Suppose we want to find out the income per household in India. So, are we going to list the income of each household? No. We will find the average income and say that is the average income per household in India. For calculating this average we use the arithmetic mean formula which is a measure of central tendency. Let us start learning more about the arithmetic mean formula in detail.

## What is Arithmetic Mean?

The arithmetic mean is the most simple and commonly used measure of central tendency. It is also very easy to calculate. We generally refer to it as Average. There are also other means of calculating averages such as Median and Mode. The arithmetic mean is the summation of all the observations or values of a data set divided by the number of observations or values. We denote it by \(\overline{X}\)

If we visualize the arithmetic mean as a balancing point on a scale, we see that half of the numerical mass of the data will be below the mean and the other half will be above it. Thus, we may say that Arithmetic mean is exactly the middle value of the data series. However, mean maybe one of the values of the data series or maybe another number or value. Also, the arithmetic mean represents the tendency of the given data. Thus, it is known as a measure of central tendency.

## Arithmetic Mean Formula

### 1] Individual Series

\(\overline{X}\) = \(\frac{1}{n}sum_{i=1}^{n} a_{i}\)= \(\frac{a_{1} + a_{2} + a_{3} + …..+ a_{n}}{n}\)

In simple terms:

\(\overline{X}\) = \(\frac{Sum of all observations}{No. of observations}\)

### 2] Discrete Series

\(\overline{X}\) = \(\frac{\sum fx}{\sum f}\)

Where,

x | Observation or the value |

f | Frequency corresponding the observation |

### 3] Continuous Series

\(\overline{X}\) = \(\frac{\sum fx}{\sum f}\)

Where,

x | Mid-value of the class |

f | Frequency corresponding the class |

Mid- value of a class = \(\frac{Upper limit + lower limit}{2}\)

### Steps to calculate Arithmetic Mean:

- Find the sum of all the observations.
- Multiply the frequency with its corresponding value and add them. This step is applicable only in the case of discrete and continuous series.
- Find the number of observations. However, in the case of discrete and continuous series, we add up all the frequencies.
- Divide the result in Step 1 or Step 2 (as the case may be) with the result in Step 3.
- The resultant figure is the Mean.

### Limitations of the Arithmetic Mean:

It is not always necessary that the arithmetic mean ideally represents the tendency of the data. It is so because if the data consists of value too high or too low, it will impact the mean. Say, for example, we need to find out the average pocket money kids of a colony get. We find that out of 20 kids, 19 receive the money ranging in between ₹100 to ₹150 per week. While the 20^{th} kid gets ₹300 per week. We can clearly see that it is almost double of the other amount. Now if we calculate the mean, this higher value will impact it and the mean will be higher.

Also, it is not much suitable for calculating the performance of investment portfolios. For this purpose Geometric mean is more suitable. Analysts also do not prefer using it to calculate present and future cash flows for making their estimates.

**Solved Examples**

Q.1. Calculate the Mean from the following data series:

Marks | No. of students |

0-10 | 6 |

10-20 | 14 |

20-30 | 42 |

30-40 | 28 |

40-50 | 10 |

Solution:

Marks | No. of students (f) | Mid-values (x) | fx |

0-10 | 6 | 5 | 30 |

10-20 | 14 | 15 | 210 |

20-30 | 42 | 25 | 1050 |

30-40 | 28 | 35 | 980 |

40-50 | 10 | 45 | 450 |

\(\sum f\) = 100 | \(\sum fx\) = 2720 |

Mid- value of a class = \(\frac{Upper limit + lower limit}{2}\)

= \(\frac{10 + 0}{2}\)

= 5

Similarly, we shall calculate all the mid-values.

\(\overline{X}\) = \(\frac{\sum fx}{\sum f}\)

= \(\frac{2720}{100}\)

= 27.20

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