### Chapter 1. Descriptive Statistics: Measures of Central Tendency

### Central Tendency and the Shape of a Distribution

Depending on the shape of the distribution, there exist some predictable relationships between the three measures of central tendency. In a *perfectly* *symmetric* distribution, the median and mean will have the exact same value.

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If a symmetric distribution is *unimodal*, then the mode will also be equal to the median and mean.

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Quite often, however, a distribution will exhibit some degree of skewness. Remember that the mean is a measure of centrality that is strongly affected by the extreme scores in the dataset. For this reason, the mean tends to be 'pulled towards' the tail-end of a skewed distribution.

For *continuous* data, the following guidelines are likely to hold for skewed distributions:

- In a
*positively skewed*distribution, the mean is larger than the median, and both of these measures are larger than the mode:- Mean > median > mode

- In a
*negatively skewed*distribution, the opposite relationship holds; the mean is smaller than the median, and both these measures are smaller than the mode:- Mode > median > mean

It should be noted that these guidelines for skewed data tend not to hold for *discrete *data.

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