Significant figures
Open the Precision and significant figures room
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01 // Why they matter
Significant figures are the digits in a measurement that carry real meaning about its precision, including all certain digits plus one estimated digit. If a graduated cylinder is marked every 1 mL, reporting a reading as 24.638 mL claims a precision the tool cannot actually provide — the reading should stop at 24.6 mL.
When you calculate with measured numbers (adding, multiplying), the answer can only be as precise as your least precise measurement. Rounding a calculated result to the right number of significant figures keeps a lab report honest about how precise the data really was.
02 // Counting rules
| Rule | Example | Sig figs |
|---|---|---|
| All non-zero digits count | 247 | 3 |
| Zeros between non-zero digits count | 1005 | 4 |
| Leading zeros never count | 0.0034 | 2 |
| Trailing zeros count only with a decimal point | 1500 | 2 (unless written 1500.) |
| Trailing zeros after a decimal point count | 2.50 | 3 |
03 // Worked example
Multiply a length by a width measured with different precision, then round the result correctly.
| Length | 12.5 cm (3 sig figs) |
| Width | 4.2 cm (2 sig figs) |
Formula For multiplication/division, round the answer to match the fewest significant figures of any measurement used.
04 // Distinguish measured digits from exact counts
Significant figures communicate how precisely a quantity was measured. A measured value includes all certain digits plus one estimated digit when the instrument and procedure allow it. Exact counts, such as three samples or twelve trials, do not have measurement uncertainty and do not limit calculated precision in the same way. Defined conversion factors may also be exact within the stated system.
Zeros can be significant or placeholders. Zeros between nonzero digits are significant; leading zeros locate the decimal point and are not significant. Trailing zeros after a decimal point communicate measured precision, while trailing zeros in a whole number can be ambiguous unless scientific notation is used. Writing 2.50 × 10² makes clear that three significant digits are intended.
05 // Apply the operation-specific rules
For multiplication and division, the result generally has as many significant figures as the input with the fewest significant figures. For addition and subtraction, the result is rounded to the least precise decimal place. These rules reflect different ways uncertainty propagates: multiplication depends on relative precision, while addition depends on the place value of the uncertainty.
For example, multiplying 2.4 by 3.12 gives a calculator result of 7.488, but the first factor has two significant figures, so the reported product is 7.5. Adding 12.1 and 0.34 gives 12.44, which is reported to the tenths place as 12.4. Follow the specific conventions taught for multi-step calculations and retain extra digits until the final step.
06 // Round only when the calculation is complete
Rounding at each intermediate step can accumulate error. Keep guard digits in the calculator, then round once at the end according to the least precise measurement and the assignment's convention. If the next digit is five or greater, common classroom rules round up; scientific software may use other tie-breaking conventions, so use the method specified by the instructor.
A displayed calculator value may contain many digits because the device computes numerically, not because the experiment supports that precision. Report a value that matches the actual instruments. A digital scale reading 12.34 grams does not justify reporting a derived mass as 12.340000 grams unless another exact operation or convention explains the extra digits. Precision in writing should follow evidence.
07 // Connect precision to uncertainty
Significant-figure rules are practical classroom tools, not a complete uncertainty analysis. A well-designed report may instead state measurement uncertainty, repeatability, instrument resolution, and confidence intervals. Two values with the same number of digits can have different quality if one instrument is calibrated and the other is not. Digits alone do not prove accuracy.
If a measured input is recorded with too few digits, useful information may be lost; if it is recorded with too many, the report implies false certainty. Read the scale carefully, record the estimated digit consistently, and include units. When a rubric specifies significant figures, apply it consistently while also explaining important limitations in plain language.
08 // Use scientific notation to show intended precision
Scientific notation removes ambiguity from trailing zeros. The value 4.2 × 10³ has two significant figures, while 4.200 × 10³ has four. Both represent the same approximate magnitude, but they communicate different measurement precision. Choose the form that matches the instrument record and use it consistently in calculations and tables.
When a result is close to a rounding boundary, keep extra digits until the final operation is complete. Then apply the class rule and report the units. If the assignment uses explicit uncertainty instead of a significant-figure rule, report that uncertainty and follow the specified convention rather than applying a second, conflicting rounding scheme.
09 // Keep precision consistent through a worked result
Suppose a rectangle measures 4.2 centimeters by 3.15 centimeters. Multiplication gives 13.23 square centimeters, but the 4.2 centimeter measurement has two significant figures, so a classroom report would normally state 13 square centimeters. Keep 13.23 in intermediate work, then round the final area once. The unit is squared because two lengths were multiplied.
If the same measurements were added, the decimal-place rule would apply instead: 4.2 plus 3.15 is reported to the tenths place as 7.4 centimeters. This contrast shows why students should identify the operation before applying a rule. Always follow the teacher's stated convention when uncertainty analysis is more detailed than the shortcut rules.
10 // Questions people ask
Do zeros always count as significant figures?
Not always. Leading zeros (like the zeros in 0.0034) never count. Zeros between other digits always count. Trailing zeros only count when there is a decimal point present, like the zero in 2.50.
How many significant figures should a final lab answer have?
For multiplication or division, match the number of significant figures in the least precise measurement used in the calculation. For addition or subtraction, match the fewest decimal places instead.
Why not just report every digit a calculator shows?
Because a calculator does not know how precise your original measurements were — it will happily give you ten digits from a measurement that was only accurate to two. Reporting more digits than your tools can support overstates your precision.