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Following are some of the important properties of arithmetic mean, which are elaborated with the help of simple problems.

Problem: A researcher conducted a research and got the observations: 50, 60, 65, 75, and 80. Using these observations explain the different properties of mean.

• The sum of the deviations of the observations from their mean is always zero.

In the above table we first calculated the mean (mean = 66) and then calculated the difference of each observation from the mean. As shown in the table the sum of deviation of these observations from mean is always zero.

• The sum of squared deviation of the observations from their mean is minimum

After taking square of deviations of the observations from their mean and adding those up it is found that sum of squared deviation of the observations from their mean is 570 which is less than 1850. We can also try other values such as 60, 65, 75 etc but sum of squared deviation of the observations from their mean will always be minimum.

• If we have one variable x and we transform this variable into y (by multiplying it with constant number and adding another constant number to it). The mean of this new variable y can be obtained by the same transformation of x bar.

It is evident from the above table that the mean (y bar) of transformed variable y can be calculated either by dividing the sum of total observations by total number or by transforming the x bar by multiplying it with 2 and adding 3 to it. This transformation is called a linear transformation.

• Mean of constant number is the constant

For example we have six constant observations e.g. 8, 8, 8, 8, 8, and 8. Now mean can be calculated by using the formula:

• The mean of a group of means called a combined mean is calculated as:

Problem: The average score of 5 batsmen is 30 runs in first match, 40 runs in second match and 25 runs in third match. Find the average score of these five batsmen in three matches.

Therefore the average score of 5 batsmen in three matches is 31.67 runs.