Step Deviation Method for Mean of Grouped Data
Grade 10 Mathematics - Period 13.4
Recall Previous Methods
Direct Method: \(\bar{x} = \frac{\sum f_i x_i}{\sum f_i}\)
Assumed Mean Method: \(\bar{x} = a + \frac{\sum f_i d_i}{\sum f_i}\) where \(d_i = x_i - a\)
Introduce Step Deviation Method
\(\bar{x} = a + \frac{\sum f_i u_i}{\sum f_i} \times h\)
Where:
\(a\) = assumed mean
\(h\) = class size
\(u_i = \frac{x_i - a}{h}\)
\(f_i\) = frequency
When to Use Step Deviation
- When \(x_i\) and \(d_i\) values are large
- To simplify calculations, especially with large numbers
Example: Daily Expenditure on Food
| Daily expenditure (in ₹) | 100-150 | 150-200 | 200-250 | 250-300 | 300-350 |
|---|---|---|---|---|---|
| Number of households | 4 | 5 | 12 | 2 | 2 |
Solution:
| Class Interval | Frequency (\(f_i\)) | Midpoint (\(x_i\)) | \(u_i = \frac{x_i - a}{h}\) | \(f_i u_i\) |
|---|---|---|---|---|
| 100-150 | 4 | 125 | -2 | -8 |
| 150-200 | 5 | 175 | -1 | -5 |
| 200-250 | 12 | 225 (a) | 0 | 0 |
| 250-300 | 2 | 275 | 1 | 2 |
| 300-350 | 2 | 325 | 2 | 4 |
| Total | \(\sum f_i = 25\) | \(\sum f_i u_i = -7\) |
\(\bar{x} = a + \frac{\sum f_i u_i}{\sum f_i} \times h = 225 + \frac{-7}{25} \times 50 = 225 + (-14) = 211\)
∴ The mean daily expenditure of a household is ₹ 211.
Practice Problem
| Literacy rate (in %) | 45-55 | 55-65 | 65-75 | 75-85 | 85-95 |
|---|---|---|---|---|---|
| Number of cities | 3 | 10 | 11 | 8 | 3 |
Find the mean literacy rate using the step deviation method.