Significant Figures on WebAssign
How does sig figs checking work?
Sig figs checking on a computer is very strict. Remember these rules regarding sig figs:
- The leftmost nonzero digit is the most significant digit.
- If there is no decimal point, the rightmost nonzero digit is the least significant digit.
- If there is a decimal point, the rightmost digit is the least significant digit, even if it is a 0.
- All digits between the least and most significant digits are counted as significant.
When using scientific notation, enter the number using the "e" notation. For example, a value of 1.3×10−2 would be entered in WebAssign as "1.3e-2". Note: The "e" must be lower case.
Here are some examples:
| 1234 | = | 4 significant figures | 5.0e2 | = | 2 significant figures | |
| 500 | = | 1 significant figure | 140e-001 | = | 2 significant figures | |
| 500. | = | 3 significant figures | 8.20000e3 | = | 6 significant figures | |
| 1300 | = | 2 significant figures | 101.001e2 | = | 6 significant figures | |
| 2.000 | = | 4 significant figures | 41003 | = | 5 significant figures |
To express a number like 1000 to 2 or 3 significant figures, you need to use scientific notation, for example, "1.00e3" would give you 3 significant figures.
Do all numerical questions check significant figures?
No. Much of the time, the questions that do check for sig figs show a little sig-fig icon (a 4.0 with a check-mark next to it). If you see this, then the sig figs are definitely checked. If you do not see it, they may be checked on the first submission, or not. It depends on the question coding.
If my answer is marked wrong, is it because the significant figures are wrong?
Mostly, no. Unless you actually see a red hint telling you that the significant figures are incorrect, then the reason for your answer being marked wrong has nothing to do with sig figs. What it means is that your submission is outside of the acceptable tolerance for being marked correct. In other words, you have a bigger problem than sig figs. Do not mess with the sig figs of your answer unless you see the hint.
What is the relationship between the "tolerance" and the number of sig figs?
The "tolerance" is the range of numbers that WebAssign will mark as correct. The tolerance can be set by the problem coder to just about anything. But by default, the tolerance is 2% of the "key". The "key" is the exactly correct value. For example, if the key were "9.81 m/s^2", with a tolerance of 2%, then any entry between "9.6138 m/s^2" and "10.0062 m/s^2" would be counted as correct. This is if there is no sig figs checking.
If sig figs are being checked, then in the question coding there is a variable called $SIGFIGS that is set to a number greater than 0. That number determines how many significant figures there must be in order for the question to be marked correct. By default, any entry that does not have exactly that number of sig figs will be marked totally wrong (0 points awarded), regardless of the value of the number.
If sig figs checking is used, the default tolerance is not a certain percent of the key but is instead usually 1 of the least significant digits. For example, if, in the previous problem, $SIGFIGS was set to 3, then the tolerance would be set to allow answers between 9.80 m/s^2 and 9.83 m/s^2.
However, both the value of $SIGFIGS and the tolerance can be altered by the question coding; this means it can change within a single problem depending on how the question was written. In many lab-related questions, $SIGFIGS is allowed to be set to a range of values, say between 1 and 4. The point is to make the question accept up to a certain maximum number of figures. As the $SIGFIGS value changes depending on what is entered, the tolerance also changes.
Sig figs in numbers with uncertainty
When a value is reported with an uncertainty, special rules apply. The uncertainty can affect the required number of significant figures in the value. A large uncertainty may require that the value be rounded to fewer figures, and a very small uncertainty may imply that the value be known to a greater number of figures. In addition, the uncertainty itself cannot have an arbitrary number of significant figures.
A number that is to be used as an uncertainty itself is fundamentally an estimate: uncertainty cannot be known precisely. Thus, by convention, you must never use more than 2 significant figures to state it.
Remember these two rules for stating a number with an uncertainty: (1) The uncertainty should be stated with 1 or 2 significant figures. (2) A value and its uncertainty should be stated to the same precision: they should have the same number of digits past the decimal point. Usually, if the leading nonzero digit in the uncertainty is greater than one, there should be only one significant digit in the uncertainty. But, if the leading digit is a 1, there may be 2 significant digits in the uncertainty. Units should always be written after the uncertainty. Note the following examples:
| Example | Comment |
|---|---|
| 2.67±0.03 m/s2 | OK |
| 2.7±0.3 m/s2 | OK |
| 2.67±0.13 m/s2 | OK (note leading "1" in uncertainty) |
| 2.672±0.016 m/s2 | OK |
| 2.67±0.0162 m/s2 | Not OK, too many digits in uncertanty. |
| 2.67342±0.02 m/s2 | Not OK, too many digits in value. |
| 2.7±0.32 m/s2 | Not OK, too many digits in uncertainty or too few in value. |
| 2.67±0.3 m/s2 | Not OK, too many digits in value or too few in uncertainty. |
| 2.60e2±0.02 m/s2 | Not OK, too few digits in value. |
| 2.6000e2±0.02 m/s2 | OK, both numbers stated to same precision. |
Because the value and uncertainty must be stated to the same precision, it may require either adjusting the value or the uncertainty to make the precision match. For example "1.56±0.028 N" would be incorrect, but either "1.56±0.03 N" or "1.558±0.028 N" would be correct, assuming both values are within tolerance of their respective keys.
Note however with 2 digits, the associated value must be stated to a greater precision in order to fall within a tighter tolerance. For example, if the key for an uncertainty were "0.028 kg*m/s", then "0.03 kg*m/s" would be accepted, but "0.030 kg*m/s" would not be accepted.
When no uncertainty is stated, the uncertainty of an experimental number is implied by the number of significant figures used to represent it. Take the value 9.81 m/s2. That there are three significant figures implies a real distinction between "9.81" and "9.82" m/s2. Given no other information, you must assume that the uncertainty in the number is 0.005 m/s2 (even though the default WebAssign tolerance would allow for a greater range of answers).
This is too much to keep track of. Tell me what I should DO!
- Assume all tolerances are 1%, regardless of sig figs.
- If you are allowed multiple submissions, enter 1 or 2 more sig figs that you think you should.
- If you get the question marked wrong, but do not see a sig figs hint, (or see a hint like Incorrect number) then you are outside of tolerance, regardless of the number of figures. Recalculate your answer from scratch, keeping all intermediate values you may have needed, to as many figures as possible.
- If you get the question marked wrong AND you ONLY see the sig figs hint, then the ONLY thing wrong is the number of significant figures. You will probably need to round to fewer if you have more than 4 figures in your submission, or extend out to more figures if you have less than 1 or 2 (although there are some that take only one sig fig). When stating numbers with uncertainty, you may need more figures rather than fewer.
ADVICE: Always fill in all blanks to a multi-part question.
Because sig figs in an answer may depend on the value of the uncertainty, it is important that these numbers be there for the program to work. If you see odd behavior, like a previously "correct" entry becoming "incorrect", it is almost certanly due to the fact that a later entry is being used to check an earlier one. In general, WebAssign questions are designed to work by reading all of the entries. Think through the whole problem, not just one number at a time.