Give three mathematical properties of Arithmetic Mean. - Vidyadip

Question

Give three mathematical properties of Arithmetic Mean.

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Answer

Properties of arithmetic mean:

The sum of the deviations of the items taken from their arithmetic mean is zero. i.e. in an individual series, $\sum\limits^{\text{n}}_{\text{i}=1}(\text{x}_\text{i}-\overline{\text{x}})=0$ in a discrete or continuous series, $\sum\limits^{\text{n}}_{\text{i}=1}\text{F}(\text{x}_\text{i}-\overline{\text{x}})=0$
The sum of the squares of the deviations of the items taken from arithmetic mean is minimum.

i.e., $\sum\limits^{\text{n}}_{\text{i}=1}(\text{x}_\text{i}-\overline{\text{x}})^2$ is minimum
i.e., $\sum\limits^{\text{n}}_{\text{i}=1}(\text{x}_\text{i}-\overline{\text{x}})^2<\sum\limits^{\text{n}}_{{\text{i}=1}}(\text{x}_\text{i}-\text{A})^2,$

Where A is any arbitrary value.

If a series of n observations consists of two components having $n_1$ and $n_2$ observations $(n_1 + n_2= n)$, and means $\overline{\text{x}}_1$ and $\overline{\text{x}}_2$ espectively then the Combined mean of n observations is given by,

Combined mean $\overline{\text{x}}=\frac{\text{n}_1\overline{\text{x}}_1+\text{n}_2\overline{\text{x}}_2}{\text{n}_1+\text{n}_2}$

Mean has a mathematical property that if items are increased or decreased by same value, then mean also increases or decreases by same value.
Similarly, if all items of a series are multiplied or divided by same number, mean also gets multiplied or divided by same number.

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