In a previous thread, I mentioned that the whole series of latitude observations that Major William Emory made to determine the astronomical latitude of a monument he established in 1852 on the US side of the Rio Grande opposite the town of Presidio del Norte presently known as Ojinaga in the State of Chihuahua, Mexico could be divided into two groups. These are the observations that were reduced using a better star catalog with more accurate values of the declinations of the stars that Emory observed and the observations that were reduced using the almanac that he had with him in the field when his work was conducted. I've called them Groups A and B and have treated them differently because one set of latitude determinations (Group B) is noisier and less certain than the other.
Group A (n=66)
Group B (n=51)
In the case of the Group A observations, Emory simply corrected the means of the latitude of his station as derived from multiple observations on the same pair of stars over various nights by applying corrections to the more accurate values of the declinations of the stars he observed to the mean values of the latitudes per each star pair that he had calculated in the field while engaged on the survey.
In the table below, I've applied the corrections that Major Emory calculated to each observation tabulated for each star pair instead of just the means.
Group A (corrected with Emory's improved declination values)
Values in parenthesis were rejected as outliers.
standard error for 66 observations (less outliers), s = 1.35”
Mean of all observations (less outliers) = 7.34”
standard error of the mean = 1.35/SQRT(66) = 0.17”
Group B (n=51)
For this set of 51 latitude observations:
mean = 6.58”
standard error for 51 observations (no outliers), s = 2.15”
standard error of the mean = 2.15/SQRT(51) = 0.30"
For : s(3), s(4) .... s(26) = the standard errors derived from the series of latitude observations
for each star pair (3) through (26) over various nights as tabulated above and
n(3), n(4) ... n(26) = the numbers of latitude observations for those star pairs over various nights
then, the pooled estimate, s(pooled) =
s(pooled) = 0.75”
This is the estimated standard error of a single latitude determination from a pair of stars observed on one night, exclusive of declination errors in the ephemeris.
The method supposes that the random errors in all observations are similarly distributed
but that the remaining declination errors in the star catalogue used to reduce the observations
result in small systematic errors for the results of each star pair.
So, the two groups give different means, each with a different uncertainty (standard error). The next step is to decide how or whether to combine the two results from the two different groups.
Okay, so just to summarize all that discussion above: all of the latitude observations appear to have the same random errors in the angle measurements. The pooled estimate of the standard error of a single latitude (exclusive of ephemeris errors) for Group A is 0.73" and for Group B, 0.75", basically identical values.
Once (apparently random) ephemeris errors are added to the results, however, the apparent standard error of a single latitude determination for Group A (with the more accurate ephemeris) is 1.35" and for Group B is 2.15".
These are not trivial differences. Obviously, Group B is a lesser class of data than Group A.
Mean of Group A:
7.34" +/- 0.17" s.e.
Mean of Group B:
6.76" +/- 0.30" s.e.
The options are (a) form a weighted mean from Group A and Group B or (b) toss out all of the Group B observations since the apparent ephemeris errors are so large (twice those of Group A).
Erroneous Assumption, That The Data Is ''Less Accurate''
When it is merely uncorrected.
Your mean of "less accurate data" is skewed by leaving in outlying observations.
Pair 7 at 4.60 and pair 9 at 3.49 are clearly outside the normal range. Removing those 2 pairs and your "less accurate data" yields a mean of 7.33. I say that it appears that the intitial corrections for the "less accuate data" was acceptable except for pairs 7 and 9.
It appears that you may know the actual stars in the pairs, so verify those 4 star corrections first.
Paul in PA
Yes, some of the latitudes are less accurate
> Pair 7 at 4.60 and pair 9 at 3.49 are clearly outside the normal range. Removing those 2 pairs and your "less accurate data" yields a mean of 7.33. I say that it appears that the intitial corrections for the "less accuate data" was acceptable except for pairs 7 and 9.
No, there clearly is no problem with the observations on star pairs 7 and 9 other than the inaccuracies in the results due to erroneous declinations having been used for those pairs. The way to properly evaluate those observations as being outliers or not is to compare the scatter of the series of observations *on each pair* to the rejection limits formed using s = 0.74" around the mean of just the series on that particular pair. This is how the ephemeris errors are removed from that analysis.
To tie the threads together:
[msg=41521]First description and pictures[/msg]
[msg=41725]Prior computatons[/msg]
The mean of the ephemeris corrections was -0.32" and its std dev 1.17"
That seems to me to be insufficient evidence for a systematic bias, so I'd just do the weighted average.
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BTW, that is an interesting trick Wendell does with the links. I pasted the complete URLs for the other threads and his software turned them into a brief reference to message numbers. Then when the message is posted, the links look like original but he has added an icon ahead of them.
Outside The Range Of Results
Yes, I agree that the data is not in itself bad, just that ephemris correction is incorrect. Therefore if you know the star pairs and get bettner corrections for those dates you can add those observations back in.
Paul in PA
> The mean of the ephemeris corrections was -0.32" and its std dev 1.17"
>
> That seems to me to be insufficient evidence for a systematic bias, so I'd just do the weighted average.
Actually, the mean of the ephemeris corrections in the Group A observations was -0.01", basically 0. That is what does lead me to think that the ephemeris errors probably have zero mean also, i.e. no systematic bias across the entire set of Group B observations.
Outside The Range Of Results
> Yes, I agree that the data is not in itself bad, just that ephemris correction is incorrect.
So what that means is you *don't* treat the observations as outliers that you suggested were.
The Incorrect Correction Makes The Pair Set Of Range
Let us just say the correction puts it out of range of the other pair sets.
Until one corrects or verifies the correction they should be excluded from the solution.
Paul in PA
The Incorrect Correction Makes The Pair Set Of Range
> Let us just say the correction puts it out of range of the other pair sets.
>
> Until one corrects or verifies the correction they should be excluded from the solution.
Actually, if you look at the series on the pair, things look fine, i.e. there is nothing abnormal about the scatter.
Naturally, there are larger declination errors in the Group B observations. There is plenty of evidence for that in the analysis of variance.
Read My Lips Kent
Yes the data has a reasonable range unto itself. The applied correction makes it unsuitably out range with the other corrected data.
Ergo, don't use it.
Paul in PA
Read My Lips Kent
> Yes the data has a reasonable range unto itself. The applied correction makes it unsuitably out range with the other corrected data.
>
> Ergo, don't use it.
Actually, if you can evaluate the standard errors of the means of the Group A and B observations, there isn't particularly any reason to exclude the Group B observations. Just because they have different uncertainties doesn't mean they can't be both used in an estimate. It is just a weighting problem.
Read My Posts Kent
I have not excluded group B as you call it, I have excluded pairs 7 and 9 and the Aug. 12 observation for pair 22.
Do the math.
Paul in PA
Selectively rejecting outliers
> I have not excluded group B as you call it, I have excluded pairs 7 and 9 and the Aug. 12 observation for pair 22.
Ah, sorry, I misread what you posted. So you want to throw out all of the observations on star pairs 7 and 9, even though they aren't outliers in the Group B data.
The problem with selectively rejecting outliers, particularly on only one side of the distribution without a sound reason, is that you can easily end up with a biased estimate. You can "steer" the estimate to some subjectively determined value unless you are very careful.
If you reject the star pairs giving means on the high side, i.e. pairs 25 and 26, lo and behold the mean of the means is nearly identical (6.93") to the mean that the means of all of the pairs with no rejections would yield (6.90"). So far, I haven't heard a good reason to consider all of the observations on 7 and 9 to be outliers.
The Observations Are Not Outliers
The corrected means for 7 and 9 are outside the range for the totality of the corrected and uncorrected means. It might seem logical to require outliers to be on both sides of the mean, but that does not always happen.
Because the uncorrected group mean with 7 and 9 removed is in agreement with the corrected group mean it is logical to assume that the uncorrected group except 7 and 9 did not require further updating. There is not enough information to determine if updated corrections for 7 and/or 9 would be suitable.
Paul in PA
The Observations Are Not Outliers
> The corrected means for 7 and 9 are outside the range for the totality of the corrected and uncorrected means.
I'm sure that you'll be able to demonstrate the truth of that proposition if that is so. The mean of the Group B observations (including those in the series on the two star pairs that you claim to think are outliers, regardless of whether you want to call them that or not) is actually not inconsistent with the mean of the Group A observations when you take the uncertainty of each into account, which is part of the point of the exercise. Unless I'm misunderstanding you, you're making the mistake of expecting the scatter of the Group B observations to resemble that of Group A reduced with a different, more accurate ephemeris.
> Because the uncorrected group mean with 7 and 9 removed is in agreement with the corrected group mean it is logical to assume that the uncorrected group except 7 and 9 did not require further updating. There is not enough information to determine if updated corrections for 7 and/or 9 would be suitable.
No, that's not logical at all when you take the actual uncertainties of the means into account. You're simply making a guess instead of actually evaluating the two different data sets on their own terms. When I get a chance, I'll work the problem more completely to show you where you went wrong.
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THANKS FOR CONTINUING THE THREAD
