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Step deviation Method for Finding the Mean with Examples

  • Difficulty Level : Medium
  • Last Updated : 17 Nov, 2020

Statistics is a discipline of mathematics that uses quantified models and representations to gather, review, analyze, and draw conclusions from data. The most commonly used statistical measures are mean, median, and mode. Variance and standard deviation are measures of dispersion in statistics and various measures of concentration including quartiles, quintiles, deciles, and percentiles. Statistics is a way more beyond the topics mentioned, but here we stop for the “Mean” by Step Deviation method. In general, there are 3 types of mean:

  1. Arithmetic mean
  2. Geometric mean
  3. Harmonic mean

This article is about the Arithmetic mean by Step Deviation method. The arithmetic mean, also called the average or average value is the quantity obtained by summing two or more numbers or variables and then dividing by the number of numbers or variables. The arithmetic mean is important in statistics. For example, Let’s say there are only two quantities involved, the arithmetic mean is obtained simply by adding the quantities and dividing by 2. Mean or Arithmetic Mean is the average of the numbers i.e. a calculated central value for a set of numbers. General Formulae for Mean is,

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Mean = Sum of observation /  Number Of Observation



Example: 

The marks obtained by 5 students in a class test are 7, 9, 6, 4, 2 out of 10. Find the mean marks for the class?

According to the formula mean marks of the class are: 

Average marks = Sum of observation / Number Of Observation

Here average marks = (7 + 9 + 6 + 4 + 2) / 5 = 28 / 5 = 5.6        

Hence the mean marks for the class is 5.6            

Derivation of Formula for Mean by Step Deviation Method

The general formula for mean in statistics is:

Mean = Σfixi / Σfi



Where,

Σfixi: the weighted sum of elements and 

Σfi: 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 fi denotes the frequency and yi denotes the class-mark of the ith class the mean is given by,

Mean = Σfiyi / Σfi

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

 Mean = A + Σfidi / Σf

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, …..un where,

ui = (yi – A) / c

Then the mean is given by the formula,



Mean = A + c x (Σfiui / Σfi)

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

Examples

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(c) = 6

Classes

Class-mark(yi)

ui = (yi – A) / c



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 + c x (Σfiui / Σfi)

          = 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(c) = 10.

Classes

Class-mark(yi)

ui = (yi – A) / c

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 + c x (Σfiui / Σfi)

          = 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(c) = 5

ClassesClass-mark(yi)ui = (yi – A) / cfrequency(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 + c x (Σfiui / Σfi)

          = 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(c) = 5

Classes

Class-mark(yi)

ui = (yi – A) / c

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 + c x (Σfiui / Σfi)

          = 17.5 + 5 x (17/80)

          = 17.5 + 1.06

          = 18.56




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