Given a Formula with an Absolute Error in of , the Absolute Error is . The
Relative Error is . If , then

(1) |

(2) |

The definitions of Variance and Covariance then give

(3) | |||

(4) | |||

(5) |

so

(6) |

(7) |

Now consider addition of quantities with errors.
For ,
and
, so

(8) |

For division of quantities with
,
and
, so

(9) |

(10) |

For exponentiation of quantities with

(11) |

(12) |

(13) |

(14) |

(15) |

For Logarithms of quantities with
,
, so

(16) |

(17) |

For multiplication with
,
and
, so

(18) |

(19) |

For Powers,
with ,
, so

(20) |

(21) |

**References**

Abramowitz, M. and Stegun, C. A. (Eds.).
*Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing.*
New York: Dover, p. 14, 1972.

Bevington, P. R. *Data Reduction and Error Analysis for the Physical Sciences.* New York:
McGraw-Hill, pp. 58-64, 1969.

© 1996-9

1999-05-25