Tutorial: Significant Figures

New (and not so new) scientists often don’t think about how many digits, or significant figures they should include when reporting numerical data.  Unless you’re using a finely calibrated instrument, most readings aren’t accurate beyond one or two percent (about two significant figures).  But that doesn’t stop a computer display or other electronic device from spitting out readings with a whole lot of digits.  As a consequence, people commonly report data with much more precision than is justified by the measurement technique they use.

Here’s the right way to report numerical data which has an error estimate (e.g. an average +/- standard deviation, or an average +/- 95% confidence interval):

1) Report error estimates with only one sigificant figure, unless the first digit is a 1 or 2.  In that case report two digits.  Round up or down as appropriate.

2) Report the average result using as many significant figures as are justified by the digits in the reported error.  Round as appropriate.

Examples:
Average = 43.695, standard deviation = 5.344, report 44 +/- 5
Average = 10.1113, standard deviation = 0.1249, report 10.11 +/- 0.12
Average = 23,765, standard deviation = 437, report 23,800 +/- 400
Average = 7,390,012, standard deviation = 251,912, report 7,390,000 +/- 250,000
Average = 0.00539, standard deviation = 0.00719, report 0.005 +/- 0.007

If your data has no error estimate and is a continuous number — say a ruler measurement  — report only as many significant figures as can be accurately and reproducibly measured.  If you can truly measure with an accuracy of 1/100th of an inch, report 17.35 inches, but if a reliable measurement can be made only to within 1/10th of an inch, report 17.4 inches.

In the case of a single discrete measurements (e.g. an accurate count of two-story houses in a town), it is correct to report the actual measurement with all digits:  13,271 houses.  However, as soon as such measurements are averaged together and you have a variability estimate for the mean, follow the rules above.  If the average over 12 communities = 13,271 houses, and the standard deviation = 598, report 13,300 +/- 600 houses (When randomly sampling a single town, this is your best guess of what the count of two story houses will be.).

Finally, when the accuracy of a measurement is  unknown, use common sense to limit the number of significant figures to two, unless the first digit is a 1 or 2, in which case report 3 significant figures. [This implies a precision of approximately 1%; very few natural measurements are more precise than this when examined across multiple individuals, locations or time points].  For example, researchers may well have accurately counted 75,456 responding patients and 23,499 non-responders, but a more sensible way to report those numbers is 75,000 and 23,500.  Similarly, 0.02987 would be reported as 0.0299 (three digits), and 0.0495 would be 0.050 (two digits).

Exceptions:  Sometimes using the same number of trailing or leading zeros is a useful technique to help facilitate comparisons.  For example, using Table 1 on the left makes it easier to compare results up and down the column, even though some entries have more significant figures than are justified by the error measurement (Table 2 has the correct number of significant figures).

Table 1                                 Table 2
Avg.         Error                    Avg.        Error
0.0123     0.0007                0.0123    0.0007
1.2405     0.5674                 1.2            0.6
0.0046    0.0030                0.005       0.003
0.0005    0.0004                0.0005    0.0004

And to make the comparison even easier, use a fixed width font!

Additional note on vocabulary:  Precision is the extent to which repeated measurements agree with each other, accuracy is the extent to which the measurements match the true value.

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