Rounding Errors, or, Why Won't WebAssign Accept My Answer?
Here is a common complaint:
"I know I am doing the problem correctly, but Webassign won't accept my answer, no matter how many significant figures I try..."Under the assumption that the student is really doing the problem in the correct way, what could the reason be that the answer is not being accepted?
Answer: ROUNDING ERRORS.
Rounding errors tend to creep into multi-part problems when one result early in the problem is used to calculate another result later, and the student uses the submitted number (which is rounded) in the later calculation.
However, it can be tricky to determine which number has been rounded "too much". This is because the criterion for acceptibility for a given submission slot is based on the percent difference between the key and the submission, and a given percent of a small number is much less than the same percent of a larger number. For example, assume that the problem requests a value and an uncertainty, e.g.,
g = ± m/s2If the value is 9.81 m/s2 and the uncertainty is 0.03 m/s2, and both are required to be within 2% of the correct value, then 9.62 to 10.00 would be accepted for the first value, but only 0.0294 to 0.0306 would be accepted for the second value. Note, in particular, that 0.029 and 0.031 would NOT be allowed to the 2% margin, even though these are "proper roundings" of the more precise values.
Now, assume that the student has entered g = 9.98±0.0295 m/s2. In the absence of sigfigs checking, both of these numbers would be accepted as correct. Next assume that what is requested is the fractional uncertainty, εg = σg/g. The correct value would be 0.03/9.81 = 0.0030581 (to way more than correct sigfigs). But if the student enters 0.0295/9.98 = 0.0029559, rounded to 0.0296, then this number will be 3.2% low, and it will be marked incorrect.
This is how a slightly wrong but still within bounds answer in one part of a problem can become a slightly wrong, but now out-of-bounds answer in another part. The student assumes that the problem is with the answer that is marked wrong, when, in fact, he or she had done that part of the question correctly.
How to avoid rounding errors
If you are having this kind of problem: you are sure you are doing the calculation correctly, and keep getting it marked wrong,
- Go back to the beginning and recheck all numbers that you have calculated. Chances are good that you have rounded something so that it is slightly off, but within the tolerance of the question.
- Keep all intermediate calculations in your calculator's memory. Save any rounding for the end of the problem.
- Enter one more sigfig than is likely to be proper. When sig figs are not checked, this will increase the likelihood that your submission is close to the correct value. When sig figs are checked, and you have too many, you will be told with the red hint, "Check significant figures".
- If you KNOW that sig figs are being checked, and you get your submission marked wrong but do NOT get the red hint, then the problem is NOT with your sig figs. You are too far from the key. Yes, it is possible that by changing the number of figures that your submission will fall under tolerance, but that is a coincidence, since the rounding just rounds your number towards the key, but it doesn't really solve the problem.
Finally, if all of the above fails, send the instructor a note. Coding errors do happen, and they can be fixed and the problem rescored.
When to round numbers in labs
In general, you should never round raw data. If your instrument reports to 4 significant figures, without fluctuations in the least digit, then that is what you want to record and use for calculations. Then, after you carry out data analysis, and work out the uncertainty, you can round things up as needed.
More simply, what you enter into WebAssign in the post-lab should be exactly what you entered into your original lab papers, unless you believe it to be in error. When WebAssign checks lab work, it often recalculates your results from the data that you enter into it, so if you use different raw data than what you originally used for your calculated results, you may get these answers marked wrong.
More generally, in terms of experimental practice, any adjustment that you make to an experimental result needs to be done for a clear reason. Rounding should only be done when you have reason to believe that the fewer digits correctly reflects the real measurement. Once you have made that choice, then all subsequent calculations should be consistent with it. Thus, if you believed that the raw data set was only correct to the number of digits that you entered, then you should recalculate standard deviations, etc., from those numbers, in order to make the results consistent.