# 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**

- 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.
- The first class interval CF is the same as the first frequency.
- 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.
- 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:**

Marks | Number of Students |
---|---|

Less than 10 | 5 |

Less than 20 | 17 |

Less than 30 | 31 |

Less than 40 | 41 |

Less than 50 | 49 |

**Solution. **

Make this table with properly defined class interval:

Marks (X) | f |
---|---|

0- 10 | 5 (Cf and F is same) |

10-20 | 17-5 = 12 (Cf present – Cf previous) |

20-30 | 31- 17 = 14 |

30-40 | 41 – 31 = 10 |

40-50 | 49 – 41 = 8 |

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

Marks(X) | Mid-points | Frequency | fm |
---|---|---|---|

0-10 | 5 | 5 | 25 |

10-20 | 15 | 12 | 180 |

20-30 | 25 | 14 | 350 |

30-40 | 35 | 10 | 350 |

40-50 | 45 | 8 | 360 |

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**

- 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.
- The first CF is the same as the total of the frequency. That is first cf = summation of frequency.
- 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 0 | 30 |

More than 2 | 28 |

More than 4 | 24 |

More than 6 | 18 |

More than 8 | 10 |

**Solution.**

Make this table with properly defined class interval:

Marks (X) | f |
---|---|

0 – 2 | 30 – 28 = Existing Cf – Next Cf |

2 – 4 | 28 – 24 = 4 |

4 – 6 | 24 – 18 = 6 |

6 – 8 | 18- 10 = 8 |

8 – 10 | 10 (Last cf = last frequency) |

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

Marks | mid points (m) | f | fm |
---|---|---|---|

0-2 | 1 | 2 | 2 |

2-4 | 3 | 4 | 12 |

4-6 | 5 | 6 | 30 |

6-8 | 7 | 8 | 56 |

8-10 | 9 | 10 | 90 |

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:

## Time to Test Yourself

- Calculate mean marks by Direct Method:

Marks | Number of Students |
---|---|

More than 0 | 50 |

More than 10 | 46 |

More than 20 | 40 |

More than 30 | 20 |

More than 40 | 10 |

More than 50 | 3 |

2. Calculate mean marks by step-deviation method:

Income | Number of Families |
---|---|

More than 75 | 150 |

More than 85 | 140 |

More than 95 | 115 |

More than 105 | 95 |

More than 115 | 70 |

More than 125 | 60 |

More than 135 | 40 |

More than 145 | 25 |

This was all about continuous series arithmetic mean.

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

- Continuous Series Arithmetic Mean
- Individual Series Arithmetic Mean
- Discrete Series Arithmetic Mean
- Introduction to the statistics.
- Functions and importance of statistics.

Thank You!

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