Forest Mensuration. Brack and Wood


Index
Overview
Help

What to measure
Definitions
Problem of error
Significant digits

Measurement ©


When you measure an object, essentially all you are doing is counting the number of standard pieces it takes to be the same size as the object. For example, the length of an Olympic swimming pool is 50 m, because 50 one metre standard lengths would be exactly the same length.

The same principle works when we deal with something that is too small to be conveniently handled by straight counting of standard pieces. For example, the heights of people or of trees are not usually an exact number of meters. So, we simply say that the standard meter is actually a group of smaller standard lengths - centimeters. Each centimeter can also be a group of still smaller objects - millimeters. So we can describe or measure anything simply by counting the number of standard or groups.

There is a slight problem when we consider fractal geometry, but for our purposes, counting of standard sized pieces leads to the mathematical description of size.

What to measure

The choice of what parameters or variables to measure is not trivial. Considerable thought and planning are needed to ensure that time, effort and money are not wasted in measuring unimportant items. It is important that the measurements made are:

Terms and definitions

Terms regularly encountered in any measurement project include:

Significant digits or figures

Note: The number of significant digits in a number is not the same as the number of decimal places!

All measurements are subject to error . This means that one is never able to quote an exact value for a measured physical quantity. If the height of a tree is estimated to lie somewhere between 38.5 and 39.5 m, then the result should be quoted as 39 m (3.9 x 10). Such a measurement is said to have been made to two significant digits.

It is important that the correct number of significant digits is reported in any measurement - a long string of digits that has no significance is misleading as it indicates a very precise measurement (or a very small error). Meaningless digits are particularly liable to arise from a series of calculations. To avoid confusion, rules for determining (and reporting) the numbers of significant digits have been developed.

Rules for writing significant digits

  1. The significant digits in a number are those reading from left to right, beginning with the first non-zero digit and ending with the last digit written which may be zero (if zero, it must not result from rounding off), e.g.

             2 sig. digits  3 sig. digits
             0.25           0.257
             0.025          2.57
             2.5            0.0257
             0.0025         25.0 (zero as read, i.e. not rounded)
    
  2. It is incorrect to record more significant digits than are actually observed, e.g. a length measurement of 8 metres taken to the nearest metre cannot be written 8.0 m.

  3. In multiplication and division, the factor with the least number of significant digits limits the number of significant digits in the product or quotient (Note parenthesised figures in the following example):
        e.g. 895.67 x 35.9 = 32 154.553.
               (5)     (3)      (8)
        The result should be written 321. x 10^2
                                     (3) 
    
    895.67 represents a measurement between 895.665 and 895.675, and 35.9 represents a measurement between 35.85 and 35.95. The products of these four limiting combinations differ in all except the first three figures:
         895.665 x 35.85 = 32109.59
         895.665 x 35.95 = 32199.156
         895.675 x 35.85 = 32109.948
         895.675 x 35.95 = 32199.516
    
    In other cases, the number of identical digits in all combinations of the product or quotient will be one less than that in the factor with the least number, e.g.:

          895.67 x 1.5 = 1343.505.
          (5)      (2)     (7)
          The result should be written 13. x 10^2 or 1.3 x 10^3
                                       (2)          (2) 
  4. The number of significant digits in an integral or whole number is infinite.
         27.146 x 50 = 1375.3
         (5)     (_)    (5)
    
  5. A good rule in a series of multiplications or divisions is to carry one more digit than the number of significant digits in the shorter factor of each set and round off to the proper number of significant digits at the end.

  6. Determining the number of significant digits in addition and subtraction requires that the numbers in the set be first aligned according to their decimal places. The number of significant digits in the result can never be greater than that of the largest of the numbers in the set, and is often less. The rule to apply is:
    1. Align the numbers according to their decimal places.
    2. Identify the largest number in the set and locate the right-most of its digits which aligns with at least the right-most digit of all the other numbers in the set.
    3. Count the number of digits to the left of and including that digit.
    4. This gives the number of significant digits in the result.
           1234.345        (largest number)
              1.002345
            567.6424
             34.45         (least right most digit)
           ===========
           1837.439745     (rounded to 1837.44)        
      
The above rules should be applied rigourously to processed data (e.g. computer printouts) because the precision implied in the computed data is almost invariably spurious. Publishing too many significant places may imply that your measurements are much better than they really are!

Rounding off

In the field

Normal practice when recording a field measurement is to round figures off to the nearest smallest unit attainable by measurement (by the instrument and technique used).

In calculations

Procedures for rounding off the result of a calculation or series of calculations must ensure that bias is not introduced. The accepted convention (using as illustration a 2-digit number n.5, where n represents the digits before the decimal place) is:

   Number <  n.5  	Round off to n, e.g.:
	                31.3-> 31.
   Number >  n.5 	Increase the ones digit by 1., e.g.:
	                31.7-> 32.
   Number = n.5 	(a)  if n is even, round off to n
                         (b) if n is odd, increase n by 1, e.g.:
	               32.5-> 32.
	               37.5-> 38.
This unbiased convention may differ from the technique you were taught in primary or secondary school, viz., if the number = n.5, increase the ones digit by 1, i.e. always round upwards.

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http://online.anu.edu.au/Forestry/mensuration/MEASURE.HTM
Cris.Brack@anu.edu.au
Mon, 14 Apr. 1997