Properties of Arithmetic Mean

Some of the properties of Arithmetic Mean are as follows:

1. The sum of deviations of the observations from their arithmetic mean is always zero : Arithmetic Mean is a point of balance. It means that the sum of positive deviations from mean and the sum of negative deviations from mean are equal. Because of this, the sum of deviations of the observations from their arithmetic mean is always zero. 

Example:

2. The sum of the square of the deviations of the items from their arithmetic mean is minimum : The sum of the square of the deviations of the items from their arithmetic mean is less than the sum of the square of deviations from any other value.

Example:

3. If each observation of a series is increased or decreased by a constant, then the arithmetic mean of the new series will also get increased or decreased by that constant respectively: For instance, the arithmetic mean of the series 4, 6, 2, 8, 10 is 6. If 2 is added to each item of the series, then the mean of the new series 6, 8, 4, 10, 12 will also get increased by 2; i.e., the new arithmetic mean will be 8.

4. If each observation of a series is multiplied or divided by a constant, then the arithmetic mean of the new series will also get multiplied or divided by that constant respectively: For instance, the arithmetic mean of the series 4, 6, 2, 8, 10 is 6. If 2 is divided by each of the items of the series, the mean of the new series 2, 3, 1, 4, 5 will also get divided by 2; i.e., the new arithmetic mean will be 3.

 5. Mean of the Combined Series: If the arithmetic mean of two or more related series is given, then it is possible to calculate the combined arithmetic mean of the series as a whole. 

6. If any two values out of arithmetic mean, number of items, or a total of the values are known, then it is possible to calculate the third value: The missing items, missing frequency, or correct mean can be determined with the help of the following formulas:

A single value used to symbolise a whole set of data is called the Measure of Central Tendency. In comparison to other values, it is a typical value to which the majority of observations are closer. The arithmetic mean is one approach to measure central tendency in statistics. This measure of central tendency involves the condensation of a huge amount of data to a single value. For instance, the average weight of the 20 students in the class is 50 kg. However, one student weighs 48 kg, another student weighs 53 kg, and so on. This means that 50 kg is the one value that represents the average weight of the class and the value is closer to the majority of observations, which is called mean. In real life, the importance of displaying a single value for a huge amount of data makes it simple to examine and analyse a set of data and deduce necessary information from it.