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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 \(x\).
2. From the data of \(x\), choose any value(preferably in the middle) as the assumed mean(\(A\)).
3. Determine the deviation (\(d = \frac{x - A}{c}\) where \(c\) is the class width) for each class.
4. Multiply the deviation and frequency of each class interval and name it as \(fd\).
5. Calculate the mean by applying the formula \(\overline X = A + \left[\frac{\sum fd}{\sum f} \times c \right]\).
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 \(c = 10\).
| Class interval | Frequency \(f\) | Midpoint \(x\) | deviation \(d = \frac{x - A}{c}\) | \(fd\) |
| \(30 - 40\) | \(124\) | \(35\) | \(-2\) | \(-248\) |
| \(40 - 50\) | \(156\) | \(45\) | \(-1\) | \(-156\) |
| \(50 - 60\) | \(200\) | \(55\) | \(0\) | \(0\) |
| \(60 - 70\) | \(10\) | \(65\) | \(1\) | \(10\) |
| \(70 - 80\) | \(10\) | \(75\) | \(2\) | \(20\) |
| Total | \(\sum f = 500\) | \(\sum fd = -374\) |
We know that the mean of the grouped frequency distribution using the step deviation method can be determined using the formula, \(\overline X = A + \left[\frac{\sum fd}{\sum f} \times c \right]\).
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 \(47.52\).
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