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

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Re: Standard Deviation Question
From: Robert Bernecky
Date: 2013 Jan 6, 10:22 -0800

>>I guess I still have the same question. When should we be using Pop and when should we be using Samp?

Marcel's answer to this question is very good. But here is more background, assuming normal distributions.

We wish to measure the index error of a sextant.

We take several measurements and get slightly different numbers. But (by assumption) the index error is not changing. The variation is due to "noise" in our measurement. In theory, we could take an infinite number of measurements and plot a histogram (# of counts at each value). If our measurements are "normally distributed", the plot will look like a bell curve. It will have a mean at the exact value of the index error, and a spread (standard deviation) that characterizes how noisy our measurement process is.

In other words, for this problem, the population (of an infinite number of measurements) is a theoretical concept. The nice property of this population distribution is that its mean is the number we want: the index error of the sextant.

We obviously do not have the infinite number of samples that make up the population, so we can't find the population's mean, so we can't find the index error. However, we can take many measurements of the index error and find the mean of that "sample". The sample mean will (almost) never match the population mean, but it will fall "near" the population mean (i.e. within a known interval about the population mean) with some confidence. To be specific, the interval defined by sample mean plus or minus two times the sample standard deviation will (with 95% confidence) contain the population mean.

To summarize, for problems of estimating a value using a noisy measurement process, the population is a theoretical, infinite number of samples. You will always be using "sample" statistics to estimate the population statistics.

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