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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: Marcel Tschudin
Date: 2013 Jan 6, 13:53 +0200
I guess I still have the same question. When should we be using Pop and when should we be using Samp?

I mentioned before that the difference relates to whether the mean value is exact or an estimate, and this depends on how one intends to interpret the measured values. I try to explain this using Greg's 20 measurements of the index error having a mean value of <x>=0.575'.

Case "Sample":
The index error was measured with the intention to correct other measurements (made under same conditions). With "other" I mean previously or in future made ones. The sample of 20 measurements were made to estimate the unknown index error knowing that it would require an indefinite number of such measurements (an indefinite large population) to obtain an exact mean value <x>. The mean of the 20 measurements is therefore an estimate and the corresponding standard deviation is Sx(Sample)=+/-0.159'. This result lets us expect that the index error in other measurements have the estimated mean of 0.575' and are with a probability of 68% between 0.416' and 0.734'.

Case "Population":
In this case we look exclusively at the 20 measured values and at the statistics these values (only) imply. Relating only to these 20 values then these measurements represent the total population and the mean of 0.575' is exact for this population. The corresponding standard deviation is Sx(Pop)=+/-0.155'. The results indicate that the 20 measurements (only) have an (exact) mean of 0.575' and that their values (only) scatter around the mean with a probability of 68% being within the range 0.420' and 0.730'.

Note that it is wrong to use the Population results for the prediction of other measurements because other measurements did not belong to the initially selected population. This error (the difference between the two cases) is largest with small numbers of measurements and reduces with an increasing number of them. For the 20 measurements the two standard deviations differ only by 2.6% (0.159' vs. 0.155'), and for an indefinite large population the difference would vanish, the two versions would provide the same result.

Hopefully both, the content and my English are understandable.

Marcel
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