# Discrete Series Arithmetic Mean

Discrete series is the second type of series. Discrete series is also known as the ungrouped frequency distribution. Under this series, we have X and F, where F is the frequency, that is, the number of times X is repeated. There are 3 methods involved under the discrete series arithmetic mean calculation. Let’s understand each method.

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## 1. Direct Method

This is the most simple method of all, its formula is:

Steps involved:

- Find FX, just multiply the X values with F values. Then add them up, to get ΣFX.
- Find ΣF
- Apply the formula; x̄ = ΣFX/ΣF

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

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

10 | 2 | 20 |

30 | 3 | 90 |

20 | 4 | 80 |

40 | 1 | 40 |

Σ | 10 | 230 |

Using the arithmetic mean formula, x̄ = ΣFX/ΣF

x̄ = 230/10 = 23

## 2. Short-cut Method

This method is also known as the assumed mean method. It’s formula is

Steps involved:

- Take assume mean, A, from X, generally I recommend selecting the middle term.
- Find d, which is the deviation from the assumed means. Use d= X-A.
- Find Fd, which is frequency multiply with deviation, d. Then find the summation of the same.
- Find the total frequency. Then use the above formula. x̄ = A + ΣFd/ΣF

Let’s try a question.

X | F | d= X-A | Fd |
---|---|---|---|

10 | 2 | -20 | -40 |

30*A | 3 | 0 | 0 |

20 | 4 | -10 | -40 |

40 | 1 | 10 | 10 |

Σ | 10 | -70 |

Now, use the formula;

x̄ = A + ΣFd/ΣF

= 30 -70/10

=23

## 3. Step-Deviation method

Since the name of the method says step deviation, we try to do the calculation faster by taking out a common factor from X. Look at the formula:

Steps involved:

- Find assume mean, A, from X. Try selecting the middle term.
- Find d, which is d= X-A
- Find d’, which is d’ = X-A/C; C is the common factor, just like an LCM taken from X.
- Find Fd’ and take the total of it. Then use the formula; x̄ = A + ΣFd’/ΣF x C

Let’s try a question.

X | F | d= X-A | d’ = X-A/C ; C = 10 | Fd’ |
---|---|---|---|---|

10 | 2 | -20 | -2 | -4 |

30*A | 3 | 0 | 0 | 0 |

20 | 4 | -10 | -1 | -4 |

40 | 1 | 10 | 1 | 1 |

Σ | 10 | -7 |

Use the formula;

x̄ = A + ΣFd’/ΣF x C

= 30 -7/10 x10

= 23

Take a look at my video explanation:

## Test Yourself

- From the following data of the marks obtained by 60 students of a class, calculate the average marks by the direct method, assume mean method, and step – deviation method.

Marks | No of students |
---|---|

20 | 8 |

30 | 12 |

40 | 20 |

50 | 10 |

60 | 6 |

70 | 4 |

2. From the following data, calculate arithmetic mean by direct method, short-cut method and step-deviation method.

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

2 | 10 |

3 | 16 |

4 | 11 |

5 | 8 |

6 | 6 |

7 | 4 |

8 | 3 |

9 | 2 |

3. Calculate average of the following discrete series. Use Assumed mean method by taking 25 as assumed mean.

Size | Frequency |
---|---|

30 | 2 |

29 | 4 |

28 | 5 |

27 | 3 |

26 | 2 |

25 | 7 |

24 | 1 |

23 | 4 |

22 | 5 |

21 | 7 |

4. Marks secured by 42 students in economics are as follows, find the average marks.

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

15 | 8 |

20 | 4 |

22 | 7 |

23 | 3 |

27 | 8 |

35 | 7 |

18 | 5 |

This was all about discrete series arithmetic mean.

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

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

Thank You!

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