**Arithmetic Mean(AM)** is the most common measure of central tendency. In the terms of a layman, the mean of the data set demonstrates an average of the given data set. A few real-life statistics examples in terms of AM are the average rainfall of a place, the average income of the employees in the firm, and more. In this article, we will be concentrating more on Arithmetic mean like definition, formula, properties, best methods to find Mean, and Questions on average (arithmetic mean).

## Definition of Arithmetic Mean (Average)

Arithmetic Mean portrays a number that is gained by dividing the sum of the elements of a dataset by the total number of values in the dataset. You can use the term Average or use the word Arithmetic Mean for making it a bit fancier. The AM can be either

- Simple Arithmetic Mean
- Weighted Arithmetic Mean

### Arithmetic Mean Formula

In short, the arithmetic mean is knowns as the average or the mean of the given numbers.

If any data set comprising of the values x_{1}, x_{2}, x_{3}, …., x_{n}, then the Arithmetic Mean A is defined as:

Also, it is denoted as:

If you are finding the mean when the frequency of the observations is given, in such a way x_{1}, x_{2, }x_{3,… }x_{n }is the recorded observations, and f_{1}, f_{2, }f_{3 }… f_{n} is the corresponding frequencies of the observations then;

Also, it is presented like:

This formula is used when the Arithmetic Mean is calculated for the ungrouped data. In order to calculate the mean or average of grouped data, we calculate the class mark. Here, the calculation of midpoints of the class intervals can be done as:

Once, you are done with calculating the class mark, the mean is solved as explained above. This method of solving the arithmetic mean is known as the direct method.

### Properties of Arithmetic Mean

A few basic properties of the arithmetic mean are as such:

- The sum of deviations of the items from their arithmetic mean is always zero, i.e. ∑(x – X) = 0.
- The sum of the squared deviations of the items from Arithmetic Mean (A.M) is minimum, which is less than the sum of the squared deviations of the items from any other values.
- If all the observations in the given data set have a value say ‘n’, then their arithmetic mean is also ‘n’.
- If every item in the arithmetic series is replaced by the mean, then the sum of these replacements will be equivalent to the sum of the specific items.
- In case each value in the score increases or decreases by a fixed value, then the mean also increases/decreases by the same number. Let the mean of x
_{1}, x_{2, }x_{3,… }x_{n}be X̄, then the mean of x_{1+k}, x_{2+k, }x_{3+k,… }x_{n+k }will be X̄+k.

### How to Find the Arithmetic Mean?

There are three ways to find the arithmetic mean. They are as follows:

Direct method

Short-cut method

Step Deviation method

#### Direct Metod for finding the Average (Arithmetic Mean)

Let x_{1}, x_{2, }x_{3,… }x_{n }be the observations with the frequency f_{1}, f_{2, }f_{3 }… f_{n}. Then, the calculation of mean can be done by using the formula:

x̄ = (x_{1}f_{1}+x_{2}f_{2}+……+x_{n}f_{n}) / ∑f_{i
}

Here, f_{1}+ f_{2} +….f_{n} = ∑f_{i} demonstrates the sum of all frequencies.

#### Short-cut Method for solving the Arithmetic Mean

An assumed mean method or change of origin method is also known as the short-cut method. The steps that should be followed to calculate the arithmetic mean are as follows:

- First, find the class marks (mid-point) of each class (x
_{i}) - Let A represent the assumed mean of the dataset.
- Calculate deviation (d
_{i}) = x_{i}– A - Now, apply the formula of the mean i.e., x̄ = A + (∑f
_{i}d_{i}/∑f_{i})

#### Step Deviation Method for Obtaining the Arithmetic Mean

The other names of this method are the change of origin or scale method. Follow the below steps and easily calculate the mean of the given data:

- Find the class marks of each class (x
_{i}). - Let’s A be the assumed mean of the data.
- Calculate u
_{i}=(x_{i}−A)/h, where h is the class size. - At last, use the formula ie., x̄ = A + h × (∑f
_{i}u_{i}/∑f_{i})

### Arithmetic Mean Examples with Solutions

**Example 1:**

Study the following example table data and find the arithmetic mean using the step-deviation method.

Class Intervals |
0-10 |
10-20 |
20-30 |
30-40 |
40-50 |
50-60 |
60-70 |
Total |

Frequency |
5 | 7 | 8 | 10 | 15 | 11 | 4 | 60 |

**Solution**:

In order to calculate the mean of the given data, we need to get the value for class marks and then decide A (assumed mean).

Let A = 35,

Here h (class width) = 10

C.I. |
x_{i} |
f_{i} |
u_{i}= x_{i}−A/h |
f_{i}u_{i} |

0-10 | 5 | 5 | -3 | 5 x (-3)=-15 |

10-20 | 15 | 7 | -2 | 7 x (-2)=-14 |

20-30 | 25 | 8 | -1 | 8 x (-1)=-8 |

30-40 | 35 | 10 | 0 | 10 x 0= 0 |

40-50 | 45 | 15 | 1 | 15 x 1=15 |

50-60 | 55 | 11 | 2 | 11 x 2=22 |

60-70 | 65 | 4 | 3 | 4 x 3=12 |

Total |
∑f_{i}=60 |
∑f_{i}u_{i}=12 |

By using the arithmetic mean formula:

x̄ = A + h × (∑f_{i}u_{i}/∑f_{i})

= 35 + 10 x (12/60)

= 35 + 2

= 37.

**Example 2:**

The runs scored by Sachin in 3 test matches are 100, 167, 215 respectively. Find the mean.

**Solution:**

Given that Sachin scored runs in 3 test matches are 100, 167, 215

Total number of test matches = 3

Mean = Sum of total runs / Number of test matches

= 100 +167 + 215 / 3

= 482 / 3

= 160

### FAQs on Average (Arithmetic Mean)

**1. What is Arithmetic Mean in Statistics?**

In statistics, Arithmetic Mean is the ratio of all observations to the total number of observations in a data set.

**2. What is the formula to find the Arithmetic Mean?**

The formula for solving the mean problems is the sum of all observations divided by a number of observations.

*Arithmetic mean formula = Σ \(\frac { X _{i} }{ n } \), where ‘i’ ranges from 1 to n*

**3. How do you solve the word problems on Arithmetic mean or average?**

Arithmetic mean can be simple arithmetic mean or weighted arithmetic mean. The simple arithmetic mean calculations vary for different data sets like for individual observations, for discrete series, and for continuous series. Go with this page and collect deep knowledge on how to calculate the arithmetic mean with examples.