## 4.1 Measures of Central Tendency

The three main measures of central tendency are mean, median, and mode. The mean reports the average value of a data set. Sometimes, it is also called arithmetic mean or average and can be expressed as follows: \[\bar{x} = \frac{1}{n} \sum_{i=1}^{N} x_i\]

The mean does not always explain data very well and there are some other measures to explore. Among them are the median and mode. The median divides the data set into two equal parts, such that half of all the values are greater than the median and half are smaller than the median. The middle term of the data set is the \((n+1)/2\)-th term. The median has the advantage that a very high or very low values do not influence the median. For example, the average taxable income in Switzerland is $56,805. The town Vaux-sur-Morges has an average taxable income of $670,046. It turns out that one very wealthy person (Andre Hoffmann from Roche Pharmaceutical) drives up the average in the village with 178 inhabitants. The mode is the value that occurs the most frequently. Consider the table below on the income distribution of ten citizens in three states. Note that the mean income in all three states is 10. Although, there is considerable variation among the citizens. The median is 10, 9.5, and 2 for state A, B, and C, respectively. The mode is 10, 10, and 2 for state A, B, and C, respectively.

A | B | C |
---|---|---|

10 | 2 | 2 |

10 | 3 | 2 |

10 | 7 | 2 |

10 | 8 | 2 |

10 | 9 | 2 |

10 | 10 | 2 |

10 | 10 | 2 |

10 | 14 | 2 |

10 | 17 | 2 |

10 | 20 | 82 |