|MadSci Network: Engineering|
'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 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 of your original measurements.
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