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Re: On potential error introduced by rounded values
From: George Huxtable
Date: 2005 Jan 12, 01:32 +0000

```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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