MadSci Network: Engineering

Re: How do you avoid round off errors when calculating to correct sig. digits?

Date: Wed Dec 2 01:21:56 1998
Posted By: David Gould, Undergraduate, Computer Engineering, University of Alabama in Huntsville
Area of science: Engineering
ID: 910938012.Eg

'Rounding to significant digits' is an approach that is unfortunately 
widespread in the educational community as a solution to the problem of 
statistical error calculation.  Often an instructor or a book will promote 
the use of significant digits because of its simplicity, while boldly 
stating that in using the technique you are correctly showing the lack of 
precision in your calculations.  Sadly, many instructors even deduct points 
for improper deviation from this flawed technique.

In fact, the inherent flaws of the technique have led you to ask the 
question, "should I wait till the end to round off, or round off at each 
step."  Ideally, you must agree, any technique for determining 'margin of 
error' would not depend on such a thing.

The concept behind significant digits is this:  

If I make a measurement valid to a certain decimal place, and then perform 
calculations with that measurement's value, the result cannot be valid to 
any more decimal places than the original measurement was.  If more than 
one measurement is involved in the calculation, the result cannot be valid 
to more decimal places than the measurement with the least number of valid 
decimal places.

Now to answer your question.  The calculations that may occur before 
arriving at the final result usually involve no extra measurements, and 
thus should not be rounded in any way.  In fact, if the final result of one 
step happens to be used within another calculation, you should use the 
pre-rounded value.  Otherwise the error (that's right!) caused by rounding 
would be cumulative throughout your calculations.

To write down every digit is obviously impossible, so you must be content 
with some limit to your accuracy, but this is not part of the use of 
significant digits.  I suggest recording to the limit of your calculator 
display, or at least 3 to 4 digits beyond the number of 'significant 

It's interesting to note that no instructor will argue that by using 
rounded values in calculations, you are cumulating an error - but many 
still insist that by using rounding at the end of the calculation you 
provide a more correct view of the result.

I urge you, if you are as disturbed by this notion as I was, to investigate 
the field of statistical error calculation, and in particular the 
calculation of margins of error.  As a final remark, I'll simply say that 
when complex operations like squaring or taking powers is performed on 
measurements, it is ludicrous to believe that precision is maintained at 
the same number of significant digits as original measurements.  In reality 
you will quickly find that your lack of precision explodes into an 
amazingly huge margin of error.  Or at least huge compared to the accuracy 
of your original measurements.

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