I am doing an experiment and I have the standard deviation but my coworker says I need the standard error. I don't know what the difference between them is. Could somebody please give me a clear distinction between the two?
The reference supplied by Fabrice Clerot is excellent and important. The standard deviation refers to the spread in the population while the standard error refers to the uncertainty of knowing the mean of the population.
The t-test is a statistical measure of the difference in the means of two populations, while the means may be considered different based on the t-test parameters, the populations may have considerable overlap.
Consider populations A and B. The mean of A = 25 and the mean of B equals 26. The means are different according to the data. The standard deviation of each is 10 so the overlap of the two populations is over 90%. If the number of subjects measured in each population is large enough, the t-test will tell us that the means are not only numerically different, but statistically different.
The means are statistically different according to the standard error, but is the difference important according to the standard deviation? The use of the standard error or the standard deviation depends upon the question you are trying to answer, not some statistical cook book.
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Standard deviations and standard errors
Douglas G Altman and J Martin Bland
http://www.ncbi.nlm.nih.gov/pmc/articles/PMC1255808/
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Standarad deviation calculated for a variable and standard deviation of a statistic or function of random variables is called standard error. The standard deviation is a descriptive statistics and standard error is an inferential statistic. Standard error speaks about the population and mostly used in testing of hypothesis and construction of confidence intervals for a population parameter.
The reference supplied by Fabrice Clerot is excellent and important. The standard deviation refers to the spread in the population while the standard error refers to the uncertainty of knowing the mean of the population.
The t-test is a statistical measure of the difference in the means of two populations, while the means may be considered different based on the t-test parameters, the populations may have considerable overlap.
Consider populations A and B. The mean of A = 25 and the mean of B equals 26. The means are different according to the data. The standard deviation of each is 10 so the overlap of the two populations is over 90%. If the number of subjects measured in each population is large enough, the t-test will tell us that the means are not only numerically different, but statistically different.
The means are statistically different according to the standard error, but is the difference important according to the standard deviation? The use of the standard error or the standard deviation depends upon the question you are trying to answer, not some statistical cook book.
"The standard error of the sample mean depends on both the standard deviation and the sample size, by the simple relation SE = SD/√(sample size). The standard error falls as the sample size increases, as the extent of chance variation is reduced—this idea underlies the sample size calculation for a controlled trial, for example. By contrast the standard deviation will not tend to change as we increase the size of our sample" (Altman and Bland, 2005).
From a different point of view: The standard error is the square root of the inverse expectation of the Fisher information at the estimate. The standard deviation is the special case for n=1.
The answer by Mark Camilleri reminds me I should have made a stronger statement following, "If the number of subjects measured in each population is large enough, the t-test will tell us that the means are not only numerically different, but statistically different." The difference between A = 25 and B = 26 may not be important, but if you add enough subjects (without changing the standard deviation) the standard error will become small enough to have statistical significance. In other words, given enough samples an insignificant difference in means can be statistically significant.
This problem of insignificant significance has let to suggestions that sample size should be limited. This may have merit in some situations, but one size does not fit all. The better course is to have the analysis fit the question asked.
A further note: Standard deviation and standard error are sometimes transformed to other units or meanings such as from total to rate (per unit time or per unit volume). Mathematically correct transformations or substitutions do not necessarily result in statistically correct transformations or substitutions. For an example see
https://www.researchgate.net/publication/259438788_Correction_to_the_Count-rate_Detection_Limit_and_SampleBlank_Time-Allocation_Methods
Article Correction to the Count-rate Detection Limit and Sample/Blan...
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