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Shortcut Method for Arithmetic Mean

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  • Last Updated : 16 Feb, 2022

Statistics, in layman’s words, is the process of gathering, classifying, examining, interpreting, and finally presenting information in an understandable manner so that one can form an opinion and, if necessary, take action. Examples:

  • A teacher collects students’ grades, organizes them in ascending or decreasing order, calculates the average class grade, or determines how many students failed and tells them so that they can begin studying more diligently.
  • Government officials are gathering census data and comparing it to previous records to see if population growth is under control.
  • Analyzing the number of persons in a country who practice a particular religion.
  • Analyzing the number of people who follow a specific religion in a country.

Arithmetic Mean

The arithmetic mean, commonly known as the average, is determined for a given collection of data by adding up the numbers in the data and dividing the sum by the number of observations. It is the most widely used central tendency approach. The direct approach of calculating the arithmetic mean is what it’s called.

What is short cut method in statistics?

Solution:

Short-Cut Method

The short-cut approach is used whenever the data values are huge and the calculation is time-consuming. When using the short- cut method to get the arithmetic mean, the stages are as follows:

  • Select one observation from the data set and use it as the series’ assumed mean. Because it is impossible to choose one observation from the class intervals while working with grouped data, one must first compute the class marks of the intervals’ mid-points and designate one as the presumed mean
  • Next, determine deviations from the expected mean (A) by subtracting the assumed mean from all other data. d = X − A.
  • Take the total of the numbers produced by multiplying the deviations obtained with the frequencies.
  • Apply the formula x̄=a+\frac{Σdf}{Σf} , where Σdf is the sum of all the deviations multiplied by respective frequencies.
  • The arithmetic mean of the given data set is the number produced in this way.

Thus the formula for the calculation of arithmetic mean by short- cut method is:

x̄=a+\frac{Σdf}{Σf}

Sample Problems

Question 1: Calculate the arithmetic mean for the following data set using the short-cut method:

Marks 

Number of students

0 – 10

5

10 – 20

12

20 – 30

14

30 – 40

10

40 – 50

5

Solution:

Marks

f

m

d = m – a

fd

0 – 10

5

5

5 – 25 = −20

−100

10 – 20

12

15

15 – 25 = −10

−120

20 – 30

14

A = 25

25 – 25 = 0

0

30 – 40

10

35

35 – 25 = 10

100

40 – 50

8

45

45 – 25 = 20

160

 

Σf = 49

 

 

Σdf = 40

x̄=a+\frac{Σdf}{Σf}

= 25 + 40/49

= 25 + 0.81

x̄ = 25.81

Question 2: Calculate the arithmetic mean for the following data set using the short-cut method:

Marks

Number of Students

10 – 20

5

20 – 30

3

30 – 40

4

40 – 50

7

50 – 60

2

60 – 70

6

70 – 80

13

Solution:

Marks

f

m

d = m – a

fd

10 – 20

5

15

−30

−150

20 – 30

3

25

−20

−60

30 – 40

4

35

−10

−40

40 – 50

7

A = 45

0

0

50 – 60

2

55

10

20

60 – 70

6

65

20

120

70 – 80

13

75

30

390

 

Σf = 40

 

 

Σdf = 280

Mean = X̄ = a+\frac{Σdf}{Σf}

= 45 + 280/40

= 45 + 7

x̄ = 52

Question 3: Calculate the arithmetic mean for the following data set using the short-cut method:

Wages

Number of Workers

0 – 10

22

10 -20

38

20 – 30

46

30 – 40

35

40 – 50

19

Solution:

Wages

f

m

d = m – a

fd

0 – 10

22

5

-20

−440

10 -20

38

15

-10

−380

20 – 30

46

a = 25

0

0

30 – 40

35

35

10

350

40 – 50

19

45

20

380

 

Σf = 160

 

 

Σdf = -90

Mean = X̄ = a+\frac{Σdf}{Σf}

= 25 + (-90)/160

x̄ = 24.44

Question 4: Calculate the arithmetic mean for the following data set using the short-cut method:

Wagesf
3-610
6-920
9-1230
12-1540
15-1850

Solution:

Wages

f

m

d = m – A

fd

3-6

10

4.5

-6

-60

6-9

20

7.5

-3

-60

9-12

30

A =10.5

0

0

12-15

40

13.5

3

120

15-18

50

16.5

6

300

 

Σf = 150

 

 

Σdf = 300

Mean = X̄ = a+\frac{Σdf}{Σf}

= 10.5 + (3000)/150

x̄ = 12.5

Question 5: Calculate the arithmetic mean for the following data set using the short-cut method: 75, 68, 80, 56, 92.

Solution: 

x

d = x – A

75

7

A = 68

0

80

12

56

-12

92

24

 

Σd = 31

Since the given series is individual and not discrete, the formula for mean using short- cut method would be as follows: 

Mean = X̄ = a+\frac{Σd}{n} , where n is the number of observations.

= 68 + 31/5

x̄ = 74.2

Question 6: Calculate the arithmetic mean for the following data set using the short-cut method. Assume that a = 8.

Deviations from assumed mean

f

-2

4

-1

8

0

13

1

20

2

12

Solution:

d

f

fd

-2

4

-8

-1

8

-8

0

13

0

1

20

20

2

11

24

 

Σf = 56

Σdf = 28

Mean = X̄ = a+\frac{Σdf}{Σf}

= 8 + (28)/56

x̄ = 8.5

Question 7: Calculate the arithmetic mean for the following data using short- cut method:

x

f

40-45

6

45-50

18

50-55

12

55-60

3

60-65

1

Solution:

x

f

m

d = m – A

fd

40-45

6

42.5

-10

-60

45-50

18

47.5

-5

-90

50-55

12

A = 52.5

0

0

55-60

3

57.5

5

15

60-65

1

62.5

10

10

 

 

Σf = 40

 

Σfd = -125

Mean = X̄ = a+\frac{Σdf}{Σf}

= 52.5 + (-125)/40

x̄ = 49.37


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