Equations

Pooled variance

 

                  

where
                   s  = pooled variance
                   s_a = variance of data set 1
                   s_b = variance of data set 2
                   n_z = size (number of observations) in data set 1
                   n_b = size (number of observations) in data set 2

z score

                  

where
                   z = z score
                   x = data value
                    = mean of data set

t value              

                  

where
                   t = t value
                   s_a = variance of data set 1
                   s_b = variance of data set 2
                   n_z = size (number of observations) in data set 1
                   n_b = size (number of observations) in data set 2
                   = mean of data set 1
                   = mean of data set 2

Effect size (d): t-Test means

                         

where
                   ES = effect size
                   S_P = pooled variance
                  = mean of data set 1
                   = mean of data set 2

Minimum sample size

                  

where
                   n = minimum sample size
                   d = effect size (see above  equation)
                  = the percentile of the unit normal curve which
                            gives the power
                  = the percentile of the unit normal curve for the significance
                              criterion for one-tailed tests, a = a₁, and for two
                               tailed tests a = a₂/2

Delta value

                    

where
                   d = delta value
                   ES = effect size (see above equation)
                   n_z = size (number of observations) in data set 1
                   n_b = size (number of observations) in data set 2

Power values

                  

where
                    = the percentile of the unit normal curve which
                            gives the power
                    = the percentile of the unit normal curve  for the
                            significance criterion for one-tailed tests, a = a₁,
                            and for two tailed tests a = a₂/2
                   d  = effect size (ES in the above formula)
                   n  = the size of each sample

References

See: http://www.mmisoftware.co.uk/pages/library/

Cohen J. Statistical Power Analysis fro Behavioral Sc iences. New York: Academic Press, 1969, 1977, 1988

Minimum, E.W, Roberts, R.C., Coladarci, T. Elements of Statistical Reasoning. Wiley, 1999

Pagano, R.P. Understanding Statistics in the Behavioral Sciences. Brooks/Cole, 1998