Question 10 of 20 Fundamentals Concept
What is the difference between a standard deviation and a standard error?
Short answer
The standard deviation describes how spread out individual measurements are. The standard error describes how uncertain an estimate such as the mean is, and equals the SD divided by the square root of the sample size, so it shrinks as n grows while the SD does not.
Why
Altman and Bland put it simply. Use the SD to say how widely scattered the measurements are, and the standard error of the mean to show the uncertainty of the estimated mean. If ten mice have tumor volumes with a mean of 400 cubic millimeters and an SD of 120, the SD tells you a typical mouse differs from the mean by roughly that much.
The SE is 120 divided by the square root of 10, about 38, and describes how much the mean itself would move if the whole experiment were repeated. Collect 100 mice and the SD stays near 120, because mice do not become more alike, but the SE falls to 12. That is why error bars showing SE look reassuringly small and why a figure must say which one it shows.
Cumming and colleagues list SD as descriptive and SE and confidence intervals as inferential, and recommend that every legend state the type of bar and the n. A 95 percent confidence interval is roughly the mean plus or minus two SEs for large samples, and wider for small ones. The SE is only meaningful when n counts independent units.
Three technical replicates of one animal give an SE that says nothing about animals.
A strong answer covers
- SD describes spread of individual values
- SE describes uncertainty of the mean
- SE equals SD divided by the square root of n
- SD stays stable as n grows, SE shrinks
- Label error bars and count independent units
Common mistakes
- Showing SE error bars to describe how variable individual samples are
- Reporting mean plus or minus a number without saying whether it is an SD or an SE
- Computing an SE from technical replicates and presenting it as biological uncertainty
The interviewer might ask next
Why does a 95 percent confidence interval need a t multiplier rather than 1.96 when n is 5?