# NavList:

## A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding

**Re: Instumental error?**

**From:**Gary LaPook

**Date:**2005 Apr 20, 14:24 -0700

The standard deviation of a
probability distribution is defined as the square root of
the
variance

(1) | |||

(2) |

where is the mean, is the second raw moment, and denotes an expectation value. The variance is therefore equal to the second central moment (i.e., moment about the mean),

(3) |

The square root of the sample variance
of a set of *N* values is the sample standard deviation

(4) |

The sample standard deviation distribution is a slightly complicated, though well-studied and well-understood, function.

However, consistent with widespread inconsistent and ambiguous terminology, the square root of the bias-corrected variance is sometimes also known as the standard deviation,

(5) |

Given only a sample of values *x*_{1},...,*x*_{n}
from some larger population, many authors
define the *sample standard deviation* by

A slightly faster way to compute the same number is given by the formula

The reason for this definition is that *s*^{2} is an unbiased estimator for the variance
σ^{2} of
the underlying population. (The derivation of this equation assumes
only that the samples are uncorrelated and makes no
assumption as to their distribution.) Note however that *s*
itself is *not* an unbiased estimator for the standard
deviation σ; it tends to underestimate the population standard
deviation. Although an unbiased estimator for "s" is known,
the formula is overly complicated and amounts to a minor correction.
Moreover, unbiasedness, in this sense of the word, is not
always desirable; see bias (statistics). Some have even
argued that the difference between *n* and *n* − 1 in the
denominator is overly complex and trivial and thus
exclude it. Without that term, what is left is the simpler expression

George Huxtable wrote:

Alex wrote-I measure some distance several times, say 7 times in 10 minutes. A star-to star distance can be considered constant for this period. The average is the arithmetic average. (x_1+x_2+x_3+x_4+x_5+x_6+x_7)/7 in this case. Here x_1...x_7 are my sextant readings. The standard deviation. I subtract the average from each x_j, square the difference add the results, divide by 6 and extract the square root. (Actually the calculator does it all for me).==================== One thing puzzles me. Why, in finding the mean of 7 squared-deviations from the average, does Alex (or his calculator) sum them and then divide by 6 rather than by 7? George. ================================================================ contact George Huxtable by email at george@huxtable.u-net.com, by phone at 01865 820222 (from outside UK, +44 1865 820222), or by mail at 1 Sandy Lane, Southmoor, Abingdon, Oxon OX13 5HX, UK. ================================================================