Posted by Anjali Kaur on Jun 01, 2021

Calculation of Arithmetic Mean in Case of Cumulative Frequency Distribution

A cumulative frequency distribution can be in the form of less than cumulative frequency distribution or in the form of more than cumulative frequency distribution. In this post, we will learn the tricks to the Calculation of Arithmetic Mean in the Case of Cumulative Frequency Distribution. But first, Feel free to join our Facebook group.

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Less than Cumulative Frequency Distribution

Frequency distribution is also known as the continuous series. So, when the class intervals are mentioned in the form of “Less than”, then it is less than the cumulative frequency distribution question. Let’s look at the common steps involved to solve such questions:

Steps Involved

  1. Remember in case the class interval is in the form of less than, the first frequency mentioned against the first class interval is the same as the cumulative.
  2. The first class interval CF is the same as the first frequency.
  3. Rest frequency is calculated by subtracting the previous cf from the existing cf, i.e., F = Cf existing – Cf previous or Cf present – Cf previous.
  4. The total frequency = the last Cf has given in the question.

Let’s apply these steps to solve a question.

Find the arithmetic mean of the following table:

MarksNumber of Students
Less than 105
Less than 2017
Less than 3031
Less than 4041
Less than 5049

Solution.

Make this table with properly defined class interval:

Marks (X)f
0- 105 (Cf and F is same)
10-2017-5 = 12 (Cf present – Cf previous)
20-3031- 17 = 14
30-4041 – 31 = 10
40-5049 – 41 = 8

Now, make this table again add a column for ‘Frequency’, ‘ Mid-points’ and ‘Frequency multiply with Mid-point’

Marks(X)Mid-pointsFrequencyfm
0-105525
10-201512180
20-302514350
30-403510350
40-50458360
Total f = 49 (same as the last cf)1265

I am applying a direct method for the calculation of Arithmetic mean, you can apply to assume mean of a step-deviation method, your answer will be the same.

Using the direct method formula:

Arithmetic mean = 1265/49 = 25.82

Take a look at my virtual explanation:

More than Cumulative Frequency Distribution

When the class intervals are mentioned in the form of “More than”, then it is more than the cumulative frequency distribution question. Let’s look at the common steps involved to solve such questions:

Steps Involved

  1. Remember in case the class interval is in the form of more than, the last frequency mentioned against the last class interval is the same as the last cumulative frequency.
  2. The first CF is the same as the total of the frequency. That is first cf = summation of frequency.
  3. Rest frequency is calculated by subtracting the next cf from the existing cf, i.e., F = Cf existing – Cf upcoming or Cf present – Cf next.

Let’s apply these steps to solve a question.

Find the arithmetic mean of the following table:

Marks (X)Number of students
More than 030
More than 228
More than 424
More than 618
More than 810

Solution.

Make this table with properly defined class interval:

Marks (X)f
0 – 230 – 28 = Existing Cf – Next Cf
2 – 428 – 24 = 4
4 – 624 – 18 = 6
6 – 818- 10 = 8
8 – 1010 (Last cf = last frequency)

Now, make this table again add a column for ‘Frequency’, ‘ Mid-points’ and ‘Frequency multiply with Mid-point’

Marksmid points (m)ffm
0-2122
2-43412
4-65630
6-87856
8-1091090
Total f = 30 (same as the first cf)190

I am applying a direct method for the calculation of Arithmetic mean, you can apply the assumed mean method or the step-deviation method, your answer will be the same.

Using the direct method formula:

Arithmetic Mean = 190/30 = 6.33

I hope it was easy to digest.

Take a look at the virtual explanation:

You can also check my YouTube channel LearnWithAnjali for the explanation of the same:

  1. Less than Cumulative Frequency
  2. More than Cumulative Frequency

Time to Test Yourself

  1. Calculate mean marks by Direct Method:
MarksNumber of Students
More than 0 50
More than 1046
More than 2040
More than 3020
More than 4010
More than 503

2. Calculate mean marks by step-deviation method:

IncomeNumber of Families
More than 75150
More than 85140
More than 95115
More than 10595
More than 11570
More than 12560
More than 13540
More than 14525

This was all about continuous series arithmetic mean.

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

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

Thank You!

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