Confidence Interval of the Mean

If 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 = 1-P; α is called the level of significance.

The confidence interval of the mean is defined by

,

with t_n-1;1-α/2 being the critical value of the t distribution with n-1 degrees of freedom and a level of significance of 1-α/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.

Hint 1:

The correct calculation of the confidence interval requires the sample to be normally distributed.

Hint 2:

In many textbooks the confidence interval of the mean is derived by discussing populations and large samples. Consequently the z-transformation is introduced without paying much attention to the t-distribution. In fact, as the t-distribution converges to the normal distribution for large sample sizes, it is much more convenient to use the t-distribution for both small and large samples.