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:
- The standard deviation (SD) is a measure of dispersion around the mean
- The SEM is the SD of the sampling distribution for the sample mean
- 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
- In a normal distribution, 68.3% of (randomly selected) values fall within +1SD, 95.4% +2SD and 99.7% +3SD
- 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?