Define the following notation:










The mean estimate , standard deviation estimate s, and standard error are computed as follows:






The confidence interval for the mean is






The t value for the test is computed as
The pvalue of the test is computed as
The equaltailed confidence interval for the standard deviation (CI=EQUAL) is based on the acceptance region of the test of that places an equal amount of area () in each tail of the chisquare distribution:
The acceptance region can be algebraically manipulated to give the following confidence interval for :
Taking the square root of each side yields the CI=EQUAL confidence interval for :
The other confidence interval for the standard deviation (CI=UMPU) is derived from the uniformly most powerful unbiased test of (Lehmann, 1986). This test has acceptance region
where the critical values and satisfy
and
where is the PDF of the chisquare distribution with degrees of freedom. This acceptance region can be algebraically manipulated to arrive at
where and solve the preceding two integrals. To find the area in each tail of the chisquare distribution to which these two critical values correspond, solve and for and ; the resulting and sum to . Hence, a confidence interval for is given by
Taking the square root of each side yields the CI=UMPU confidence interval for :
The DIST=LOGNORMAL analysis is handled by logtransforming the data and null value, performing a DIST=NORMAL analysis, and then transforming the results back to the original scale. This simple technique is based on the properties of the lognormal distribution as discussed in Johnson, Kotz, and Balakrishnan (1994, Chapter 14).
Taking the natural logarithms of the observation values and the null value, define




First a DIST=NORMAL analysis is performed on in place of . The geometric mean estimate and CV estimate of the original lognormal data are computed as follows:




The t value and pvalue remain the same. The confidence limits for the geometric mean and CV on the original lognormal scale are computed from the confidence limits for the arithmetic mean and standard deviation in the DIST=NORMAL analysis on the logtransformed data, in the same way that is derived from and is derived from s.