Contents	Introduction	Descriptive	Hypothesis	Tables	Appendix

Four criteria are used to estimate the appropriate sample size for a study. Sometimes called a power analysis, the primary goal  is to collect enough cases to ensure you don't make a Type II error (Beta). Beta is the probability of not rejecting a null hypothesis when it should have been rejected.  Power is 1- beta and is defined as the probability of correctly finding statistical significance.  A common value for power is .80 or an 80% chance that a random sample will find a statistically significant difference when there truly is a difference in the population. The discussion that follows is focused specifically on 2-tailed tests between two independent random samples. What follows are two examples of power analysis.  There are many other approaches and equations for estimating sample size.

Criteria

Significance (alpha): This is the threshold for finding statistical significance. Normally this is set at .05, a 5% chance of rejecting a null hypothesis when there is in fact no significant difference or relationship between the underlying populations. As alpha gets smaller, sample size requirements increase.

Statistical power (1-beta):  The probability of finding a statistically significance difference or relationship when there truly is one in the underlying populations.  This is also known as statistical power. A power of .80 is common.   As power increases so does the sample size requirements.

Expected difference (effect size):  This is the expected difference or relationship between two independent samples. Also known as the effect size.  The obvious questions is how do we know what difference we will find if we have not yet conducted the sampling?  If possible, it may be useful to use the effect size found in prior studies.  In many, if not most cases, the effect size is determined by literature review, logical assertion, and conjecture.

Variability in the population:  This is the expected variability in the samples.  As with effect size, this must either be based on prior knowledge or logical assertion. In the examples that follow, standard deviations will be used for the variability measure. The formulas presented here assume the variability in the the two samples are the the same (homogeneous).
 

Sample size estimation for tests between two independent sample proportions

Formula

where

N= the sample size estimate
Zcv=Z critical value for alpha (.05 alpha has a Zcv of 1.96)
Zpower=Z value for 1-beta (.80 power has a Z of 0.842)
P1=expected proportion for sample 1
P2=expected proportion for sample 2

Sample size estimation for tests between two independent sample means

Formula

where

N= the sample size estimate
Zcv=Z critical value for alpha (.05 alpha has a Zcv of 1.96)
Zpower=Z value for 1-beta (.80 power has a Z of 0.842)
s=standard deviation
D=the expected difference between the two means