Weights and the Olympic Bell

[Carl Scott Zimmerman]

It appears that there is a need for some re-education in measurement 
and conversion arithmetic.  Here it is.

First, some terminology:
   accuracy = how close a measurement is to reality
   precision = the smallest unit of measurement used
Accuracy is limited by precision, but not determined by it; accuracy 
can be much less than precision.  (When that happens, it's also 
confusing!)

It is theoretically possible to measure the weight of the Olympic 
Bell to the nearest microgram, which would be a precision of 11 
significant digits.  But given the practical limitations of ordinary 
measuring equipment, it is unlikely that it could be measured with an 
accuracy of more than four or five significant digits.

The best weight we have for the Olympic Bell up to now is Dickon's 
figure of 22.91 metric tonnes.  In the absence of any accompanying 
statement as to the accuracy of that measurement, we must take both 
the accuracy and the precision of it as "correct to the fourth 
significant digit" (which is also, in this case, the second decimal 
place).  Then the exact weight, if one could measure it with greater 
precision, must be in the range
      22.91 plus or minus .005 metric tonnes
or (adding non-significant trailing zeros and moving the decimal point)
      22910 plus or minus 5 kilograms
In other words, we presume that it has been measured (weighed) to the 
nearest 10 kilograms.

Since many of us were originally educated in societies which did/do 
not use the metric system of measurement, we are more comfortable 
with something else - the Imperial system of weights in the UK and 
closely-related countries, and the avoirdupois system of weights in 
the USA.  So conversion becomes necessary.

Conversion factors are widely known, based on international standards 
for units of measurements.  A few are perfectly precise; as in the 
conversion from metric tonnes to kilograms above, they do not cause 
any loss of accuracy, because they are based on the definitions of 
the units of measurement involved.  But most have some limitation on 
the precision that they can carry.  For example, the conversion 
factors in the online calculator which Chris Pickford referenced have 
precisions which vary from 8 to 11 significant digits.  Those were 
the best values that I could find when I developed that calculator 
several years ago (based on a similar calculator developed by someone 
else).

When converting a measurement from one system to another, it is 
important to use a conversion factor which has at least as much 
precision as the measurement being converted, in order to avoid 
inadvertent loss of accuracy.  In practice, one additional 
significant digit is usually enough, though it never hurts to use 
more as long as you keep track of accuracy.  The online calculator 
helps in this regard through the option to specify the number of 
significant figures (digits) to be used in presenting the results. 
Behind the scenes, it calculates from the initial input value with 
the maximum possible precision, and then rounds the results according 
to the specified number of significant figures.

The conversion factor between kilograms and pounds is 0.45359237 kg/lb.
To convert kilograms to pounds, divide by that factor; to convert 
pounds to kilograms, multiply by that factor.  To keep track of the 
known or presumed accuracy, the "plus or minus" value must be 
similarly converted.  (The online calculator does NOT do this 
automatically; you must figure it for yourself.)

Then 22910 plus or minus 5 kilograms (or 22910 +/- 5 kg)
converts to 50507.9043 +/- 11.0231 pounds.
Since the size of the accuracy range is so large, it makes no sense 
to retain the decimal parts of these values; rounding both results to 
the nearest pound, we get
    50508 plus or minus 11 pounds (50508 +/- 11 lb).

Since an Imperial ton is 2240 pounds (a precise conversion), this is 
equivalent to
    22-10-3-24 +/- 11 in ton-cwt-qtr-lbs form, or
    22.548 +/- 0.005 Imperial tons in decimal form.
To drop the accuracy range, this should then be stated as
    22.55 Imperial tons

In summary, when converting measurements, it is important to 
understand (a) what is known or implied about the accuracy and 
precision of the original value, (b) what the original system of 
measurement was, (c) what conversion factor should be used between 
the original system and the target system, and (d) how to apply that 
factor to determine the converted value without losing any accuracy 
or implying false accuracy.  Most of the errors which have been made 
in this forum's discussion of the weight of the Olympic Bell have 
probably arisen from confusion about (b) or (c).

If this essay is TMI (Too Much Information), I shan't apologize, 
because you could have hit the Delete button at any time.  But I 
would welcome correction of any misstatements.

Carl
           
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