Properties of the Mean – Explained with Examples

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In this article, we list the eight most important properties of the mean. The arithmetic mean of a given set of data values x_1,x_2,\ldots,x_n is denoted by the symbol \bar{x}.

1. If we add a constant to the given data set, then the new mean is obtained by adding the same constant to the old mean. For example, if we have a set of values 1,2,3, and 4 whose mean is 2.5 then, if we add 8 to each of the data values the new mean is equal to 8+2.5=10.5. This can be verified as follows: \text{New Mean }=\frac{9+10+11+12}{4} = \frac{42}{4} =10.25.
2. If we multiply the given data set by a fixed constant, then the new mean is obtained by multiplying the old mean by the same constant. For example, if we have a set of values 2,3,5, and 6 whose mean is 4 then, if we multiply each of the data values by 3 the new mean is equal to 3*4=12. This can be verified as follows: \text{New Mean }=\frac{6+9+15+18}{4} = \frac{48}{4} =12.
3. The sum of deviations of the items from the arithmetic mean is always zero. This means that given set of data values x_1,x_2,\ldots,x_n we have that: \sum_{i=0}^{n}(x_i-\bar{x})=0.
4. The mean is affected by the value of every item in the series.
5. It is reliable in the sense that it does not vary too much when repeated samples are taken from one and the same population.
6. Since the mean depends on each and every value of the series, it is greatly impacted by the presence of outliers. The outliers are extreme values which can make the mean either extremely large or extremely small.
7. The sum of squared deviations of the items from arithmetic mean is minimum, that is, less than sum of squares of deviations from any other value. This means that for any value ‘A’ we have that: \sum_{i=0}^{n}(x_i-\bar{x})^2\leq \sum_{i=0}^{n}(x_i-A)^2 . This is a very important property in regression analysis.
8. If we have two groups of sizes n_1\text{ and }n_2 and means \bar{x}_1\text{ and }\bar{x}_2 then, we can calculate the combined mean/grand mean using the formula, \bar{x}=\frac{n_1\bar{x}_1+n_2\bar{x}_2}{n_1+n_2}.

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