Shortcut Method for Arithmetic Mean

In layman's words, statistics is the process of gathering, classifying, examining, interpreting, and finally, understandably presenting information to 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 is what it's called.

What is the shortcut 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:

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	5	45	45 - 25 = 20	100
	Σf = 46			Σdf = -20

x̄=a+\frac{Σdf}{Σf}
= 25 -20/46
= 25 - 0.4348
x̄ = 24.57

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

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:

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:

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 shortcut 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.

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:

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

Practice Question on Shortcut Method for Arithmetic Mean

Question 1: Find the mean using the shortcut method: 48, 50, 52, 54, 46

Question 2: Find the arithmetic mean of: 95, 100, 105, 90, 110

Question 3: Using 200 as the assumed mean, calculate the average of: 195, 205, 210, 190, 200

Question 4: Find the mean of the following numbers using 70 as assumed mean: 65, 68, 72, 74, 71

Question 5: Find the mean of these values using the shortcut method: 145, 150, 155, 160, 140

Question 6: If the values are: 75, 78, 72, 74, 71, find the mean using 74 as the assumed mean.