According to Gravetter and Wallnau (2013), standard error is “the standard deviation of the distribution of sample means” (p. 207). The standard error is important for research with sample distributions because “it provides a measure of how much difference is expected from one sample to another” (Gravetter & Wallnau, 2013, p. 206). For a research with unknown mean and unknown standard deviation of the population, the standard error becomes a valuable measure because the value of standard error reveals how accurate the sample mean is for estimation of its population mean.
As stated in Gravetter and Wallnau (2013), the standard error is computed as the population’s standard deviation divided by the square root of the sample size:
Standard Deviation of Population = SD
Sample Size = n
Standard Error = SD/SQUAREROOT(n)
This calculation reflects the law of large numbers which states that “the larger the sample size (n), the more probable it is that the sample mean is close to the population mean” (Gravetter & Wallnau, 2013, p. 207). For example, a population the research subject has a mean of µ=90 and a standard deviation of σ = 21.
If we take a sample with size n = 9, then the standard error is calculated as:
Standard Error = 21/SQUAREROOT(9) = 21/3 = 7
If our sample size is bigger: n = 49, then the standard error is calculated as:
Standard Error = 21/SQUAREROOT(49) = 21/7 = 3
The above result indicates that larger samples generate more accurate estimate of population mean due to smaller standard error. As sample size (n) increases from 9 to 49, the size of the standard error decreases from 7 to 3, thus the mean of distribution of sample means for sample size 49 is closer to the population mean.
In summary, the standard error describes the distribution of sample means and measures how accurate the population mean may be estimated by a sample mean. The law of large numbers indicates that the standard error is reduced by increasing the sample size, thus the sample mean is closer to the population mean when researchers select a sample of larger size.
Gravetter, F. J. & Wallnau, L. B. (2013). Statistics for the behavioral sciences (9th ed.). Belmont, CA: Wadsworth.