# Centrist bias

**Centrist bias** is a political spectrum statistic that measures the tendency for elected candidates to be near the mean voter.

## Definition

The centrist bias of a set of winning candidates with respect to a given political spectrum is equal to Pearson's correlation coefficient between the distance of a voter from the mean voter and the distance of that voter from the nearest winning candidate.

## Example

Assume a one-dimensional political spectrum with the voter distribution

- 15% at position 0
- 20% at position 0.25
- 30% at position 0.5
- 20% at position 0.75
- 15% at position 1

If the candidate set {0.25, 0.75} is elected, then

voters | position | distance from mean | nearest winner | distance from winner |
---|---|---|---|---|

15% | 0.00 | 0.50 | 0.25 | 0.25 |

20% | 0.25 | 0.25 | 0.25 | 0.00 |

30% | 0.50 | 0.00 | either | 0.25 |

20% | 0.75 | 0.25 | 0.75 | 0.00 |

15% | 1.00 | 0.50 | 0.75 | 0.25 |

To compute the centrist bias, we consider the (distance from mean, distance from winner) pairs:

- (0.50, 0.25) with frequency 30%
- (0.25, 0.00) with frequency 40%
- (0.00, 0.25) with frequency 30%

The correlation, and thus the centrist bias, is zero.