The factors determining the number of replications:

• The variance
• The size of the difference that has physical significance, between two means .
• The significance level of the test , or the probability of Type I error.
• The power of the test (the probability of detecting difference ) where is the probability of Type II error.

The required replication number for each treatment group , for two-sided alternatives, can be estimated with the following formula:

(1)

where is the value of the standard normal variate exceeded with probability and is exceeded with probablity . Typical values are and , yielding the power of the test of . Let us comment about the Stevens measurement classification:

• nominal (also categorical or discrete)
• ordinal
• interval
• ratio

The formula (1) applies to interval measurements, such as temperature in the Fahrenheit or Celsius scale. When we are dealing with a ratio measurement, such as temperature in the Kelvin scale, we may modify the above discussion by using relative analogues of and . We assume that the expected value of the measurement is on the ratio scale. The percent coefficient of variation is defined as follows:

where is the true standard deviation and is the true expected value of the mean. Similarly, we may define the measure of the difference as percentage of the overall expected mean:

The values and may be used in place of and in formula (1) if one prefers the ratio scale. The factors that result in increase of ;

• The variance, or increases.
• The size of the difference, or , decreases.
• The significance level of the test decreases.
• The power of the test increases.

This figure illustrates why the replication number formula works well. It was generated with R, and it uses somewhat realistic values of the parameters. An attached document contains a proof of a related inequality. (Click on the image to magnify.)