# APPENDIX B

## PROPAGATION OF ERRORS IN CALCULATIONS

# PROPAGATION OF ERRORS IN CALCULATIONS

While discussions of accuracy, error, and precision may be complicated,
an understanding of these concepts will provide the foundation for
evaluation of methods and technologies given in the report. In addition,
introductions of sloped measurement issues and significant figures will
provide basis for discussion when evaluating various area measurement
procedures.

## Accuracy

Measurement of discrete objects (cars, dogs, etc.) provides exact count
of the number of objects in a group. Measurement of continuous values
(distance, temperature, etc.) is not exact, and thus it is subject to a
level of accuracy dependent upon the instrument used. "It can be
unconditionally stated that (1) no measurement is exact, (2) every
measurement contains errors, (3) the true value of a measurement is
never known, and therefore, (4) the exact error present is always
unknown (Wolf and Brinker, 1994).\" Instruments measuring distance and
angles have a given level of accuracy. The greater the accuracy of the
equipment, the greater potential for approaching the "true" value of the
measurement. For example, the finer the divisions on a ruler the more
accurate the measurement will be.

## Precision

Multiple measurements of the same variable provide an estimate of a
method\'s or instrument's precision. Stated another way, the precision
of a method or instrument is a measure of its ability to give the same
answer when making multiple measurements of the same variable. Precision
is not synonymous with accuracy. For example, we may make five (5)
measurements with a tape measure, reading 10.09 ft, 10.10 ft, 10.11 ft,
10.11 ft, and 10.10 ft. The average value is 10.102 ft. The difference
between measurements is 0.02 ft. If the answer only needs to be to the
nearest foot, this appears to be a very precise process. Later, it is
discovered that the actual distance measured is 9.10 ft. So while the
tape is very precise, it is inaccurate. Inaccuracy and lack of precision
are a result of mistakes and errors.

## Mistakes and Errors

Mistakes are blunders usually caused by misunderstanding the problem,
carelessness, fatigue, missed communication, or poor judgement (Wolf and
Brinker, 1994). Eliminating mistakes is essential if accurate and
precise measurements are to be obtained. Mistakes are identified and
corrected by employing a system of checks in the measurement process.
Common approaches include the following:

-   Using standard and reliable methods and procedures

-   Have another individual review the work

-   Comparing results to other, independent, estimates

-   Insuring that the sum of the parts equals the whole

While the results of errors in one\'s work may be similar to making
mistakes, they derive from a fundamentally different basis. Errors
derive from nature and the limits of human abilities. Errors in
measurement arise from several sources:

-   Nature‑Variations in wind, temperature, magnetic declination, etc.

-   Instrumentation‑Imperfection in the construction or adjustment of
    instruments

-   Personal‑Limitations of the human senses

Errors can be classified as either systematic or random. Systematic
errors are those which are constant for given conditions. Since these
errors are "consistent", the measurements can be corrected for these
types of errors. For example, changes in the length of a steel tape due
to changes in temperature can be estimated and results can be corrected
for the variation.

Random errors remain after systematic errors and mistakes are
eliminated. Because they are "random", they are generally compensating
in nature. These errors tend to be smaller and cluster near the mean
value. Correction for random errors is often handled by sharing the
error among all the possible sources of error. A technique used to
correct these errors is called the "least squares" method (Moffitt and
Bouchard, 1992).

Errors in measurements are propagated in calculated quantities such as
area. The general equation for propagation of error is:

E~total~ = ±(E~a~^2^ + E~b~^2^ + E~c~^2^ + \...+ E~n~^2^ )^1/2^

Where:

E~a~, E~b~, E~c~, \...E~n~ = Error associated with each measurement

a, b, c, \...n = Lengths of lines or angles measured

Application of error propagation is given in the following examples.

### Example 1

A 100 ft. tape is known to have an error of 0.005 ft. each time it is
used. What would be the error it the tape measures 1200 ft?

This is an error of a sum problem.

E~total~ = ±(E~1~^2^ + E~2~^2^ + E~3~^2^ + \...+ E~n~^2^ )^1/2^

If E is the same for each measurement,

E~total~ = ±(E~1~^2^ + E~2~^2^ + E~3~^2^ + \...+ E~n~^2^ )^1/2^ =
±(nE^2^) = ±E(n)^1/2^

Thus,

E~total~ = ± 0.005(12)^1/2^ = ± 0.017 ft

We can rearrange the equation to answer the question,

 How accurate do my measurements need to be if I want to limit my error
to a fixed percent?

### Example 2

An odometer wheel is used to measure the lengths of a rectangle. Side A
is 425 ft and side B is 105 ft long. The error in side A is ± 0.5 ft and
side B is ±0.15 ft. What is the error in the area?

This is a propagation error of a product problem.

E~area~ = ±(A^2^E~b~^2^ + B^2^E~a~^2^)^1/2^

E~area~ = ±(425^2^ x 0.15^2^ + 105^2^ x 0.5^2^)^1/2^

E~area~ = ±82.58 ft^2^ or 0.18%

## Errors Associated with Closed Traverse

Measuring lengths and angles of perimeter lines is termed *closed
traverse*. It is considered *closed* since the beginning and ending
points are the same. It is considered *traverse* because it
circumscribes the area. The closed traverse method is a very common
surveying technique to document the perimeter of an area. Reconstructing
the area on paper using measurements taken from the field will show that
the beginning and ending points do not coincide. The distance between
these points is called the *linear error*. Table 1 summarizes linear
accuracy needed in a traverse to ensure appropriate accuracy in the
area.

Summary of errors in a closed traverse

  ----------------------- ----------------------- -----------------------
  Linear Accuracy                                 Area Accuracy

  Relative\*              Percent                 Percent

  1/5000                  0.020%                  0.028%

  1/3000                  0.033%                  0.047%

  1/1000                  0.100%                  0.141%

  1/500                   0.200%                  0.283%

  1/100                   1.000%                  1.414%

  1/50                    2.000%                  2.828%
  ----------------------- ----------------------- -----------------------

\*Relative accuracy is the amount of error per unit of measurement.

Example: 1/5000 = 1 ft of error in 5000 ft of traverse length.

### Example 3

A closed traverse is completed using a theodolite and EDM (Total
Station) on a landscaped area. The area is calculated to be 230,450
ft^2^. The traverse has an accuracy of 1/5000, which is to say that for
every 5000 ft of measurement the survey is off 1 ft.

E~area~ = (area)(relative accuracy)(2)^1/2^

E~area~ = (230,450 ft^2^)(1/5000)(2)^1/2^

E~area~ = 65.18 ft^2^ or 0.03%

If the same survey were completed with a compass and odometer wheel,
with a relative accuracy of 1/75 the error would be:

E~area~ = (area)(relative accuracy)(2)^1/2^

E~area~ = (230,450 ft^2^)(1/75)(2)^1/2^

E~area~ = 4,345.41 ft^2^ or 1.89%

## Sloped vs. Horizontal Measurements

Traditional surveying methods considers distances and areas based on
*horizontal* planes. Measurements based on sloped distances between
points will have greater areas than those based on planes. Areas based
on sloped distances appropriately represent landscaped areas.

Sloped versus horizontal measurements.

Measurement techniques such as global positioning system (GPS) and
aerial photographs produce horizontal measurements. Although sloped
measurements can be obtained through various software programs, it
generally is not done. Using a total station or compass and tape to
measure distances between points can be done either on a sloped or
horizontal basis. However it is less complicated to measure the sloped
distance with a compass and wheel.

The following Table estimates the effect of ground slope on the
differences of sloped and horizontal areas. Slope effects are less than
1% for a 10% slope in each direction. The increase in surface areas is
approximately 2.5% as the ground slope approaches 16%. This implies that
the affect of ground slope is of very little importance except in those
parks where the slopes exceed 16% on a significant portion of the park.

Ground slope effects on area measurement.

  ---- ------------ ----------- ----------- ----------- ----------- -----------
       Slope in                                                     
       Horizontal                                                   
       (X)                                                          
       Direction                                                    
       (%)                                                          

       0            1           2           4           8           16

  0    0.00%        0.005%      0.02%       0.080%      0.319%      1.272%

  1    0.005%       0.01%       0.02%       0.08%       0.32%       1.28%

  2    0.02%        0.02%       0.04%       0.10%       0.34%       1.29%

  4    0.08%        0.08%       0.10%       0.16%       0.40%       1.35%

  8    0.32%        0.32%       0.34%       0.40%       0.64%       1.60%

  16   1.27%        1.28%       1.29%       1.35%       1.60%       2.56%
  ---- ------------ ----------- ----------- ----------- ----------- -----------

## Significant Figures

A word of caution about the use of significant figures in calculated
results. Calculators and computers will add, subtract, divide and
multiply two numbers together and give the answer with as many numbers
as it has space to display. Unfortunately this can give the impression
that there are more accuracy in the results than can be justified. As in
the example below a calculator might show the results to four or more
decimal places:

4356 X 13 = 56,628.0000

The conclusion might be that this answer is accurate to four (4) decimal
places. The truth is that the answer has the same number of significant
figures as the *least number* of significant figures used in the
computation.

**Example**

  ----------------------------------- -----------------------------------
  *Number*                            *Significant Figures*

  4356                                4

  13                                  2
  ----------------------------------- -----------------------------------

>  Which number (from above) has the least number of significant
> figures?
>
> 13
>
>  How many significant figures does it have?
>
> 2
>
>  Therefore, the product of these two numbers will only have *two*
> significant figures.
>
> 4356 x 13 = 56,628
>
>  Therefore, the product of these two numbers should be rounded to:
>
> 57,000

A surveying or mathematics book contains further discussion on this
topic (Brinker and Minnick, 1987).