I shot four sets of 5 targets, per Kent's previous thread on the subject. I reduced them, meaner them, and calculated the Standard Deviation (The "Standard Error) for each observation, as shown in the spreadsheet. But I'm confused about the last steps of the calculation.
Is the overall standard deviation the average of the 5 different observations' standard error, or 4.4"?
> I shot four sets of 5 targets, per Kent's previous thread on the subject. I reduced them, meaner them, and calculated the Standard Deviation (The "Standard Error) for each observation, as shown in the spreadsheet. But I'm confused about the last steps of the calculation.
>
> Is the overall standard deviation the average of the 5 different observations' standard error, or 4.4"?
Oops. Spell check not working (or working too well)...."meaned" them....
Anyway, would it be more appropriate to take ALL of the readings (40 total), then sum the squares of ALL of them together (450)
And take the square root of that divided by N-1.
That comes out to 3.39" Is that a better way to determine the overall accuracy of the instrument?
You need to walk through his steps in the old thread again. First, I think he was finding the statistics of a D+R pair, so there are no left and right residuals.
Find the average angle for each point. That's what you subtract from your bold column to get residuals v. Then find v' so the mean of each set is 0. Then find the overall sum of squares and root.
However, something went badly wrong with target #3. I'd discard its data entirely and try to figure out if something made it move or if it is really an instrument problem. Without that target, I get 6.8 seconds as the standard error of a D+R pointing.
EDIT: Never mind. I ASSumed that your reduced angles were ok. Kent wisely recomputed them himself.
From your observations, the best estimate of the standard error of a direction taken as the mean of Face Left and Face Right with your instrument is +/-16". The high residuals on Target No. 3 suggest that something was problematic about the target, though. If Target No. 3 is rejected as an outlier, the standard error is significantly less than 16".
Here is how to calculate the standard error of a direction taken as the mean of Face Left and Face Right from those observations. I'm posting this before double-checking the arithmetic to give others something to do.
[Edit: on rechecking the arithmetic, I see that in Set 2 the mean direction to Target No. 3 was 135-28-00, not 135-31-00. There definitely was something wrong with Target No. 3. I'd recompute the standard error tossing Target No. 3, but ideally the test should be run with targets without whatever problem attached to Target No. 3}
[pre]
S e t 1
-----------------------------------------------
Obj. Reduced v v' Sum(v')^2
1 0-00-00 00 -0.4
2 76-41-50 -05 -5.4
3 135-31-05 +19 +18.6
4 217-34-05 -05 -5.4
5 289-57-15 -07 -7.4
------ -----
Mean +0.4 0.0 459.2
S e t 2
-----------------------------------------------
Obj. Reduced v v' Sum(v')^2
1 0-00-00 0 +3.6
2 76-41-55 -10 -6.4
3 135-31-00 +24 +27.6
4 217-34-20 -20 -16.4
5 289-57-20 -12 -08.4
------ -----
Mean -3.6 0.0 1155.2
S e t 3
-----------------------------------------------
Obj. Reduced v v'
1 0-00-00 0 -0.4
2 76-41-40 +5 +4.6
3 135-31-45 -21 -21.4
4 217-33-50 +10 +9.6
5 289-57-00 +8 +7.6
------ -----
Mean +0.4 0.0 629.2
S e t 4
-----------------------------------------------
Obj Reduced v v'
1 0-00-00 0 -3.4
2 76-41-35 +10 +6.6
3 135-31-45 -21 -24.4
4 217-33-45 +15 +11.6
5 289-56-55 +13 +9.6
------ -----
Mean +3.4 0.0 877.2
Mean directions from Sets 1 - 4
-----------------------------------------------
Obj Reduced
1 0-00-00
2 76-41-45
3 135-31-24
4 217-34-00
5 289-57-08
C o m p u t a t i o n o f S t a n d a r d E r r o r
-----------------------------------------------------------
Sum of (v')^2 = 3121
n = number of sets
t = number of targets
s.e. = SQRT [3121/((n-1)(t-1))]
= SQRT [3121/(3 x 4)]
= SQRT [3121/12]
= 16"
[/pre]
If the directions to Target No. 3 are considered to be blundered, the standard error computed from the remaining observations would be +/6.8".
The blunder to Target No. 3, however, indicates that some reexamination of either target design or technique is in order since test measurements should not include blunders.
[pre]
S e t 1
-----------------------------------------------
Obj. Reduced v v' Sum(v')^2
1 0-00-00 00 +4.25
2 76-41-50 -05 -0.75
3
4 217-34-05 -05 -0.75
5 289-57-15 -07 -2.75
------ -----
Mean -4.25 0.0 26.8
S e t 2
-----------------------------------------------
Obj. Reduced v v' Sum(v')^2
1 0-00-00 0 +10.5
2 76-41-55 -10 +0.5
3
4 217-34-20 -20 -9.5
5 289-57-20 -12 -1.5
------ -----
Mean -10.5 203.0
S e t 3
-----------------------------------------------
Obj. Reduced v v'
1 0-00-00 0 -5.75
2 76-41-40 +5 -0.75
3
4 217-33-50 +10 +4.25
5 289-57-00 +8 +2.25
------ -----
Mean +5.75 0.0 56.8
S e t 4
-----------------------------------------------
Obj Reduced v v'
1 0-00-00 0 -9.5
2 76-41-35 +10 +0.5
3
4 217-33-45 +15 +5.5
5 289-56-55 +13 +3.5
------ -----
Mean +9.5 0.0 133.0
Mean directions from Sets 1 - 4
-----------------------------------------------
Obj Reduced
1 0-00-00
2 76-41-45
3
4 217-34-00
5 289-57-08
C o m p u t a t i o n o f S t a n d a r d E r r o r
-----------------------------------------------------------
Sum of (v')^2 = 420
n = number of sets
t = number of targets
s.e. = SQRT [420/((n-1)(t-1))]
= SQRT [420/(3 x 3)]
= SQRT [420/9]
= 6.8"
[/pre]
Note that the reduced mean direction for target 3 changed only 5" between the 1st and 2nd set, and not at all between 3rd and 4th, but had a huge change between 2nd and 3rd.
The fact that the other directions didn't change nearly that much rules out the instrument moving (assuming measurements done in tabulated order).
This is suggestive, but not conclusive, that the target moved in the middle of the procedure. I would compute from distance*tan(42.5") how much of a movement that would be, and ask myself if there was anything going on that would explain it.
Like all good research projects, one of the conclusions must be "need more research".
> The fact that the other directions didn't change nearly that much rules out the instrument moving (assuming measurements done in tabulated order).
>
> This is suggestive, but not conclusive, that the target moved in the middle of the procedure. I would compute from distance*tan(42.5") how much of a movement that would be, and ask myself if there was anything going on that would explain it.
>
> Like all good research projects, one of the conclusions must be "need more research".
Well, in this case, the solution is fairly straightforward. It should go without saying that flat targets are a rule free of many errors inherent in targets of other shapes, particularly round targets. So just make up five targets and attach them to stable objects more than 100 ft. from the instrument station. Problem solved.
One easy target design is a piece of yellow corrugated plastic of the sort used in yard signs, cut into 6"x6" pieces, masking off a "V" with black tape, the apex of the Vee as the target object.
> Note that the reduced mean direction for target 3 changed only 5" between the 1st and 2nd set, and not at all between 3rd and 4th, but had a huge change between 2nd and 3rd.
I would look mainly at the residuals (v') column since there is nothing magical about the 0-00-00 direction. It is subject to random error as well.
Good point.
In this case, those numbers tell pretty much the same story.
> Good point.
>
> In this case, those numbers tell pretty much the same story.
My best guess would be that if the instrument is given a proper test, that the standard error of a direction (taken as the mean of F Lt and F Rt) will most likely be found to be closer to 6" than to 16". However, routinely obtaining that accuracy in field measurements will require more attention to target design, i.e. using flat, high contrast targets.
Error in Original Data Set
Okay, the original data set posted contains large arithmetic errors in the reductions of the observed directions to Target No. 3 in Set 2.
Assuming the Face Left and Face Right observations are correct, the correctly reduced mean directions give a standard error of +/-6.8" for a direction taken as the mean of F Lt and F Rt (the same value obtained above from all of the observed directions except for those to Target No. 3).
If that is the standard error of a direction, then an angle between two directions taken as the means of both faces will have a standard error of SQRT(2) x 6.8" = +/-9.6", or +/-10" for most practical purposes. I would use +/-10" in Star*Net for the standard error of angles measured with that same instrument (taken as the mean of both faces) until shown otherwise.
[pre]
S e t 1
-----------------------------------------------
Obj. Reduced v v' Sum(v')^2
1 0-00-00 00 +5.6
2 76-41-50 -05 +0.6
3 135-31-05 -11 -5.4
4 217-34-05 -05 +0.6
5 289-57-15 -07 -1.4
------ -----
Mean -5.6 0.0 63.2
S e t 2
-----------------------------------------------
Obj. Reduced v v' Sum(v')^2
1 0-00-00 0 +9.6
2 76-41-55 -10 -0.4
3 135-31-00 -6 +3.6
4 217-34-20 -20 -10.4
5 289-57-20 -12 -2.4
------ -----
Mean -9.6 0.0 219.2
S e t 3
-----------------------------------------------
Obj. Reduced v v'
1 0-00-00 0 -7.4
2 76-41-40 +5 -2.4
3 135-30-40 +14 +6.6
4 217-33-50 +10 +2.6
5 289-57-00 +8 +0.6
------ -----
Mean +7.4 0.0 111.2
S e t 4
-----------------------------------------------
Obj Reduced v v'
1 0-00-00 0 -8.4
2 76-41-35 +10 +1.6
3 135-30-50 +4 -4.4
4 217-33-45 +15 +6.6
5 289-56-55 +13 +4.6
------ -----
Mean +8.4 0.0 157.2
Mean directions from Sets 1 - 4
-----------------------------------------------
Obj Reduced
1 0-00-00
2 76-41-45
3 135-30-54
4 217-34-00
5 289-57-08
C o m p u t a t i o n o f S t a n d a r d E r r o r
-----------------------------------------------------------
Sum of (v')^2 = 551
n = number of sets
t = number of targets
s.e. = SQRT [551/((n-1)(t-1))]
= SQRT [551/(3 x 4)]
= SQRT [551/12]
= 6.8"
[/pre]
Error in Original Data Set...YES
> Okay, the original data set posted contains large arithmetic errors in the reductions of the observed directions to Target No. 3 in Set 2.
>
Absolutely correct.
The blunder was not in the targeting or instrument moving. I checked my field data and found that my errors are in the transfer of numbers from field book to excel spreadsheet. All the math of going from H.ms to H, then subtracting, then converting H back to H.ms is absolutely fraught with danger of mistakes. Set3 Target3 is "233", not "283" in the "Left" and "Right" columns, and I fixed Set2 Target3 minutes in the "Reduced" columns to be "31'", not "28'".
It's no wonder no one does this math by hand anymore, but instead rely on data collectors (possibly going directly into star net). My HP 11C is heating up with all the stupid arithmetic.o.O
That said, I'm uncertain why the type of target affects the results, unless you're saying it needs to be sharp enough to pick the precise point every time. My targets consisted of a survey stake and painted finish nail:
I may have been too close, though. You suggest 100' or more; I was targeting at 20'. The double recticle lines in the Topcon were almost exactly as far apart as the nails were wide. I thought I got the dead center of the nail every time.
But for further testing and analysis (which I intend to do), I've GOT to solve the workflow problem, or I'm going to drive myself crazy.
What software do you use, and how do you get all your numbers from instrument to paper?
Where does v' come from?
> Find the average angle for each point. That's what you subtract from your bold column to get residuals v. Then find v' so the mean of each set is 0. Then find the overall sum of squares and root.
S e t 1
-----------------------------------------------
Obj. Reduced v v' Sum(v')^2
1 0-00-00 00 -0.4
2 76-41-50 -05 -5.4
3 135-31-05 +19 +18.6
4 217-34-05 -05 -5.4
5 289-57-15 -07 -7.4
------ -----
Mean +0.4 0.0 459.2
I'll chalk this one up to being super dense, but if I don't ask, I can't learn:
Where the heck does v' come from?
Like for: 1 0-00-00...v' = -0.4.
How are these calculated?
Error in Original Data Set...YES
> That said, I'm uncertain why the type of target affects the results, unless you're saying it needs to be sharp enough to pick the precise point every time. My targets consisted of a survey stake and painted finish nail:
>
>
My original comment about targets came from what appeared to be a blunder in the directions to Target No. 3. That was before I realized that there was just an arithmetic mistake in the reduction of the observed directions. I think that the final standard error of a direction of +/-6.8" is probably fairly realistic for that instrument. That means that angles measured with the instrument, taken as the mean of both faces, will have a standard error of +/-10".
> I may have been too close, though. You suggest 100' or more; I was targeting at 20'. The double recticle lines in the Topcon were almost exactly as far apart as the nails were wide. I thought I got the dead center of the nail every time.
The reason why round objects generally do not make the best targets is that they tend to have a highlighted side which makes bisecting the target more difficult than with a good flat target. My sense
The object in testing is to use as close to optimal conditions as possible, to determine what the limits of the instrument's accuracy are.
> What software do you use, and how do you get all your numbers from instrument to paper?
Most surveyors, myself included, use electronic data loggers to record measurements and to transfer them to software such as Star*Net for further computations. For a test such as this, though, I'd just log it in a field book and reduce the directions in the field book.
Where does v' come from?
[pre]
S e t 1
-----------------------------------------------
Obj. Reduced v v' Sum(v')^2
1 0-00-00 00 -0.4
2 76-41-50 -05 -5.4
3 135-31-05 +19 +18.6
4 217-34-05 -05 -5.4
5 289-57-15 -07 -7.4
------ -----
Mean +0.4 0.0 459.2
[/pre]
> Where the heck does v' come from?
> How are these calculated?
The best estimates of the true values of the directions to the targets are obtained by taking the grand means of Sets 1 through 4. The residuals, v, are then the grand means - the value for the set.
In the case of Set 1, the grand means were:
[pre]
Mean directions from Sets 1 - 4
-----------------------------------------------
Obj Reduced
1 0-00-00
2 76-41-45
3 135-30-54
4 217-34-00
5 289-57-08
[/pre]
So subtracting the reduced directions (mean of F Lt and F Rt) for Set 1 from those grand means that are the best estimates of the true values of the directions to each target gives the residuals, v.
However, the mean of that column of residuals, v, is +0.4". So, the residuals are reduced by adding -0.4" to each to bring their mean to 0.0. These reduced residuals are in the v' column. That last operation is equivalent to making the mean of the directions to targets 1-6 in Set 1 the same as the mean of the directions taken as the means of Sets 1 - 4, but is easier to handle arithmetically.
Target distance.
A reason for using the greatest convenient distance is to minimize the effects of anything moving.
If, for instance, a tripod leg sinks enough to move the instrument a mere 1/32 inch sideways, the direction to a target at 50 ft will read 10 seconds different.
Also remember that with care and practice, surveyors of prior times did every bit of calculation with pencil, paper, and function tables and got reliable results.
As a geezer myself, I was on the cusp of the change. I went through high school trig and engineering college with a slide rule and tables. The first scientific pocket calculator appeared on campus shortly before I graduated. I have now been using calculators, spreadsheets, and other applications so long that I'm slow at mental math but I try to retain the ability. I'm slightly embarrassed to admit that after adding up my checkbook (another geezer item) I use a spreadsheet to check my math.
TS Direction Accuracy - Two Questions
>
One thing that I should have asked about is how the direction measurements were made. For example, the Face Left and Face Right directions to Target No. 1 are identical for all four sets. That suggests that the instrument was rezeroed and set to a particular value to begin the Face Right series.
In my opinion, best practice would have been not to do that, but to just change face and *read* the direction to Target No. 1 and the others. The directions will differ by nominally 180 degrees, but that should not present any problem. I would expect that there are errors introduced in the rezeroing and resetting between faces and it is easy enough to test whether the accuracy is improved by omitting that bit.
The other question is how the circle was advanced between sets as would appear from the observations. Was the circle actually rotated by the increments implied by the changes in the Face Left directions to Target No.1 from set to set? Was the centering of the instrument over the ground mark altered in any wasy in the process?
Estimation of Variance
Bill, possibly you can give a better explanation of the process of estimating the variance of the five directions in the four sets than I have. The essential part of the problem is, of course, how to compare the directions in each set to the best estimates of their true values (the grand means).
That would be done by reducing the sets so that the average of the five directions is equal to the average of the five mean directions, but that is arithmetically heavier to do. Reducing the raw residuals, v, to yield a set, v', whose mean is 0 gives the same result, but much more painlessly, arithmetically speaking.
Error in Original Data Set...YES
The whole number might not matter. It's possible you could truncate everything to the seconds or possibly convert everything to seconds (ie 1 degree equals 3600 seconds) and work from there. Nails are tricky targets too because if it is not perfectly plumb and you sight a little higher or lower you'll introduce error. Flat targets on stationary objects like a building wall with a clear horizontal and vertical target just bold enough to resolve in the scope would be ideal.
Error in Original Data Set...YES
> The whole number might not matter. It's possible you could truncate everything to the seconds or possibly convert everything to seconds (ie 1 degree equals 3600 seconds) and work from there.
Well, if a person wants to avoid degrees-minutes-seconds notation, the obvious choice is just to set the instrument to read in grads. You can calculate the s.e. in centicentigrads (cc), just as you would for arc-seconds, and then convert the result to seconds.