Error Analysis and Significant Figures in Chemical Measurement
A result reported without an idea of its uncertainty is not a measurement, and the arithmetic for handling that uncertainty is short.
Two kinds of error
| Systematic | Random | |
|---|---|---|
| Effect on results | Shifts all in the same direction | Scatters both ways |
| Reduced by repeating? | No | Yes, as the average of more trials |
| Detected by | Comparison with a standard or another method | Spread of repeated measurements |
| Typical cause | Miscalibrated instrument, method bias | Reading fluctuations, small variations in technique |
Accuracy and precision
Accuracy is closeness to the true value; precision is agreement among repeated measurements. They are independent, so all four combinations occur, and describing a result requires both.
Significant figures
| Rule | Statement |
|---|---|
| Non-zero digits | Always significant |
| Zeros between non-zeros | Significant |
| Leading zeros | Never significant — they only locate the decimal point |
| Trailing zeros after a decimal point | Significant |
| Trailing zeros with no decimal point | Ambiguous — use scientific notation |
In calculations
- Multiplication and division: the result has as many significant figures as the least precise input.
- Addition and subtraction: the result has as many decimal places as the input with fewest decimal places.
- Round only at the end, never at intermediate steps, since rounding early accumulates error.
- Exact numbers — counts and defined conversions — do not limit significant figures.
The distinction between the two rules matters: multiplication counts significant figures, addition counts decimal places, and applying the wrong one is a common error.
Error propagation
When measured quantities are combined, their uncertainties combine too, and the rule depends on the operation.
Adding in quadrature means squaring, summing and taking the square root. Uncertainties do not simply add, because random errors are as likely to partly cancel as to reinforce.
A practical consequence worth noting: the largest relative uncertainty dominates the result. Improving a measurement that is already the most precise gains almost nothing, so effort should go to the weakest link.
Describing a set of measurements
The mean is the best estimate of the true value in the absence of systematic error. The standard deviation measures spread, and the standard error of the mean — standard deviation divided by the square root of the number of measurements — measures how well the mean is determined.
Because the square root appears, quadrupling the number of measurements only halves the uncertainty in the mean. That diminishing return is why very large numbers of repeats are rarely worthwhile.
Rejecting an outlier
An apparently anomalous result should not be discarded because it looks wrong. Statistical tests exist for deciding whether a value lies far enough from the others to be rejected at a stated confidence level.
Discarding data without such a test, simply because it is inconvenient, is a serious methodological fault, and questions on this topic often make the point deliberately.
Frequently asked questions
Can a precise result be inaccurate?
Yes, and it is common. A systematic error gives tightly clustered results that are all wrong by the same amount.
Why round only at the end?
Because rounding at each step introduces error repeatedly, and those errors accumulate. Carrying extra digits internally and rounding once avoids it.
Why do uncertainties add in quadrature?
Because random errors are independent and as likely to cancel as to reinforce. Simple addition would assume they always reinforce, overestimating the total.
How many measurements are enough?
Enough that the standard error is small compared with the precision required. Because the improvement goes as the square root, there is a point beyond which more repeats are not worth the effort.
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