Statistics- Class- 10: Step Deviation Method-Learning Objective (LO-4): Students will be able to find the mean of grouped data using the step deviation method.

Step Deviation Method for Mean of Grouped Data

Grade 10 Mathematics (AP/TS/NCERT) - Period 13.4

Lesson Overview

Learning Objective (LO-4): Students will be able to find the mean of grouped data using the step deviation method.

Lesson Structure:

• Introduction & Hook Activity (5 minutes)
• Explicit Teaching/Teacher Modeling (15 minutes)
• Group Work (10 minutes)
• Independent Practice (10 minutes)
• Closing & Assessment (5 minutes)

1. Introduction & Hook Activity

5 minutes

Real-world Scenario

"Imagine we want to calculate the average daily allowance of students in our class. How would we do it if we had the data grouped in ranges?"

"We've learned about direct and assumed mean methods. Today we'll learn an even more efficient method."

Think-Pair-Share

Think about this question for 1 minute, then discuss with a partner for 2 minutes:

"Why might we need a different method when dealing with large numbers?"

• Large numbers make calculations more difficult
• Step deviation simplifies the process
• It reduces the chance of calculation errors

Vocabulary Review:

• Mean - The average of a dataset
• Frequency - How often a value occurs
• Class interval - A range of values
• Class mark (midpoint) - The center value of a class interval
• Assumed mean - A guessed average that simplifies calculations
• Deviation - Difference between a value and the assumed mean

2. Explicit Teaching/Teacher Modeling (I Do)

15 minutes

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 1: 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

Step-by-step solution:

1 Find midpoints (\(x_i\)): 125, 175, 225, 275, 325

2 Choose assumed mean (\(a\)): 225 (middle value)

3 Determine class size (\(h\)): 50

4 Compute \(u_i = \frac{x_i - a}{h}\):

• \(\frac{125-225}{50} = \frac{-100}{50} = -2\)
• \(\frac{175-225}{50} = \frac{-50}{50} = -1\)
• \(\frac{225-225}{50} = \frac{0}{50} = 0\)
• \(\frac{275-225}{50} = \frac{50}{50} = 1\)
• \(\frac{325-225}{50} = \frac{100}{50} = 2\)

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	0	0
250-300	2	275	1	2
300-350	2	325	2	4
Total	\(\sum f_i = 25\)			\(\sum f_i u_i = -7\)

Apply formula:

\(\bar{x} = a + \frac{\sum f_i u_i}{\sum f_i} \times h = 225 + \frac{-7}{25} \times 50 = 225 + (-14) = 211\)

∴ The mean daily expenditure of a household is ₹ 211.

Experience and Reflections

Teacher will think aloud during the modeling process:

• "I'm choosing 225 as the assumed mean because it's the middle value."
• "Notice how the u_i values become simpler integers instead of large numbers."
• "This method reduces calculation errors when dealing with large values."

3. Group Work (We Do)

10 minutes

Divide into groups of 4. Work together to solve this problem:

Example 2: Literacy Rate of Cities

Literacy rate (in %)	45-55	55-65	65-75	75-85	85-95
Number of cities	3	10	11	8	3

Guide questions:

1. What are the midpoints of each class?
2. What assumed mean would be appropriate?
3. What is the class size?
4. Calculate the \(u_i\) values for each class.
5. Complete the table with \(f_i u_i\) values.
6. Apply the step deviation formula to find the mean literacy rate.

Group Roles

• Facilitator: Keeps the group on task
• Recorder: Writes down the group's solutions
• Calculator: Performs calculations
• Reporter: Shares the group's findings

Mean literacy rate ≈ 69.43%

4. Independent Practice (You Do)

10 minutes

Solve this problem individually:

Practice Problem: Daily Wages of Workers

Daily wages (in ₹)	200-300	300-400	400-500	500-600	600-700
Number of workers	5	8	12	7	3

Create a complete solution table and find the mean daily wage using the step deviation method.

Mean daily wage ≈ ₹435.71

Quick Check

1. What does the "midpoint" of a class interval represent?

The average of the upper and lower limits

The difference between class limits

The frequency of the class

The size of the class interval

2. Why do we use midpoints when calculating the mean of grouped data?

Because they are easier to calculate

Because we don't have individual values

Because they are always integers

Because textbooks require it

3. What is the formula for the mean of grouped data using step deviation?

\(\bar{x} = \frac{\sum f_i x_i}{\sum f_i}\)

\(\bar{x} = a + \frac{\sum f_i d_i}{\sum f_i}\)

\(\bar{x} = a + \frac{\sum f_i u_i}{\sum f_i} \times h\)

\(\bar{x} = \frac{\sum x_i}{n}\)

5. Closing & Assessment

5 minutes

Summary of Key Points:

• Step deviation method simplifies calculations with large numbers
• Formula: \(\bar{x} = a + \frac{\sum f_i u_i}{\sum f_i} \times h\)
• Important to choose appropriate assumed mean (usually middle value)
• Reduces calculation errors when dealing with large values

Homework:

Problems 6 to 9 in Exercise 13.1 from your textbook

Create one original problem that can be solved using the step deviation method

Exit Ticket

On a piece of paper, answer:

"What is one advantage of the step deviation method over the assumed mean method?"

Submit your answer before leaving class.