac{\sum fu}{\sum f} \times h \right]\).
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 \(47.52\).