# On the use of Error bars

The following article of**Geoff Cumming,Fiona Fidler,1 and David L. Vaux: Error bars in experimental biology, Journal of Cell biology, 177, 1, 2007 7-11,**has momentarily the second highest readership in Mendeley: it addresses the meaning of error bars in scientific papers. It mentions the following rules:

- Rule 1: When showing error bars, describe what they are.

- Rule 2: Indicate the number of experiments.
- Rule 3: Use error bars and statistics only for independently repeated experiments, not replicates.
- Rule 4: Show the standard error SE=SD/n
^{1/2}or confidence interval CI rather than standard deviation SD. - Rule 5: SE can be doubled in width to get 95 percent CI if n is 10 or more.
- Rule 6: A gap of SE indicates P value 0.05 ands statistic significance if n is 10 or more.
- Rule 7: with 95 CI and n=3, overlap of one full arm indicates P=0.05
- Rule 8: For repeated measurements of same groups, CI, SE are irrelevant for comparisons within the same group.

**standard deviation SD**and the

**standard error SE**(which is SD divided by the square root of the number of experiments). An other important quantity in scientific experiments is the

**P-value**: Assume we have a random variable X over some probability space and we measure X=c. The question is, whether this experiment is significant or not assuming a null-hypothesis (which stands for the setup of our probability space). The P-value of the experiment is defined as

p = P[ X > c]By convention (note this is arbitrary and therefore a bit controversial), one calls p smaller than 0.05 a

**statistically significant**result and a P-value smaller than 0.01 a

**highly significant**result. For example, if you see 10 times head when throwing a coin 30 times. What is the P value? If X is the number of heads, then P[X smaller or equal to 10] = F[10], here computed with Mathematica

f=CDF[BinomialDistribution[30,0.5]]; f[10]is 0.0493. This is considered statistically significant. With this assumptions we would considered it a significant test that the null hypothesis (the coin is fair and different coin experiments are independent) is rejected. However, if see 11 heads, then the P-value is slightly larger than 0.1 and the test is not significant. You see how easy it is to cheat here: Just repeat your coin flipping experiment a lot until you reach an instance with a statistically significant deviation result. This will eventually happen. Suppress the other experiments as "test trials" and publish the result.