### 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
Message:
```
'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
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.

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
digits'.

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

```

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