Fundamentals of Statistics contains material of various lectures and courses of H. Lohninger on statistics, data analysis and chemometrics......click here for more. 
Home Univariate Data Measures of Location Mean Confidence Interval of the Mean  
See also: mean, t distribution  
Confidence Interval of the MeanIf the mean is calculated from a sample, one wants to know the probability that the true mean lies within certain limits around the calculated mean. These limits are established by the confidence interval. The width of the confidence interval depends on the probability P that the true mean is found within the limits. The probability P is usually specified as α = 1P; α is called the level of significance. The confidence interval of the mean is defined by with t_{n1;1α/2} being the critical value of the t distribution with n1 degrees of freedom and a level of significance of 1α/2. The factor is called the standard error of the mean and can be regarded as some kind of standard deviation of the mean, because the mean of samples shows a variation which is controlled by the central limit theorem of statistics; its standard deviation decreases with the square root of n. When reading publications one often finds the precision of measurements specified in different (and confusing) ways: some authors indicate the confidence interval as defined above, others specify the standard error only, and still others use the standard deviation of the sample to indicate the uncertainty of a measurement. In general, one should always specify the confidence interval (and its level of significance) together with the mean. The standard error overestimates the actual precision of the measurement (i.e. the standard error is alway narrower than the confidence interval) while the standard deviation of the sample underestimates it.


Home Univariate Data Measures of Location Mean Confidence Interval of the Mean 