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What is the standard error of the measurement?
The standard error is the standard deviation of the sampling distribution of a statistic.
For example, suppose you are estimating the mean height of the population of eastern white pines. You select a sample of 100 trees, measure their height, and calculate a mean. Any given sample mean will be a function of the population mean, AND the random unique characteristics of the individual trees in the sample. Thus, if I were to take another sample of 100 trees, that mean would be a little different, and so would the mean of a third sample, and so on. If I calculated means for a very large number of samples of the same size, this sample of sample means would themselves have a mean value and a standard deviation. The mean of this "sampling distribution" would be the population mean, and the standard deviation is the standard error of the measurement. In this case the measurement is the mean, but it can be any sample statistic. The standard error tells us how much we can expect any given sample statistic to deviate from the population parameter we are estimating.
Just like a sample standard deviation from our tree example above tells us how much we can expect each tree to deviate from the mean of its sample, the standard error tells us how much we can expect any given statistic to deviate from its sampling mean, and remember, the mean of the sampling distribution is the actual population parameter value. The standard errror thus allows us to create confidence intervals and test hypotheses at a specified level of uncertainty, (e.g., 95% percent confidence, alpha: p<0.05, that sort of thing).
The problem is that we never actually collect a large number of samples, but often only one. So we have to estimate the standard error. The formulas for the estimate of the standard error can be simple or complex, but fortunately, there are computer programs to do this for us these days.
A good reference for this topic is Hays, W.L, (1981). Statistics, Third Edition. New York: Holt, Reinhart & Winston. See Chapter 5, Sampling Distributions and Point Estimation.
If you have further questions, send E-mail to stats@ssc.utexas.edu.