1. Sample Size
Copyright  2008 by The McGraw-Hill Companies. This material is intended for educational purposes by licensed users of LearningStats. It may not be copied or resold for profit.

2. Sample Size For a mean: For a proportion: where
n = the required sample size
e = the desired error
z = z value for the desired confidence level.
Comment Since s is often unknown, a small-scale pilot study may be required to estimate s using s.
Comment Since p is unknown, a conservative assumption is to set p = This will guarantee a large enough sample.

3. Sample Size for a Proportion
Problem How large a sample should be taken to estimate the proportion of dissatisfied HMO patients with an error of  0.03 and 98% confidence?
Solution Since p is unknown, set p = For 98% confidence we use z = For a desired error of  0.03, the required sample size is 1503.
Note If an estimate p is available from a preliminary sample, use it (especially if you are pretty sure that p is not 0.50).

4. Effect of Confidence Level
Problem How large a sample should be taken to estimate the proportion of dissatisfied HMO patients with an error of  0.03? The sample size increases with the desired confidence level.
Comment Samples sizes for a proportion are often quite large. For example, national political polls that require error rates of 2-3% often survey over 1000 people.

5. Sample Size for a Mean
Problem How large a sample should be taken to estimate the mean waiting time of patients at a clinic with an error of  1.5 minutes and 95% confidence?
Solution The formula requires the standard deviation, so we take a small preliminary sample and obtain s = 8.37 minutes. For 95% confidence we use z = For a desired error of 1.5, the required sample size is 120.
Note To estimate s, a small preliminary sample is probably an inevitable pre-task.

6. Effect of Confidence Level
Problem How large a sample should be taken to estimate the mean waiting time of patients at a clinic with an error of  1.5 minutes? The sample size increases with the desired confidence level.

7. Tradeoffs in Sample Size
Problem The client says it would be too expensive and time-consuming to survey 1503 patients. The statistician is asked for an alternative.
Solution The statistician suggests lowering the confidence level to 90% (using z = 1.645) and raising the desired error to 5%. The resulting sample size is The client agrees that this will suffice, and appreciates the reduced cost of data gathering.

8. But what if you can’t guess R?
The Empirical Rule
Problem How large a sample should be taken to estimate the mean waiting time of patients at a clinic with an error of  1.5 minutes and 95% confidence? We do not know s.
Solution Instead of taking a small preliminary sample, if we know the approximate range (R) we can set s = R/6, since for a normal distribution the range is about m  3s. If we assume that patients wait between 0 and 60 minutes, we could set s = 60/6 = 10 (a “good guess” about s). For 95% confidence and a desired error of 1.5, the required sample size is 171.
But what if you can’t guess R?

9. Problems with the Empirical Rule
Problem Recent research suggests that the rule for estimating s = R/6 is often too conservative, i.e., it may result in sample sizes that are too small. A better rule might be to set s = R/4. Alternatively, just take a small preliminary sample and use s as an estimate of s.
Suggested Reading Richard H. Browne, “Using the Sample Range as a Basis for Calculating Sample Size in Power Calculations,” The American Statistician, Vol. 55, No. 4, November, 2001, pp

10. Finite Populations For a mean: For a proportion:
Finite population adjustment:
Comment If the population is finite and sampling is without replacement, the sample size should be adjusted. However, this adjustment will have little effect if the sample size is less than 5% of the population (i.e., you can ignore the adjustment if n/N < 0.05).

11. Other Sample Size Issues
The scientific aura of sample size formulas may divert attention from more fundamental issues such as:
Goals of the proposed study
Statistical significance vs. importance
Practical and ethical criteria
However, these formulas can help structure the dialogue between statistician and client.
Suggested Reading Russell V. Lenth, “Some Practical Guidelines for Effective Sample Size Determination,” The American Statistician, Vol. 55, No. 3, August, 2001, pp

12. Commercial Software
Multi-purpose software packages (e.g., SAS, SPSS, Minitab, Statistica) will perform sample size and power calculations. You might also try specialized vendors such as these:
Biostat
NCSS
nQuery
StatXact
Source Recent copies of Amstat News. This list is not intended to be suggestive, not comprehensive.