1.4 Independent and identically distributed random variables with random sample size.In regression analysis, the term "standard error" refers either to the square root of the reduced chi-squared statistic, or the standard error for a particular regression coefficient (as used in, say, confidence intervals). In other words, the standard error of the mean is a measure of the dispersion of sample means around the population mean. Therefore, the relationship between the standard error of the mean and the standard deviation is such that, for a given sample size, the standard error of the mean equals the standard deviation divided by the square root of the sample size. This is because as the sample size increases, sample means cluster more closely around the population mean. Mathematically, the variance of the sampling distribution obtained is equal to the variance of the population divided by the sample size. This forms a distribution of different means, and this distribution has its own mean and variance. The sampling distribution of a mean is generated by repeated sampling from the same population and recording of the sample means obtained.
If the statistic is the sample mean, it is called the standard error of the mean ( SEM). The standard error ( SE) of a statistic (usually an estimate of a parameter) is the standard deviation of its sampling distribution or an estimate of that standard deviation. These statistical techniques are more appropriate to large-scale studies with many calculated results to average. For a value that is sampled with an unbiased normally distributed error, the above depicts the proportion of samples that would fall between 0, 1, 2, and 3 standard deviations above and below the actual value. A more rigorous method for treating repeated measurements is to calculate standard deviations and standard errors (the standard deviation divided by the square root of the number of trials).
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