How do you sum a set of order-of-magnitude values? For example, if my personal library contains E0 dieting books, E0 classical literature volumes, E1 textbooks, and E2 fantasy books, then how many books do I have total? Clearly the fantasy books dominate and I have E2 total books. So to get the sum total, I take the maximum E-value of the set?

Sometimes, but this is not true when there are many of the lower terms. For example let's look at the world population by country. To get a picture of it, we take the values from wikipedia and group them by their order-of-magnitude:

OOM | Count |
---|---|

E1 | 0 |

E2 | 1 |

E3 | 6 |

E4 | 13 |

E5 | 41 |

E6 | 44 |

E7 | 92 |

E8 | 41 |

E9 | 3 |

To get the contribution of each order-of-magnitude, we multiply with their count (i.e. add with their count's E-value):

OOM | Count | Contribution |
---|---|---|

E1 | 0 | 0 |

E2 | 1 | E2 |

E3 | 6 | E4 |

E4 | 13 | E5 |

E5 | 41 | E7 |

E6 | 44 | E8 |

E7 | 92 | E9 |

E8 | 41 | E10 |

E9 | 3 | E9 |

So now we take the maximum of the contributions and we get E10 from the E8 countries (take that, China, India, and U.S.A!), which is in fact ~ the world population at 7.8E9.

I wouldn't expect to use this technique all that much since it's a bit laborious for what I want, but it's nice to keep in mind and serves as a conceptual sanity check.

Follow-up: What are some pernicious examples? What is the process of binning by E-value? This probably has a common name.

NOTE: v ~ \(E(round(log_{10}(v)))\)