Too much math at one time is....
> Just keep in mind that the deviations from the mean would only be part of the range errors. They are not the measure of the actual error in the range.
Absolutely understood. I'm on to Cyclic errors next (after a cold beer starts working on my headache).
Thanks for all your help.:-)
Standard error vs. standard deviation
> For the direct series, 0.0013 (0.0012732 to 7 decimal places) is the sample standard deviation of the 20 observations, not the standard error.
Actually, for n=20, the difference between the standard deviation (n=20 degrees of freedom) and the standard error (n=19 degrees of freedom) is negligle, but
s.d. = 0.00124 ft. and s.e. = 0.00127 ft. (= 0.0013)
"Standard error" is the more common usage term in surveying since typically we are dealing with small samples, not entire populations, and so are estimating the standard deviation based upon n-1 degrees of freedom.
> Your calculation of the standard error for 3 observations is correct. Note that it is a larger number than rfc's which demonstrates the superiority of the mean of a larger number of observations.
Actually, what it demonstrates is that 100 observations are a waste of time considering that the random errors of centering completely override the miniscule improvement in the apparent standard error (inner precision) of the mean of the range measurements.
Note that the scatter of repeat range measurements to a fixed prism is not a good estimator of the actual standard error of a range measurement. What the scatter merely does is to measure the internal precision of the instrument's ability to resolve the phase of the carrier.
Superior but Useless
> think I'm satisfied that given that I will be turning at least 3 sets of D+R angles (still TBD) for my upcoming traverses, and that I'll be recording distance for each of those observations, I can sleep at night, setting the instrument to record a single distance observation for each angle observation.
Keep in mind that the step I think you've skipped has been testing the standard error of your target centering. That is an extremely important parameter that bears both upon angle and distance uncertainties, particularly as distances between control points shorten.
Superior but Useless
Well, yes. The key point is to understand the meaning of the standard error. There is less variability in the means of several samples than in the individual items that make up the population. And the variation is reduced by including more items in each sample.
But you reach a point of diminishing returns and that's what Kent pointed out. In this case, it comes quickly because the sample standard deviation is small to begin with.
Incidentally, your direct observations, the only numbers I analyzed, may not be normally distributed. If you do a simple five-number summary, the observations are skewed to the right; ie, toward the larger measurements.
Here are the 5 numbers:
Minimum: 19.586
Quartile 1: 19.5865
Median: 19.588
Quartile 3: 19.588
Maximum: 19.591
Note that the median and the third quartile are the same number. In a normally distributed sample, the median would be half way between the first quartile and the third quartile.
This is a tight distribution, so it's of little practical consequence, but if you're after a statement of accuracy, you may not be correct down to the millimeter.
Standard error vs. standard deviation
Differences in terminology are pervasive in the world of statistics and really multiply in the applied world. You already know all of the following, but it's nice to think about fundamentals sometimes.
In the mathematics world, there are standard deviation (sigma) and sample standard deviation (s). In rfc's case, if there were only the twenty observations that he made, they would comprise the population and there would be twenty degrees of freedom. If these twenty are only a sample, then the number of degrees of freedom is reduced to 19 and the resulting sample standard deviation becomes an estimate of the population standard deviation.
Standard error, in the math world, is shorthand for standard error of the mean. It represents the standard deviation of the distribution of the means of samples of various sizes taken from a population. That is, it provides a way to determine how far from the population mean a sample mean is likely to be.
If you have either sigma or s, you can estimate the standard error.
These are three different, but deeply entwined, concepts. Often the difference between them is small, but sometimes it matters.
Your counsel, as always, is wise.
Standard error vs. standard deviation
> Differences in terminology are pervasive in the world of statistics and really multiply in the applied world. You already know all of the following, but it's nice to think about fundamentals sometimes.
>
> In the mathematics world, there are standard deviation (sigma) and sample standard deviation (s). In rfc's case, if there were only the twenty observations that he made, they would comprise the population and there would be twenty degrees of freedom. If these twenty are only a sample, then the number of degrees of freedom is reduced to 19 and the resulting sample standard deviation becomes an estimate of the population standard deviation.
Except, observations of any measured quantity such as angles and distances are not the entire population. They are merely a sample drawn from the universe of possible measurements that a particular process might produce. rfc might as easily have logged forty ranges, for example. The mean is merely an estimate of what the "true" value obtained from the means of an even larger series of repeated measurements would converge toward.
In measurement science, the term "standard deviation" has meaning in the case of testing procedures where the number of observations, n, is so large that the difference between the estimates of standard deviation based upon n degrees of freedoms and n-1 degrees of freedom may be overlooked. A sample size of 20 is generally regarded as about that cutoff point, but nonetheless the mean of a sample size of 20 will have some uncertainty and so treating it as error free will underestimate the variance.
The convention of calling the estimate of the standard deviation based upon n-1 degrees of freedom the "standard error" is that given by J.E. Jackson in his Sixth Edition of David Clark's "Plane and Geodetic Surveying, Vol. Two Higher Surveying", and for the reasons stated above.
Standard error vs. standard deviation
Differences in terminology are pervasive. Out of curiosity, what does Jackson call sigma divided by the square root of n?
Standard error vs. standard deviation
> Differences in terminology are pervasive. Out of curiosity, what does Jackson call sigma divided by the square root of n?
Where sigma is estimated from n-1 degrees of freedom,
sigma/SQRT(n) is uniformly called "the standard error of the mean" in the surveying literature I'm familiar with, Jackson included.
Standard error vs. standard deviation
> Standard error, in the math world, is shorthand for standard error of the mean. It represents the standard deviation of the distribution of the means of samples of various sizes taken from a population.
Math Teacher, you hit the nail on the head. Guilty as charged. I put down Ghilani and Wolf, when I was stumped by Kent's ever increasing spiral of nuance, and picked up an old math text on statistics, that instructed me exactly how to derive the "Standard Error"!
This stuff is challenging enough for me, just learning the language in one area of expertise.
>
> Your counsel, as always, is wise.
So true.
Standard error vs. standard deviation
> > Standard error, in the math world, is shorthand for standard error of the mean. It represents the standard deviation of the distribution of the means of samples of various sizes taken from a population.
>
> Math Teacher, you hit the nail on the head. Guilty as charged. I put down Ghilani and Wolf, when I was stumped by Kent's ever increasing spiral of nuance, and picked up an old math text on statistics, that instructed me exactly how to derive the "Standard Error"!
Well, let's not lose sight of the fact that the term "standard error" in the surveying literature means the small sample standard deviation (based upon n-1 degrees of freedom). It does NOT mean the "standard error of the mean". More importantly, the central concept that the standard deviation of small samples is estimates differently than of entire populations goes uncontested.
Standard error vs. standard deviation
Note that for a sample size of one, that is, a single observation, the standard error of the mean and the sample standard deviation are the same. In that case, s over the square root of n is equal to s, where s is the sample standard deviation.
In measurement applications, it's never true that we have the population mean or the population standard deviation, sigma. However, if we did have those, the standard error would be calculated using sigma. We would not need an estimate because we have the actual value.
So it seems to me that, in the section you're referencing, Jackson is calculating statistics for a single observation and assuming correctly that sigma will never be available. Somewhere else he evidently addresses the standard error of the sample mean and discusses statistics for sample sizes larger than one.
It's true that in rfc's case little to nothing is gained by increasing the number of measurements of one line beyond 3 or so. That is a very tight distribution.
Each of these statistics answers a different question. The trick in statistical analysis of any data is to match the proper statistic to the question we want to answer. Terminology is always subject to customary usage and that makes learning more difficult.
Standard error vs. standard deviation
> Note that for a sample size of one, that is, a single observation, the standard error of the mean and the sample standard deviation are the same. In that case, s over the square root of n is equal to s, where s is the sample standard deviation.
Except that is a completely trivial case. Nobody attempts to estimate the standard error of a measurement process from a sample size of 1.
> In measurement applications, it's never true that we have the population mean or the population standard deviation, sigma.
Yes, obviously. This is why the standard error is estimated with n-1 degrees of freedom: because the mean is uncertain and assuming otherwise underestimates the variance.
> So it seems to me that, in the section you're referencing, Jackson is calculating statistics for a single observation and assuming correctly that sigma will never be available.
No, that is pretty wide of the mark. In small sample statistics, one inevitably uses the standard error based upon n-1 degrees of freedom as the most efficient estimator of the standard deviation of the population sampled.
> It's true that in rfc's case little to nothing is gained by increasing the number of measurements of one line beyond 3 or so. That is a very tight distribution.
In the case of rfc's instrument, I'm pretty sure that there are other sources of uncertainty in both the instrumental range measurement and in the resulting distance measurement between control points that completely flood the small variances of his samples. In the case of the instrumental range measurement, there are warm-up effects for example that probably figured into his test. In the case of the computation of the resulting distance, the contribution of target centering erros are likely the single largest source of uncertainty if the manufacturer's claim is for an s.e. of +/-(2mm + 2ppm) for a range measured with his instrument.
Standard error vs. standard deviation
We're a lot closer than it might seem. First, as you stated, Jackson's use of the term standard error for sample standard deviation is standard usage in lots of the physical sciences. That statistic is the input for calculating a host of different standard errors such as the standard error of a proportion. Thank you for supplying the reference and the insight. I have references that say the same thing, but I lost sight of that.
When I used the term sample of one, I didn't mean it literally. What I wanted to do is connect Jackson's standard error to the standard error of the mean of samples of increasing sizes. There is a standard error for a single observation, and it is the standard deviation of the distribution. Similarly, there are standard errors of the mean for samples of two, three, four, and so on.
Using s, the n-1 estimate, is applicable only to small samples. You mentioned a sample size of 20 as a dividing line; my sources say 30, but if you had a sample of 100, s would not be the proper estimate of the population standard deviation. That's moot since there is nothing to be gained from such a large sample in this case, but it's good to know.
As always, you supply correct information and wise counsel.
