# Inclusive and Exclusive Series – Arithmetic Mean

Exclusive series is the most common one, where we have class intervals like; 0-10, 10-20, etc. Inclusive series is the one where the class intervals are like; 0-9, 10-19, 20-29, etc. We can find the arithmetic mean using inclusive series directly, but sometimes it is asked to convert it into exclusive series and then find the arithmetic mean. Let’s learn ‘Inclusive and Exclusive Series – Arithmetic Mean’.

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## Calculation of Arithmetic Mean in case of Inclusive series

Under this, we will learn how to find the Arithmetic mean in the case of inclusive series only.

Question. The following table shows the monthly pocket expenses of the student of a class. Find out the average pocket expenses.

Pocket Expenses | 20-29 | 30-39 | 40-49 | 50-59 | 60-69 |

Number of Students | 10 | 8 | 6 | 4 | 2 |

Solution.

Calculation of arithmetic mean of inclusive series is the same as of exclusive series.

Let pocket expenses be X, that is, class interval.

Frequency is number of students, F.

Recreating the question box, to solve for the arithmetic mean.

X | f | Mid Point(M) | fM |
---|---|---|---|

20-29 | 10 | 24.5 | 245 |

30-39 | 8 | 34.5 | 276 |

40-49 | 6 | 44.5 | 267 |

50-59 | 4 | 54.5 | 218 |

60-69 | 2 | 64.5 | 129 |

Total | 30 | 1135 |

Here, I have applied the direct method, under continuous series:

Using the above formula.

Arithmetic Mean = 1135/30

**Arithmetic Mean = 37.833**

## Calculating Arithmetic Mean by converting Inclusive series to Exclusive series.

Now, we will do the conversion first of the given series. Then we will apply the formula to arrive at the arithmetic mean.

Question.

A polling agency interviewed 200 persons. The age distribution of those persons was recorded as under:

Age (in years) | 80-89 | 70-79 | 60-69 | 50-59 | 40-49 | 30-39 | 20-29 | 10-19 |

Frequency | 2 | 2 | 6 | 20 | 56 | 40 | 42 | 32 |

Solution.

Step 1. Arrange them in ascending order. Make sure frequency is arranged accordingly.

Step 2. Find the difference between class interval and divide it by 2.

Step 3. Subtract the difference of the lower limit and Add the difference to the upper limit.

Step 4. Find the arithmetic mean.

X | X’ | f | Mid point (M) | fM |
---|---|---|---|---|

10-19 | 9.5-19.5 | 32 | 14.5 | 464 |

20-29 | 19.5-29.5 | 42 | 24.5 | 1029 |

30-39 | 29.5-39.5 | 40 | 34.5 | 1380 |

40-49 | 39.5-49.5 | 56 | 44.5 | 2492 |

50-59 | 49.5-59.5 | 20 | 54.5 | 1090 |

60-69 | 59.5-69.5 | 6 | 64.5 | 387 |

70-79 | 69.5-79.5 | 2 | 74.5 | 149 |

80-89 | 79.5-89.5 | 2 | 84.5 | 169 |

200 | 7160 |

Using the direct method of continuous series.

**Arithmetic Mean = 7160/200 = 35.8 Years**

**You can take a look at the below video for this topic explanation by me. **

## Test Yourself

Question. The following table shows the wages of the workers. Calculate the average wage of the workers.

Wages (In INR) | Number of Workers |
---|---|

10-19 | 8 |

20-29 | 9 |

30-39 | 12 |

40-49 | 11 |

50-59 | 6 |

**Answer should be 34.07. **

**Good Luck!**

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

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

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