The TTEST Procedure
Arithmetic and Geometric Means
The arithmetic mean (more commonly called simply the mean) of the distribution of a random variable X is its expected value, 
. The arithmetic mean is the natural parameter of interest for a normal distribution because the distribution of the difference
of normal random variables has a known normal distribution, and the arithmetic mean of a normal difference is equal to the
difference of the individual arithmetic means. (No such convenient property holds for geometric means with normal data, with
either differences or ratios.)
The usual estimate of an arithmetic mean is the sum of the values divided by the number of values:
![\[ \mbox{arithmetic mean} = \frac{1}{n} \sum _{i=1}^ n y_ i \]](images/statug_ttest0026.png){kind=small} |
The geometric mean of the distribution of a random variable X is 
, the exponentiation of the mean of the natural logarithm. The geometric mean is the natural parameter of interest for a lognormal
distribution because the distribution of a ratio of lognormal random variables has a known lognormal distribution, and the
geometric mean of a lognormal ratio is equal to the ratio of the individual geometric means. (No such convenient property
holds for arithmetic means with lognormal data, with either differences or ratios.)
The usual estimate of a geometric mean is the product of the values raised to the power 
, where n is the number of values:
![\[ \mbox{geometric mean} = \left( \prod _{i=1}^ n y_ i \right)^\frac {1}{n} \]](images/statug_ttest0029.png) |