Period 13.4: Mean of Grouped Data
Step Deviation Method | Grade 10 AP/TS/NCERT
Lesson Overview
Learning Objective (LO-4)
Students will be able to find the mean of grouped data using the step deviation method
Duration
45 minutes
Grade Level
10 AP/TS/NCERT
Key Formula
Key Vocabulary
Phase 1: Explicit Teaching/Teacher Modelling (I Do)
Teacher-Led Example: Daily Wages of Workers
Consider the following distribution of daily wages of 50 workers of a factory.
| Daily wages (in ₹) | 500 - 520 | 520 - 540 | 540 - 560 | 560 - 580 | 580 - 600 |
|---|---|---|---|---|---|
| Number of workers | 12 | 14 | 8 | 6 | 10 |
Step-by-Step Solution:
| Class Interval | Frequency (fi) | Class Mark (xi) | ui = (xi - A)/h | fiui |
|---|---|---|---|---|
| 500 - 520 | 12 | 510 | -2 | -24 |
| 520 - 540 | 14 | 530 | -1 | -14 |
| 540 - 560 | 8 | 550 (A) | 0 | 0 |
| 560 - 580 | 6 | 570 | 1 | 6 |
| 580 - 600 | 10 | 590 | 2 | 20 |
| Total | 50 | - | - | -12 |
Given: A = 550, h = 20, Σfi = 50, Σfiui = -12
Interpretation: The mean daily wage of workers is ₹545.20
Phase 2: Group Work (We Do)
Group Activity: Daily Expenditure on Food
The table below shows the daily expenditure on food of 25 households in a locality.
| Daily expenditure (in ₹) | 100 - 150 | 150 - 200 | 200 - 250 | 250 - 300 | 300 - 350 |
|---|---|---|---|---|---|
| Number of households | 4 | 5 | 12 | 2 | 2 |
Group Task: Work in groups of 4 to find the mean daily expenditure on food using the step deviation method.
Guiding Questions:
- What should be the class size (h)?
- Which class mark would make a good assumed mean (A)?
- How will you calculate ui for each class?
- What is the formula for finding the mean?
| Class Interval | Frequency (fi) | Class Mark (xi) | ui = (xi - A)/h | fiui |
|---|---|---|---|---|
| 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 | 25 | - | - | -7 |
Given: A = 225, h = 50, Σfi = 25, Σfiui = -7
Interpretation: The mean daily expenditure on food is ₹211
Phase 3: Independent Work (You Do)
Individual Practice: Heartbeats of Women
Thirty women were examined in a hospital by a doctor and the number of heartbeats per minute were recorded and summarised as follows. Find the mean heartbeats per minute for these women.
| No. of heartbeats per minute | 65-68 | 68-71 | 71-74 | 74-77 | 77-80 | 80-83 | 83-86 |
|---|---|---|---|---|---|---|---|
| Number of women | 2 | 4 | 3 | 8 | 7 | 4 | 2 |
Instructions:
- Work individually to solve this problem
- Choose your own assumed mean
- Show all calculation steps clearly
- Check your answer for reasonableness
| Class Interval | Frequency (fi) | Class Mark (xi) | ui = (xi - A)/h | fiui |
|---|---|---|---|---|
| 65-68 | 2 | 66.5 | -3 | -6 |
| 68-71 | 4 | 69.5 | -2 | -8 |
| 71-74 | 3 | 72.5 | -1 | -3 |
| 74-77 | 8 | 75.5 (A) | 0 | 0 |
| 77-80 | 7 | 78.5 | 1 | 7 |
| 80-83 | 4 | 81.5 | 2 | 8 |
| 83-86 | 2 | 84.5 | 3 | 6 |
| Total | 30 | - | - | 4 |
Given: A = 75.5, h = 3, Σfi = 30, Σfiui = 4
Interpretation: The mean heartbeats per minute is 75.9
Assessment & Check for Understanding
Challenge Problem: Find Missing Frequency
The following distribution shows the daily pocket allowance of children of a locality. The mean pocket allowance is Rs 18. Find the missing frequency \( f \).
| Daily pocket allowance (in ₹) | 11 - 13 | 13 - 15 | 15 - 17 | 17 - 19 | 19 - 21 | 21 - 23 | 23 - 25 |
|---|---|---|---|---|---|---|---|
| Number of children | 7 | 6 | 9 | 13 | \( f \) | 5 | 4 |
| Class Interval | Frequency (fi) | Class Mark (xi) | ui = (xi - A)/h | fiui |
|---|---|---|---|---|
| 11 - 13 | 7 | 12 | -3 | -21 |
| 13 - 15 | 6 | 14 | -2 | -12 |
| 15 - 17 | 9 | 16 | -1 | -9 |
| 17 - 19 | 13 | 18 (A) | 0 | 0 |
| 19 - 21 | f | 20 | 1 | f |
| 21 - 23 | 5 | 22 | 2 | 10 |
| 23 - 25 | 4 | 24 | 3 | 12 |
| Total | 44 + f | - | - | -20 + f |
Given: A = 18, h = 2, Mean = 18, Σfi = 44 + f, Σfiui = -20 + f
Solution: The missing frequency \( f \) is 20
Lesson Summary
- Step deviation method simplifies calculation when dealing with large numbers
- The method uses normalized deviations (ui) instead of actual deviations
- Choose assumed mean (A) as a convenient class mark, preferably in the middle
- Class size (h) is the width of each class interval
- The formula transforms back to the original scale by multiplying by h and adding A