Now we explain the three different ways of defining the center of a distribution. All three are called measures of central tendency.

**Balance Scale**

One definition of central tendency is the point at which the distribution is in balance. Figure 2 shows the distribution of the five numbers 2, 3, 4, 9, 16 placed upon a balance scale. If each number weighs one pound, and is placed at its position along the number line, then it would be possible to balance them by placing a fulcrum at 6.8.

**Smallest Absolute Deviation**

Another way to define the center of a distribution is based on the concept of the sum of the absolute differences. Consider the distribution made up of the five numbers 2, 3, 4, 9, 16. Let's see how far the distribution is from 10 (picking a number arbitrarily). Table 2 shows the sum of the absolute differences of these numbers from the number 10.

Values | Absolute difference from 10 |
---|---|

2 | 8 |

3 | 7 |

4 | 6 |

9 | 1 |

16 | 6 |

Sum | 28 |

The first row of the table shows that the absolute value of
the difference between 2 and 10 is 8; the second row shows
that the difference between 3 and 10 is 7, and similarly for
the other rows. When we add up the five absolute
differences, we get 28. So, the sum of the absolute
differences from 10 is 28. Likewise, the sum of the absolute
differences from 5 equals

We are now in position to define a second measure of central tendency, this time in terms of absolute differences. Specifically, according to our second definition, the center of a distribution is the number for which the sum of the absolute differences is smallest. As we just saw, the sum of the absolute differences from 10 is 28 and the sum of the absolute differences from 5 is 21. Is there a value for which the sum of the absolute difference is even smaller than 21? Yes. For these data, there is a value for which the sum of absolute deviation is only 20. See if you can find it. A general method for finding the center of a distribution in the sense of absolute difference is provided in the document Absolute Differences Simulation

**Smallest Squared Deviation**

We shall discuss one more way to define the center of a distribution. It is is based on the concept of the sum of squared differences. Again, consider the distribution of the five numbers 2, 3, 4, 9, 16. Table 3 shows the sum of the squared differences of these numbers from the number 10.

Values | Squared differences from 5 |
---|---|

2 | 9 |

3 | 4 |

4 | 1 |

9 | 16 |

16 | 121 |

Sum | 151 |

The target that minimizes the sum of squared differences provides another useful definition of central tendency (the last one to be discussed in this section). It can be challenging to find the value that minimizes this sum. We'll show you how to do it in the upcoming document Squared Differences Simulation