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Re: On potential error introduced by rounded values
From: Peter Fogg
Date: 2005 Jan 12, 16:29 +1100

"I don't see how he has calculated that number"

There are six chances that a number rounded to, say, 24, could have been rounded from: 23.5, 23.6, 23.7, 23.8, 23.9, 24.0 - so one chance in six that the number was 23.5. As there are 6 values in the sight reduction table, the probability of a maximum error of 3.0 (6 numbers of something.5 each) is 1 in 6 to the power of six, 0.000021433 expressed as a decimal. Thus a negligible chance. Quoting a remote possibility of error as a major shortcoming to be taken seriously is what I understand by disingenuous - just to further merge not so very dissimilar topics.

"We have just agreed that the multiple roundings in the calculation we have considered are unlikely to exceed 2'. Even so, that could imply that the true position may not be in the unique 1' rectangular box that was the stated result of the calculation, but could quite likely be in one of the four adjacent 1' boxes that surround it, and with a smaller chance, in one of the adjacent 1' boxes that surround that. That was just the result of the agreed rounding errors in the calculation process, and errors in the observation itself haven't even been considered yet."

Oh dear, George, you need to read the argument carefully before rushing into print.

" ...even if the fix is entirely accurate). It means that the position is somewhere within a rectangle..." to quote myself.

We are assuming that this fix is accurate. Being accurate means that the position is somewhere within the rectangle indicated by the fix.

"I don't follow what Peter is saying here (or if I do, then I disagree)."

Wonderful stuff! Disagree first, understand later!

-----Original Message-----

From: Navigation Mailing List [mailto:NAVIGATION-L@LISTSERV.WEBKAHUNA.COM] On Behalf Of George Huxtable

Sent: Wednesday, 12 January 2005 12:32 PM

Subject: Re: On potential error introduced by rounded values

There are two separate topics merged into Peter Fogg's recent pair of mailings, for both of which George Bennett's "Complete on-board Celestial Navigator has provided examples.

The topics are

1. The effect of multiple roundings on precision of the result, which I will respond to here under the threadname "On potential error introduced by rounded values."

2. The defects of the formula for obtaining azimuth from its sine and the errors that it gives rise to in Bennett's lookup table for azimuth, which I'll deal with elsewhere under the threadname "Sight Reduction Formula Question".

I wrote-

>"If we take 6 such roundings, then

>

>in theory the maximum amount they could possibly (but most unlikely)

>

>contribute to an error in position is ±3 miles..."

and Peter Fogg replied-

>And the chance of that most unlikely event is one in six to the power

>of six, or 0.00002, or 0.002%?

I don't see how he has calculated that number, and perhaps Peter will explain. The probability of the rounding errors combining to exceed ±3 miles is precisely zero. And the probability of the error being exactly 3 miles is also zero.

I think that there's a conceptual problem here. For a contiuous distribution such as we're discussing, the probability of the error being ANY exact number is always zero. You have to express it as the probability of being in some bracket between two values, or exceeding some limit, for the probability to have meaning.

>If this is correct, the results from models similar to the Excel sample

>are unlikely to show an error result of more than 2 minutes of arc in a

>significant number of cases.

I don't think it is correct, but even so, I agree with that conclusion.

> When a

>fix is expressed to whole minutes of arc it doesn't mean the boat is at

>that intersection (it is almost certainly not, even if the fix is

>entirely accurate). It means that the position is somewhere within a

>rectangle (a square at the equator) bounded by the halfway points to

>the next intersection of minutes of arc of lat/long.

I don't follow what Peter is saying here (or if I do, then I disagree). The true position isn't necessarily in that one rectangle. You could only state with certainty that the true position was in that one 1' rectangle if you knew that the fix was in itself entirely accurate, in terms of both observation and calculation, except for some final single rounding operation that expressed the result to the nearest whole minute.

We have just agreed that the multiple roundings in the calculation we have considered are unlikely to exceed 2'. Even so, that could imply that the true position may not be in the unique 1' rectangular box that was the stated result of the calculation, but could quite likely be in one of the four adjacent 1' boxes that surround it, and with a smaller chance, in one of the adjacent 1' boxes that surround that. That was just the result of the agreed rounding errors in the calculation process, and errors in the observation itself haven't even been considered yet.

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.

================================================================

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