# Interpreting numerical value of standard error of the mean

I am unclear about how to interpret the value of the Standard Error of the Mean (SEM) directly. For example, when a mean is reported as 5.00 + 0.50SEM, how do you directly relate the 0.50 to 5.00?

To quickly run through the basic theory concerning the standard error:

1. The standard deviation (SD) is a measure of dispersion around the mean
2. The SEM is the SD of the sampling distribution for the sample mean
3. This sampling distribution is derived from the means of an infinite number of samples from a statistical population and is normally distributed according to the Central Limit Theorem
4. In a normal distribution, 68.3% of (randomly selected) values fall within +1SD, 95.4% +2SD and 99.7% +3SD
5. The SEM decreases with increasing sample size

QUESTION 1: Given points 2) and 4) is it correct to interpret the SEM in the same way as SD as a descriptive statistic? That is, for a sample with mean 5.00 and SEM 0.50, is it correct to conclude the true population mean lies between 4.50 and 5.50 with probability 68.3%?

QUESTION 2: For a given statistical population being sampled, does the sampling distribution change with sample size (given point 5)? i.e. should point 3) be: "...derived from the means of an infinite number of samples of a given size from a statistical population..."?

QUESTION 3: Since the SEM is not calculated directly but estimated from the SD of a sample, what effect does departure from a normal distribution of the sample have on calculation of the SEM? Put another way, if a sample has for example a highly skewed distribution, calculating the SD is inappropriate (it is a property of a normal distribution only), so can the SEM still be estimated reliably from the SD?

• There's a lot of misinformation here, so to help you along, note that #3 is incorrect except when the original population has a normal distribution itself (the CLT does not apply, because we're talking about a finite sample); therefore #4 is not always applicable--although it's a pretty good rule of thumb nonetheless. #5 is only a general tendency--the SEM can increase for a while as the sample size goes up. Re question 3: the SD is a property of most distributions, not just the normal distributions. – whuber Jan 29 '13 at 8:33
• I'm confused why you refer to a finite sample for point #3. Isn't the sampling distribution of the mean theoretical, and thus describes an infinite population and the CLT applicable? I'm a biologist, not a statistician, so everything I've read is pitched at the expected level of understanding. Nevertheless, everything about the CLT I've read states point #3 except I should have added the word "approximately normal". If you could suggest a text that may help me, as a biologist, get my head around this theory and answer the 3 questions I posed that would be FANTASTIC! – DeanP Feb 8 '13 at 1:32
• I like the Freedman, Pisani, Purves book and I think you might enjoy it, too. – whuber Feb 8 '13 at 3:10