[at-l] Math & the measurement of the trail, was Georgia AT - post number one

[Arthur Gaudet]

(Ah, Thanks Greenbriar, for the correction! Here's a post that perhaps no one at
all will read! I hope Pitts will be interested and perhaps contribute as well.)

Precision, repeatability, accuracy, and also average, mean, median - they all
have definitions, but many don't know the differences.

The best measurement system is the one that is both precise and accurate. The
distribution of numbers is tightly clustered around a value, and the value
happens to be the correct one, known through a previous calibration procedure.

Precision means a tight statistical distribution around a number. Repeatability
is another description of this property of precision. In statistics the standard
deviation (sigma)is a measure of the "tightness or looseness" of samples around
the mean. From memory - one sigma includes about 66% of the samples and 3*sigma
includes 99% of all the samples (for Normal Distibutions, let's not get fuzzy
with all of the other distributions and the Chebyshev's Inequality)  

Accuracy is always in relation to a known quantity and has little/nothing to do
with precision. It's how closely a new measured distance comes to the known
number. This is an interesting topic because it's the method used for
calibration of a measurement wheel or for a bicycle computer. You find two
places that are a known distance apart and measure it to determine how accurate
the wheel measures this known distance. For bicycle computers you use this "roll
out" method and then adjust an internal Cal setting until you get the number you
want. I've been able to calibrate my bicycle so tightly that I know when my tire
pressure has dropped even a few psi. (Lower psi affects the tire diameter,
creating a smaller wheel circumference. A smaller wheel circumference causes the
computer to count more wheel rotations per mile and therefore reads a higher
value than expected for a known distance. I have a system that is accurate to 1%
and it takes a lot of work to achieve this level of reliability.) 

I've often wondered what calibration method(s) was used by MacKaye when the
trail was first measured. One of his wheeels is now at the Harpers Ferry Office,
mounted on the wall. The mechanism for measuring clicks is hidden from sight. I
can't see how to make adjustments to it, so perhaps it's a fixed mechanism.

Here's some Math to illustrate the definitions:
Given a set of numbers: 5, 7, 8, 8, 9
The median is 7               (half way between 5 and 9)
The average or mean is 7.4    (add the numbers, divide by 5) (*Note there are
many other types of means besides the        
                               arithmetic mean.)
Scenario #1:
Pittsburgh travels back & forth between Springer & Gooch Gap Shelter 7 times and
reads these distances on his wheel: 14.96, 15.08, 14.85, 15.13, 14.96, 15.00,
15.02.
The average or mean is 15.00 and the standard deviation (sigma) is 0.09.
This set of data has a pretty tight range and 60% of the data falls within +/-
.09, or between 14.91 and 15.09.

Scenario #2:
Pittsburgh travels back & forth between Springer & Gooch Gap Shelter 7 times and
reads these distances on his wheel: 13.79, 15.48, 14.19, 15.15, 14.96, 15.21,
16.22.
The average or mean is still 15.00 but now the standard deviation (sigma) is
0.81.
This set of data has a pretty loose range and now the one sigma range is +/-
.81, or between 14.19 and 15.81.

My Personal Opinions:
In scenario #1 I know the actual distance is "more than likely" between 14.9 and
15.1. It makes sense to me to round represent this number as 15.0, implying the
precision is within 0.1 miles. To call the number 15.00 would imply that I
believe the number is "more than likely" between 14.99 and 15.01. This is an
implied precision ten times more than we have determined from the sample set and
through the calculation of standard deviation.

For scenario #2 I know the distance is "more than likely" between 14.2 and 15.8.
My choice is to represent this number at 15, implying that the precision is less
than half a mile.

Discussion:
With all the ups & downs, lefts & rights along the trail an extremely good wheel
will measure to about 1% precision or to within 53 feet in a mile. In 15 miles
this becomes about 800 feet. This generates values between 14.925 and 15.075,
similar to scenario #1. This assumes no other sources of error: mis-reading the
dials, skipping/sliding along the trail, change of tire pressure from day to
day, mis-handling the wheel, no changes with temperature, etc. (Note this is not
related to the calibration of the wheel which can be thrown off by other
mis-haps!)

Since the trail is a pretty harsh environment, esp. when travelling
day-after-day, I think that the reality is closer to Scenario #2 when measuring
these distances. Or perhaps it is somewhere between #1 and #2. I might be
convinced that a precision of measurement of .1 miles is achievable, but I'd
have to see the data!

Perhaps some of this is alleviated by breaking up a long distance into a series
of short measurements? Well, yes & no. But perhaps this is a topic for another
time or contributor.

--Arthur aka RockDancer

-----Original Message-----
From: Gary Ticknor [mailto:garyticknor at verizon.net] 
Sent: Tuesday, July 22, 2008 8:44 AM
To: rockdancer97 at comcast.net
Subject: Re: [at-l] Georgia AT - post number one...

Or is that 2 decimals of precision? ;)

- Greenbriar

Arthur Gaudet wrote:
> Hi Pitts,
> I noticed that the distances are using 2 decimal digit accuracy. 
>
> An online AT distance calculator is at...
>