# Continuous Series Arithmetic Mean

Continuous series is the third series followed by individual and discrete 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:**

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

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

X | F |
---|---|

0-10 | 2 |

10-20 | 5 |

20-30 | 3 |

Solution;

X | F | m | Fm |
---|---|---|---|

0-10 | 2 | 5 | 10 |

10-20 | 5 | 15 | 75 |

20-30 | 3 | 25 | 75 |

10 | 160 |

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:**

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

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

X | F |
---|---|

0-10 | 2 |

10-20 | 5 |

20-30 | 3 |

Solution;

X | F | m | d= m-A | fd |
---|---|---|---|---|

0-10 | 2 | 5 | -10 | -20 |

10-20 | 5 | 15(A) | 0 | 0 |

20-30 | 3 | 25 | 10 | 30 |

10 | 10 |

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:

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

Let’s try a question;

X | F |
---|---|

0-10 | 2 |

10-20 | 5 |

20-30 | 3 |

Let’s find its solution-

X | F | m | d= m-A | d’ = d/C | Fd’ |
---|---|---|---|---|---|

0-10 | 2 | 5 | -10 | -1 | -2 |

10-20 | 5 | 15(A) | 0 | 0 | 0 |

20-30 | 3 | 25 | 10 | 1 | 3 |

10 | 1 |

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:

- Average age of the people of a country is shown in the following table:

Age(Years) | People(‘000) |
---|---|

10-20 | 30 |

20-30 | 32 |

30-40 | 15 |

40-50 | 12 |

50-60 | 9 |

Find out the mean age by the direct method.

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

Class Interval | Frequency |
---|---|

10-20 | 4 |

20-40 | 10 |

40-70 | 26 |

70-120 | 8 |

120-200 | 2 |

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

Class Interval | Frequency |
---|---|

20-25 | 10 |

25-30 | 12 |

30-35 | 8 |

35-40 | 20 |

40-45 | 11 |

45-50 | 4 |

50-55 | 5 |

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

Items | Frequency |
---|---|

10-8 | 10 |

8-6 | 8 |

6-4 | 6 |

4-2 | 4 |

2-0 | 2 |

5. Sachin made the following runs in different matches:

Runs | Frequency |
---|---|

5-15 | 10 |

15-25 | 12 |

25-35 | 17 |

35-45 | 19 |

45-55 | 22 |

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

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

Class | Frequency |
---|---|

less than 10 | 5 |

10-20 | 12 |

20-30 | 18 |

30-40 | 22 |

40-50 | 6 |

50-60 | 4 |

more than 60 | 3 |

This was all about continuous series arithmetic mean.

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

- Individual Series Arithmetic Mean
- Discrete Series Arithmetic Mean
- Introduction to the statistics.
- Functions and importance of statistics.
- Important questions on Measures of Central Tendency
- Finding Missing Frequency in case of Median
- Median under less than cumulative frequency distribution

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