Standard Error - Assumptions and Usage

Assumptions and Usage

If the data are assumed to be normally distributed, quantiles of the normal distribution and the sample mean and standard error can be used to calculate approximate confidence intervals for the mean. The following expressions can be used to calculate the upper and lower 95% confidence limits, where is equal to the sample mean, is equal to the standard error for the sample mean, and 1.96 is the .975 quantile of the normal distribution:

Upper 95% Limit =

Lower 95% Limit =

In particular, the standard error of a sample statistic (such as sample mean) is the estimated standard deviation of the error in the process by which it was generated. In other words, it is the standard deviation of the sampling distribution of the sample statistic. The notation for standard error can be any one of, (for standard error of measurement or mean), or .

Standard errors provide simple measures of uncertainty in a value and are often used because:

• If the standard error of several individual quantities is known then the standard error of some function of the quantities can be easily calculated in many cases;
• Where the probability distribution of the value is known, it can be used to calculate a good approximation to an exact confidence interval; and
• Where the probability distribution is unknown, relationships like Chebyshev's or the Vysochanskiï–Petunin inequality can be used to calculate a conservative confidence interval
• As the sample size tends to infinity the central limit theorem guarantees that the sampling distribution of the mean is asymptotically normal.