# Number of significant figures to report for a confidence interval

I'm a TA in stat for med students and when discussing confidence intervals together with the one-sample t-test, I got doubts on the decimals that should be retained. Specifically, we tested a data set of the body height of 3400 people against the null hypothesis of the average being equal to 170 cm. R gave a sample mean of 172.3974 with a confidence interval from 172.0846 to 172.7102. Sample SD was 9.8, the input heights were given in whole numbers.

A participant noted that giving seven significant figures would equal faking precision. I agree, but with such a narrow CI, should it really only be three like in the input data?

Random generation of 3400 body heights up to microns with the same mean and SD indicated that the first five significant figures of the upper and lower bound remained the same after cropping the four simulated decimals.

What's more, if both bounds of the interval rounded to the same integer, as could easily have been the case, the CI would be a single number.

• Why do you have an arbitrary null hypothesis about the mean height? Is it a hypothesis made up in order to be able to use a hypothesis test? If so then consider whether you really need to test a hypothesis. Most likely the estimation provided by the confidence interval is all that you will need. Jan 13, 2018 at 20:40
• The hypothesis was made up to demonstrate the relationship between p values and confidence intervals, and a point was made that the latter are at times more intuitive. Pedagogically it would certainly be preferable to replace it with an H0 that corresponds to some actual prior belief in the future, but the same question could still arise. Jan 14, 2018 at 12:19

The standard error of the mean (to 1 or 2 significant digits, I chose 2) is 9.8/sqrt(3400) = 0.17. The mean should be reported with the number of significant digits that makes it consistent with its standard error, i.e., 172.40. The result should be reported as 172.40 +/- 0.17 and you should identify it as being the estimated value of the height and its standard uncertainty. The Guide to Uncertainty in Measurement (GUM ) takes us this far. I have not seen explicit guidance on the number of significant digits to give in the upper and lower coverage intervals that are constructed with the mean, standard uncertainty, and coverage factor, but is seems reasonable to express them to the same number as the standard uncertainty, i.e., (172.07, 172.73) if we assume normality (coverage factor = 1.96) and want a 95% coverage interval.

To report the confidence interval I would use one extra digit of precision beyond the precision in the data. If the data are reported to full centimetres, use one decimal for the confidence interval.

• The first paragraph is good advice, but the second paragraph is very sloppily worded and would support the dangerous conflation of statistical significance within a model and a real world truth. Jan 13, 2018 at 20:41
• Is there any authoritative statement (like the GUM above, which doesn't seem to cover CIs) to refer to? Jan 14, 2018 at 20:40

You use the lowest number of sig figs in the calc. Since the lowest number of sig figs is associated with the measured height (again assuming that you had to ensure this is the case by actually measuring the height of the participant), then the sig figs in 170 cm will rule. Here that's 2.

This makes sense as I am not aware of a measuring device that for this application that is precise to 4 or 5 decimal places as your confidence interval suggests.

A typical measurement used for this application would typically be reported to one decimal place such as 170.0. If this is the case, then you have 4 sig digits.

Lastly if you are using the SD in the calc and it is 0.17, again you are at 2 sig figs.

170 has 2 sig figs 170.01 has 5 sig figs

simply convert to scientific notation if in doubt 1.7 x 10^2 vs 1.7001 x 10^2

Please note that my comments are directly related to sig figs and not the process applying various statistical tests and/or what is included in those stat calcs.