Motivating Properties
The arithmetic mean has several properties that make it useful, especially as a measure of central tendency. These include:
- If numbers have mean X, then . Since is the distance from a given number to the mean, one way to interpret this property is as saying that the numbers to the left of the mean are balanced by the numbers to the right of the mean. The mean is the only single number for which the residuals defined this way sum to zero.
- If it is required to use a single number X as an estimate for the value of numbers, then the arithmetic mean does this best, in the sense of minimizing the sum of squares (xi − X)2 of the residuals. (It follows that the mean is also the best single predictor in the sense of having the lowest root mean squared error.)
- For a normal distribution, the arithmetic mean is equal to both the median and the mode, other measures of central tendency.
Read more about this topic: Arithmetic Mean
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