# Individual Series Arithmetic Mean

Arithmetic mean or mean or average is the number that is obtained by adding the values of all the items of a series and dividing it by the number of observations. In this post, we will understand the individual series of the arithmetic mean. Let’s have a look at its most basic and popular formula, this is used under the direct method.

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## Methods of calculating arithmetic mean in case of individual series

Individual series or ungrouped data means the same. There are 3 methods of calculating arithmetic mean in case of individual series:

### 1. Direct Method

This is the simplest and the most common method of finding the mean. All you have to do is follow the below steps:

a. Add up all the items of the given series, use Σ x

b. Find the total number of items, use N for it.

c. Use the formula; x̄ = Σ x/N

Try 1 question; Pocket allowance of 10 students is 15, 20, 30, 22, 25, 18, 40, 50, 55 and 65. Find out the average pocket allowance.

Solution. Add all the pocket allowances;

Σ x = 15+ 20+ 30+ 22+ 25+ 18+ 40+ 55+ 50+ 65 = 340

N = 10

x̄ = Σ x/N = 340/10 = 34

### 2. Assumed Mean Method

The assumed mean method is also known as the short-cut method. See, whenever the word assume means appears, you must find an assumed mean(A) from the series. Generally, we choose any middle term as an assumed mean. Let’s see the steps involved in this method:

a. Find A from the given data or series, try taking the middle term.

b. Find deviations, which is denoted by ‘D’ = X – A

c. N remains the same as the number of the observations.

d. Use the formula as;

Try 1 question; Pocket allowance of 10 students is 15, 20, 30, 22, 25, 18, 40, 50, 55 and 65. Find out the average pocket allowance.

Solution.

Pocket Allowance, X | Deviation = X-A |
---|---|

15 | -10 |

20 | -5 |

30 | 5 |

22 | -3 |

25 *(A) | 0 |

18 | -7 |

40 | 15 |

50 | 25 |

55 | 30 |

65 | 40 |

Σ D= 90 |

Now, use the formula

x̄ = 25 + 90/10 = 25 + 9 = 34

### 3. Step-deviation method

This is the 3rd method under individual series. Again we will find the assumed mean from the given data and we also need to find a common factor like a LCM, that makes it step deviation. Let’s see the steps involved under this method;

a. Find A from the data, preferably the middle term.

b. Find deviation, D = X-A

c. Find D’= D/C, C stands for the common factor in the data.

d. Use the below formula

Now try 1 question; find the arithmetic mean from the following data:

Monthly Expenditure X | D = X-A | D’ = D/C C=10000 |
---|---|---|

50000 | -30000 | -3 |

60000 | -20000 | -2 |

40000 | -40000 | -4 |

70000 | -10000 | -1 |

80000* A | 0 | 0 |

20000 | -60000 | -6 |

10000 | -70000 | -7 |

30000 | -50000 | -5 |

90000 | 10000 | 1 |

100000 | 20000 | 2 |

-25 |

Use the formula;

= 80000 + (-25 x 10000)/ 10

= 80000 – 25000 = 55,000

Now, try the following exercise on your own.

## Test Yourself

- Following is the monthly income of 8 families in a locality; 70, 10, 500, 75, 13, 250, 8, 42. Find the arithmetic mean using the direct method and assumed mean method.
- Students of class XII secured the following marks in their statistics paper; 20, 44, 65, 28, 45, 67, 30, 50, 68, 39, 53, 70, 40, 60, 75. Calculate the arithmetic mean using the direct and the assumed mean method.
- Eight workers earn the following income; 30, 36, 34, 40, 42, 46, 54, 62. Find out the arithmetic mean.
- The pocket allowance of 5 students is 125, 75, 150, 175, 200. Find out the arithmetic mean.
- Following is the height of 10 students: Calculate arithmetic mean using direct and assumed mean methods:

Students | A | B | C | D | E | F | G | H | I | J |

Height(cm) | 155 | 153 | 168 | 160 | 162 | 166 | 164 | 180 | 157 | 165 |

6. Weight of 15 persons is as follows:

20, 28, 34, 39, 42, 50, 53, 54, 59, 64, 72, 74, 74, 78, 79

Find out mean weight, using direct as well as assumed mean method.

This was all about individual series arithmetic mean.

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

Thank You!

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