There appear to be two different definitions of the standard error.

The standard error of a sample of sample size  is the sample's standard deviation divided by . It therefore estimates the standard deviation of the sample mean based on the population mean (Press et al. 1992, p. 465). Note that while this definition makes no reference to a normal distribution, many uses of this quantity implicitly assume such a distribution.

The standard error of an estimate may also be defined as the square root of the estimated error variance  of the quantity,

(Kenney and Keeping 1951, p. 187; Zwillinger 1995, p. 626).

REFERENCES:

Kenney, J. F. and Keeping, E. S. Mathematics of Statistics, Pt. 1, 3rd ed. Princeton, NJ: Van Nostrand, 1962.

Kenney, J. F. and Keeping, E. S. "Standard Error of the Mean." §6.5 in Mathematics of Statistics, Pt. 2, 2nd ed. Princeton, NJ: Van Nostrand, pp. 110 and 132-133, 1951.

Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, 1992.

Zwillinger, D. (Ed.). CRC Standard Mathematical Tables and Formulae. Boca Raton, FL: CRC Press, 1995.