Posted by Anjali Kaur on May 17, 2021

Continuous Series Arithmetic Mean

Continuous series is the third series followed by individual and discreet series. It is also known as the frequency distribution.

Under this series, we have X and F, where F is the frequency, that is, the number of times X is repeated. But X is given in the form of an interval or range; like 0-10 or 10-20. There are 3 methods involved under the continuous series arithmetic mean calculation. Let’s understand each method.

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1. Direct Method

It is a simple method, you find the mid-point from X, multiply them with F, then add them up to find summation fm, where m refers to the mid-point from X, finally, the sum is divided by the summation F.

Steps involved:

  1. Find the midpoints from the class interval(X) given in the question.
  2. Find ‘fm’ which is calculated by multiplying frequency with the mid-point, then find the sum of fm calculated
  3. Find the sum of frequency and apply the above formula.

Let’s try a question; if X and F is given in the question.

XF
0-102
10-205
20-303

Solution;

XFmFm
0-102510
10-2051575
20-3032575
10160

How did I calculated mid- points?

m = (lower limit + upper limit)/2

In this question; 0+10/2 = 5, 10+20/2 = 15 and 20+30/2 = 25.

Now, let’s use the direct method formula under continuous series

X̄ = Σfm/Σf

X̄ = 160/10 = 16

2. Short-Cut Method/Assumed Mean Method

Whenever the name shortcut or assume mean comes, remember we need to select ‘A’ which is the assumed mean. Under continuous series, we will select it from the midpoints calculated.

Steps involved:

  1. Find the mid-points from the class intervals given in the question.
  2. Now, select A(assume mean) from the midpoints you calculated.
  3. Find ‘d’ which is the deviation from the assumed mean; d = m-A
  4. Find ‘fd’ and total it.
  5. Find the total frequency and apply the formula.

Let’s try a question; if X and F is given in the question.

XF
0-102
10-205
20-303

Solution;

XFmd= m-Afd
0-1025-10-20
10-20515(A)00
20-303251030
1010

Let’s apply the formula now;

X̄ = 15 + 10/10 = 15 + 1 = 16

3. Step-Deviation method

This is the 3rd method to find the arithmetic mean under continuous series. Its formula is:

Steps involved:

  1. Find the mid-points from the class intervals given in the question.
  2. Now, select A(assume mean) from the midpoints you calculated.
  3. Find ‘d’ which is the deviation from the assumed mean; d = m-A
  4. Find d’ which is d divided by common factor; common factor will be selected based on the midpoints.
  5. Find Σfd’
  6. Find the total frequency and apply the formula.

Let’s try a question;

XF
0-102
10-205
20-303

Let’s find its solution-

XFmd= m-Ad’ = d/CFd’
0-1025-10-1-2
10-20515(A)000
20-303251013
101

Here, the common factor is selected based on the midpoints. As midpoints, the common factor is 10. So, C = 10

Now use the formula:

X̄ = 15 + (1/10) x 10 = 15 + 1 = 16

I hope it was easy to digest. Take a look at the virtual explanation on my YouTube Channel:

Test Yourself:

  1. Average age of the people of a country is shown in the following table:
Age(Years)People(‘000)
10-2030
20-3032
30-4015
40-5012
50-609

Find out the mean age by direct method.

2. Calculate the arithmetic mean of the following frequency distribution by direct method:

Class IntervalFrequency
10-204
20-4010
40-7026
70-1208
120-2002

3. Calculate arithmetic mean from the following data by assume mean method:

Class IntervalFrequency
20-2510
25-3012
30-358
35-4020
40-4511
45-504
50-555

4. Find out arithmetic mean from the following using assume mean method:

ItemsFrequency
10-810
8-68
6-46
4-24
2-02

5. Sachin made the following runs in different matches:

RunsFrequency
5-1510
15-2512
25-3517
35-4519
45-5522

Calculate the average mean of the runs by step-deviation method.

6. Calculate arithmetic mean of the following frequency distribution by all 3 methods:

ClassFrequency
less than 105
10-2012
20-3018
30-4022
40-506
50-604
more than 603

This was all about continuous series arithmetic mean.

I hope it was helpful, you can refer more posts related to the statistics.

  1. Individual Series Arithmetic Mean
  2. Discrete Series Arithmetic Mean
  3. Introduction to the statistics.
  4. Functions and importance of statistics.

Thank You!

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