Arithmetic Mean Formula of Grouped Data Using Step Deviation Method | Solved Examples, Important Questions, FAQs

Arithmetic Mean of Grouped Data Using Step Deviation Method | Class 11 Notes

Written by Neeraj Anand | Published by ANAND TECHNICAL PUBLISHERS

Introduction:

The Step Deviation Method is an advanced technique for calculating the Arithmetic Mean of grouped data. It simplifies the process by using deviations from an assumed mean, divided by a common factor (class width). This method is particularly useful when data values are large or complex.

For Class 11 students preparing for Board Exams, JEE Mains, and Advanced, mastering the Step Deviation Method provides a faster and more efficient way to compute the arithmetic mean.

Advantages of the Step Deviation Method:

✔️ Reduces calculations by working with smaller numbers.
✔️ Ideal for large data sets with uniform class widths.
✔️ Useful in Board Exams and JEE Mains for quick and accurate calculations.

Derivation of Formula for Arithmetic Mean by Step Deviation Method

The general formula for mean in statistics is:

Mean = Σf_ix_i / Σf_i

Where,

Σf_ix_i: the weighted sum of elements and 

Σf_i: the number of elements

In the case of grouped data, assume that the frequency in each class is centered at its class-mark. If there are n classes and f_i denotes the frequency and x_i denotes the class-mark of the i_th class the mean is given by,

Mean = Σf_ix_i / Σf_i

When the number of classes is large or the value of f_i and x_i is large, an approximate (assumed) mean is taken near the middle, represented by A and deviation (d_i) is taken into consideration. Then mean is given by,

 Mean = A + Σf_id_i / Σf_i 

In the problems where the width of all classes is the same, then further simplify the calculations of the mean by computing the coded mean, i.e. the mean of u₁, u₂, u₃, …..u_n where,

u_i = (x_i – A) / h

Where h = Class Interval or Class Width

Then the mean is given by the formula,

Mean = A + h x (Σf_iu_i / Σf_i)

This method of finding the mean is called the Step Deviation Method.

Step Deviation Method

Consider the Example as given below.

Class Interval	10-25	25-40	40-55	55-70	70-85	85-100
Number of Students	2	3	7	6	6	6

In the step deviation method, we will add one more column to the table to find the mean, which is u_i = (x_i – a)/h

Where “a” is the assumed mean and “h” is the class size, which is equal to 15 (i.e) width of the class interval.

Class Interval	Number of students (fi)	Class Mark (xi)	di = xi – 47.5 di = xi – a	ui =(xi – a)/h (h=15)	fiui
10-25	2	17.5	-30	-2	-4
25-40	3	32.5	-15	-1	-3
40-55	7	47.5	0	0	0
55-70	6	62.5	15	1	6
70-85	6	77.5	30	2	12
85-100	6	92.5	45	3	18
Total	Σfi = 30				Σfiui = 29

Therefore, we obtained

\(\begin{array}{l}\bar{u}=\frac{\sum f_{i}u_{i}}{\sum f_{i}}\end{array} \)

The relation between \(\begin{array}{l}\bar{u}\end{array} \) and \(\begin{array}{l}\bar{x}\end{array} \) is:

\(\begin{array}{l}\bar{x}=a+h\frac{\sum f_{i}u_{i}}{\sum f_{i}}\end{array} \)

Now, substitute the values of a, h,Σf_i , and Σf_iui in the above formula to get the mean,

x̄ = 47.5 + 15(29/30)

x̄ = 47.5 + 15(0.967)

x̄= 47.5+ 14.5

x̄ = 62

Hence, the mean of the marks scored by the students = 62.

Difference between Direct, Assumed Mean and Step Deviation Method

Direct Method, Assumed Mean Method, and Step Deviation Method are three different techniques for calculating the mean (average) of a dataset. Each method has its own way of simplifying the calculations. Some of the common differences among these methods are:

Direct Method	Assumed Mean Method	Step Deviation Method
Best method for simple problems and small data set.	Best method for large data sets with potentially large values.	Best method for large data sets with uniform class intervals.
This method is straightforward and easy to understand.	It simplifies large arithmetic calculations using the assumed mean.	Further simplifies calculations and arithmetic complexity by working with smaller numbers.
In this method we calculate midpoints(xi) of each class intervals and multiply them with corresponding frequency(ƒ), i.e., ƒixi .	In this method we calculate deviation(di) of each data point and multiply them with corresponding frequency(ƒ), i.e., ƒidi .	In this method we calculate step deviation(ui) of each class intervals and multiply them with corresponding frequency(ƒ), i.e., ƒiui .
The formula of this method isMean = ∑ƒixi / ∑ƒi	Formula for assumed mean method isMean = a + ∑ ƒidi / ∑ ƒi	Formula for step deviation method isMean = a + h × ∑ ƒiui / ∑ ƒi

---

Solved Examples For Arithmetic Mean of Grouped Data Using Step Deviation Method

Question 1: Find the mean for the following frequency distribution?

Class Intervals	84-90	90-96	96-102	102-108	108-114
Frequency	8	12	15	10	5

Solution:

Applying the Standard Deviation Method,

We take the assumed mean to A = 99, and here the width of each class(h) = 6

Classes	Class-mark(xi)	ui = (xi – A) / h	frequency(fi)	fiui
84-90	87	-2	8	-16
90-96	93	-1	12	-12
96-102	99	0	15	0
102-108	105	1	10	10
108-114	111	2	5	10
Total			50	-8

Mean = A + h x (Σf_iu_i / Σf_i)

          = 99 + 6 x (-8/50)

          = 99 – 0.96

          = 98.04

Question 2: Find the mean for the following frequency distribution?

Class Intervals	20-30	30-40	40-50	50-60	60-70	70-80
Frequency	10	6	8	12	5	9

Solution:

Applying the Standard Deviation Method,

Construct the table as under, taking assumed mean A = 45, and width of each class(h) = 10.

Classes	Class-mark(xi)	ui = (xi – A) / h	frequency(fi)	fiui
20-30	25	-2	10	-20
30-40	35	-1	6	-6
40-50	45	0	8	0
50-60	55	1	12	12
60-70	65	2	5	10
70-80	75	3	9	27
Total			50	23

Mean = A + h x (Σf_iu_i / Σf_i)

          = 45 + 10 x (23/50)

          = 45 + 4.6

          = 49.6

Question 3: The weight of 50 apples was recorded as given below

Weight in grams	80-85	85-90	90-95	95-100	100-105	105-110	110-115
Number of apples	5	8	10	12	8	4	3

Calculate the mean weight, to the nearest gram?

Solution:

Construct the table as under, taking assumed mean A = 97.5. Here width of each class(h) = 5

Classes	Class-mark(xi)	ui = (yi – A) / h	frequency(fi)	fiui
80-85	82.5	-3	5	-15
85-90	87.5	-2	8	-16
90-95	92.5	-1	10	-10
95-100	97.5	0	12	0
100-105	102.5	1	8	8
105-110	107.5	2	4	8
110-115	112.5	3	3	9
Total			50	-16

Mean = A + h x (Σf_iu_i / Σf_i)

          = 97.5 + 5 x (-16/50)

          = 97.5 – 1.6

          = 95.9

Hence the mean weight to the nearest gram is 96 grams.

Question 4: The following table gives marks scored by students in an examination:

Marks	0-5	5-10	10-15	15-20	20-25	25-30	30-35	35-40
Number of students	3	7	15	24	16	8	5	2

Calculate the mean marks correct to 2 decimal places?

Solution:

Construct the table as under, taking assumed mean A = 17.5. Here width of each class(h) = 5

Classes	Class-mark(xi)	ui = (xi – A) / h	frequency(fi)	fiui
0-5	2.5	-3	3	-9
5-10	7.5	-2	7	-14
10-15	12.5	-1	15	-15
15-20	17.5	0	24	0
20-25	22.5	1	16	16
25-30	27.5	2	8	16
30-35	32.5	3	5	15
35-40	37.5	4	2	8
Total			80	17

Mean = A + h x (Σf_iu_i / Σf_i)

          = 17.5 + 5 x (17/80)

          = 17.5 + 1.06

          = 18.56

---

Practice Problems – Step Deviation Method

Problem.1 : Find the mean for the following frequency distribution:

Class-Interval	Frequency
10-20	3
20-30	5
30-40	8
40-50	4
50-60	2

Problem.2. Calculate the mean for the given frequency distribution:

Class Interval	Frequency
5-15	7
15-25	10
25-35	15
35-45	6
45-55	2

Problem.3. Calculate the mean for the given frequency distribution:

Class Interval	Frequency
1-5	6
5-9	8
9-13	10
13-17	4
17-21	5

Problem.4. Determine the mean for this frequency distribution:

Class Interval	Frequency
2-6	3
6-10	6
10-14	9
14-18	7
18-22	5

Problem.5. Find the mean for the given frequency distribution:

Class Interval	Frequency
3-8	4
8-13	10
13-18	6
18-23	8
23-28	2

Problem.6. Calculate the mean for the following frequency distribution:

Class Interval	Frequency
0-4	5
4-8	8
8-12	10
12-16	7
16-20	3

---

Frequently Asked Questions (FAQs) on Mean of Grouped Data

What are the three methods used to find the mean of grouped data?

The three methods used to find the mean of the grouped data are:
Direct method
Assumed mean method
Step deviation method

What is meant by class-mark?

The classmark is also called the midpoint of the class intervals, which can be found by taking the average of its upper-class limit and lower-class limit.
Class Mark = (upper class limit + lower class limit)/2

Can we get the same mean value for both the grouped and ungrouped data?

The mean value of grouped data slightly differs from the ungrouped data because of the midpoint assumption.

What does “h” mean in the step-deviation method?

In the step-deviation method, “h” represents class size.

How to choose the value of “a” in the assumed mean method?

In the assumed mean method, the value of “a” can be chosen which lies in the centre of x₁, x₂, . . ., x_n.

What is the step deviation method for finding the mean?

The step deviation method is a simplified approach for calculating the mean of a frequency distribution when the data is large. It involves using assumed mean and step deviations (class intervals) to reduce the arithmetic calculations.

How do you select the assumed mean (A) in the step deviation method?

The assumed mean (A) is usually chosen from the class midpoints of the distribution. It can be any midpoint, but usually the midpoint of the central class interval is selected to simplify calculations.

What are the step deviations and how are they calculated?

Step deviations d_i are calculated using the formula

d_i = (x_i – A) / h

, where x_i is the class midpoint, A is the assumed mean, and h is the class width.

How do you choose the class width (h) in the step deviation method?

The class width (h) is typically chosen based on the uniform width of the class intervals in the frequency distribution.

What are the advantages of using the step deviation method?

The step deviation method simplifies calculations, especially when dealing with large data sets or data with large values. It reduces the magnitude of the numbers involved, making arithmetic operations easier and less error-prone.

---

📥 Download PDF Notes:

Get Class 11 Statistics Notes on Arithmetic Mean using Step Deviation Method from ANAND CLASSES for Board and JEE exam preparation.