TWiki> CrySP Web>AppliedStatistics (2021-03-29, JustinTracey)

# Applied Statistics

This page is intended to be a CrySP FAQ for statistics in papers.

## How many sigfigs do I report?

Report standard deviation to 1 sigfig (or, if you're feeling bold, 2 sigfigs if you've run at least 51 experiments, or 3 sigfigs if you've run at least 5001 experiments). Report summary statistics to the precision of the first sigfig of the standard deviation, or the measurement error, whichever is less precise. E.g., if your stddev is 0.02, you can report your mean to two decimal places.

The point of significant figures is to represent the precision to which results are being reported. There are two limits to precision: measurement error (e.g., "how many lines are on my ruler and how accurate are they", "how many times does my clock tick in a second and how evenly spaced are they"), and variability in the distribution of the phenomenon being measured. The former requires some simple rules, which you likely learned in high school science class. The latter involves some math.
But if you do want to actually know the number of sigfigs in your stddev, the answer is you just repeat the process: you report to the precision of the first sigfig of the estimated stddev of your stddev. The formula for estimating the stddev of stddev of a normal distribution is Δs=s/√(2n-2), where Δs is the estimated stddev of the stddev, and n is the number of samples. (There is a more precise formula at this stats stack exchange post , the formula given here is an estimator derived independently in the comments on that post and at this statistics for physics students site .) So suppose you ran 51 experiments. Then Δs=s/10, which means that you can report the standard deviation to 2 digits (the first digit of Δs will be one to the right of the first digit of s). We get our third digit at 5,001 experiments, our fourth at 500,001 experiments, etc., but we've long exited human interpretability at this point.