Activity Feed › Discussion Forums › Strictly Surveying › Simple angle error/sig digit question

Simple angle error/sig digit question
Posted by am95405 on February 13, 2019 at 4:42 amSurvey student here ??.Brown’s Evidence and Procedures for Boundary Location, sixth edition.
On page 150 he states, “Built into the expression of any numerical value is an error of half the magnitude of the unit of that last significant figure.”
Then on page 153, he has this example:
“An angle of 5 1/2 degrees is readily translated by surveyors, attorneys, and the courts to 5 degrees and 30 minutes. In evaluating and applying evidence, this is not so. Applying significant numbers 5 1/2 degrees ranges from 5 3/8 to 5 5/8 degrees; while 5 degrees and 30 minutes ranges from 5 degrees 29.5 to 5 degrees 30.5 minutes, or a much smaller spread. “So how does sig figure apply to 5 ?«? Would it not be 5 ?« + ?¬ = 5 ?? and 5 ?« – ?¬ = 5 ?¬ ?
I understand and appreciate the concept of paying attention to sig digits, especially when converting units (chains to ft and so forth), but am not following the math for the angle example. I can see why the 30 minutes becomes 29.5 to 30.5, but not how the 1/2 goes to 3/8 and 5/8. I expected 1/4 and 3/4. Is it a simple typo?
aharris replied 5 years, 7 months ago 10 Members · 15 Replies 
15 Replies

I think that the text is conceptualizing it this way: 1/2 of 1/2?ø is 1/4?ø. 1/4?ø of range is then normalized (centered) around 5 1/2?ø (notice that the range between 5 3/8?ø and 5 5/8?ø is 1/4?ø). All that being said, do what you gotta do to get an A in your class, but you’ll find much better resources for statistics than Brown.

When I see a description of 5 1/2 degrees I cannot assume the precision is to the nearest 1/2 degree, as usually the case is to the nearest 1/4 degree, since the next angle may be just as well as 5 degrees as 5 1/4 or 3/4 degrees. If I see 5?ø 30′ I can assume little, as it may have been a 5′ or 10′ reading circle. I may be able to extrapolate a value after seeing a much longer series of descriptions.
Paul in PA

Significant digit disagreements have been at the heart of many a test score argument in physics, computer science and a host of other courses. In physical measurements, two things are at play: the limitations of the equipment and computational mathematics.
From the computational viewpoint, I’ve always looked to rounding rules for guidance. For example, 5.45 and 5.5499999… both round to 5.5. The number 5.5 was given to only one decimal point, so we assume that it is exact within the limits of that one decimal point. Thus, we can say that the true result lies here: 5.45 <= 5.5 < 5.55. Had the number been given as 5.50, our limits would be 5.495 <= 5.50 < 5.505.
With 5 1/2, the situation is more complicated. First, in computational math, fractions are exact. You can test this crudely with your HP 50g. It has an exact mode that returns fractions as answers and an approximate mode that returns decimal answers. Second, we know that 5 3/8 < 5 1/2 < 5 5/8, but generalizing from the 5.5 example, it’s also true that 5 9/20 < 5 1/2 < 5 11/20. Any number of denominators can be used to narrow or widen the range of answers.
What Brown did here was to assume a standard deviation. Notice that changing the number from 5.5 to 5.50 implies a smaller standard deviation.
When you add instrument and personal errors, things get more complicated. but as @FrozenNorth said, there are better references on this subject than Brown.

What was the standard for the equipment used in the day? What were the other angles listed to?
51/2.
I would be happy with anything between 5 and 6 but I would not expect it. How does the figure close?
But you are likely looking at an out of context example. Brown seems to have used 1/8 graduation on the compass. How would you know?
So… where are the monuments…

Thank you all. This wasn’t a question on a test or related to grade. In any other book, I would have let it go. It’s just that Brown’s book seems to be so wellknown I thought the chances of a mistake are small. The idea of a 1/4 as the spread makes sense (that was my instructor’s thought), except it’s not applied to the 30 second the same way.
I asked because I thought maybe there is something unique in surveying/use of surveying equipment and the 1/8 th. spledues’ comment on a compass’ 1/8 graduation may be the reasoning here. Brown’s book just gives this as an example, and no other information is supplied.

Posted by: MathTeacher
What Brown did here was to assume a standard deviation. Notice that changing the number from 5.5 to 5.50 implies a smaller standard deviation.
Isn’t it that in his case this doesn’t work? From 5 1/2 to 5 30 he shows a larger standard deviation, even if you accept the 3/8 and 5/8 (1/4 degree) and 30.529.5 (1 degree).

I would echo spledeus. Interpret things in context. If angles/distances in a document appear to be mostly expressed to some precision and that is compatible with practices at the time and place it was generated, go with that interpretation of the questionable one.
An example where context might indicate otherwise would be a 90 degree angle (not given as 90 00) on a subdivision plat. It was probably held as exactly as computations allowed even though other angles were rounded to whole minutes or whatever.
. 
@am95405 Maybe?? Interpreting 5 1/2 degrees as 5d 30m gives an interval of 5d 29.5m <= 5d 30m < 5d 30.5m, which is 1 minute wide or 1/60 of a degree. That’s significantly narrower than the 1/4 degree that Brown first used. So, in the sense of implied accuracy, 5 1/2 and 5d 30m are not strictly interchangeable. Going from 5 1/2 degrees to 5d 30m changes the units of measurement from degrees to minutes and the implied accuracy from fractions of a degree to fractions of a minute.
So here’s a question about using your Garmin. From a computation perspective, is the decimal degree display more accurate than the degrees, minutes, seconds display, or is it the other way around?

You’re correct. I forgot the unit difference, so it’s 15 minutes versus 1 minute.
On the Garmin, doesn’t the precision depend on how many digits you display in the DD? I don’t have a Garmin, but DMS would give you to 1/3600 of a degree. If you set the DD to dd.ddd, then it’s 1/1000 th of a degree I would think.

The choices on my Garmin are five place degrees (0.036 second increments), which is more precise than 3 place minutes (0.06 second increments), or one place seconds (0.1 second increments). In latitude those correspond to roughly 3.6, 6, and 10 ft. So the display rounding error (ignoring measurement error) is half those increments or less.
The actual accuracy of a Garmin at a reasonable confidence level doesn’t approach those values until you have averaged sessions of some length on multiple days, because of the iono/tropo propagation errors that WAAS doesn’t model perfectly. In my old 76S unit there are other computational roundoffs internal to the solution that cause 2 ft errors even if you averaged the readings every day for months. And of course the conversion from WGS84 to NAD83 is the null transformation which ignores a 3+ ft difference.
. 
I agree with the text that “this is not so”. They don’t make the point very well though. From an evidentiary standpoint, significant digits is the wrong analysis when applied to older writings. Normally what I’ve seen in older deeds with a number like that is that all are either 1/2 or even degree. It’s telling you that equipment and procedures were used to try and achieve nearest 1/2 degree (scrivener probably never heard of significant digits). And that leads to the fact that they probably were not Andrew Ellicott and you can expect those bearings to vary much more than 1/2 degree. Simply put, I would testify that just about any evidence on the retracement is going to be more reliable than the bearings in this case. And there really is no “expected” range we could justify by using significant digits. You might come up with an “expected” range based on completed retracements in similar situations at similar time and place. But have to consider where it is also. In upstate NY, MI, MN lots of local attraction, in TX maybe not so much. One I liked to use in class was from Missouri and looked like bearings but turned out to be mariners compass (old sea dogs getting off the boat surveying and writing land descriptions because they had some education in navigation on the Ocean/Mississippi?).
BTW, I think it’s that text and edition that did have some typo’s in example on double or single proportions PLSS. Maybe it was boundary control book, but anyway it does happen.

Don’t forget this is a boundary book, not a measurement book. Don’t spend to much time worrying about his math. After all, the next course in the 5 1/2 degree document could be 20 1/4, and all the angles in the 30 minute document could be reported to the nearest 1/2 degree, so the 5 1/2 degrees could actually be more precise. I think, reading the whole passage in an earlier addition, the point he is trying to make is that within a document variations in the reported precision have a significance to the retracing surveyor.
Very few boundary documents take significant figures into account. Most people here probably report to the nearest second, but they probably don’t measure better than the nearest 1/4 minute, or 10 seconds.

Directions given on old Surveys seem to have not been derived from very big effort or given much concern. Right angles can mean anything from 60 to 120 degrees.
Say you have an old survey with two lines which intersect at a corner. Line 1 has a reported bearing of N 01?ø15′ W and Line 2 has a reported bearing of S 89?ø30′ E. There’s a bunch of monuments marking both lines including one at the intersection. So you set up on the intersection monument and zero on line 1, looking over all the monuments which are arrow straight, and you turn to line 2 (also an arrow straight line of monuments), you may turn 91?ø45′ right but I wouldn’t count on it. I’ve seen several degrees difference in this situation many times. The reason is it turns out the bearings given are record from earlier Deeds or Plats and the Surveyor didn’t want to report the differences or deal with the conflict. It’s not like their client had a clue if the angle was 3?ø off.
The first time it happened to me I was a 20something kid, I set up on the monument, sighted down the side line then turned the record angle to look down the back of the lot. I was shocked, there was at least half a mile of fences and utility poles looking that way several degrees to the left of where I just turned. There were several surveys in the area (a big rectangle of lots) and they all agreed on what the eastwest bearing was and the northsouth bearing was. I went up the back line and found the client’s other corner, turned to it and it was right in line with all the fences and power poles running for a mile or so up that way LOL.

Thank you all for your responses (hopefully this forum doesn’t frown upon thank yous). I appreciate both the theoretical, and practical responses.

I work some areas that the existing public records are compass survey description.
It is not uncommon for boundaries less than 300 varas and especially not more than 50 varas to not even be close to their correct bearing and 3/4 of them are very close considering they were following compass directions.
I’d rely upon the distance far more than the compass direction to find record monuments.
As they are passed on to the next conveyance and so on, the scribes will not understand the correct way to type or display a compass direction from a voice recorded transcript or dictated in person that get mangled into some code to decipher into what the surveyor actually wrote in long hand decades to centuries before.
Compass headings are a guide and not an absolute. I have seen many 20?ø in error from North.
Log in to reply.