Calculating Sample Size
In order to be able to accurately project the results of a survey question from the sample to the
entire population of the target market, the correct sample size must be used. The correct sample
size can be calculated using this formula:
Z 2  2
n
E2
in which:
n = Sample Size
Z = Level of Significance (Expressed as a Z-Score)
 = Population Standard Deviation (2 = Population Variance)
E = Acceptable Amount of Sampling Error
The Z-score in this equation can be looked up from a table that shows the probability of a sample
error. An example of this is the table of Z-values contained in the back of the textbook.
Generally, for this class and in practice we always use the probability of a sample error to be
0.10 or 0.05. A probability of a sample error of 0.10 has an associated Z-score of 1.645. A
probability of a sample error of 0.05 has an associated Z-score of 1.96.
The acceptable amount of sampling error is something that must be determined by management.
The more accurate management wants the survey results to be, the smaller the amount of
sampling error has to be. You may suggest an amount of sampling error, but the final decision
on this value should generally be left up to management. One reason for this is that the smaller
the amount of sampling error is, the larger the sample size will need to be. A bigger survey costs
more to conduct, and management needs to weigh this cost in determining the amount of
sampling error.
The population standard deviation is a number you will need to estimate. This is the most
difficult part of determining the sample size. There are several ways to estimate the population
standard deviation. The best ways are to use the results from a prior survey or to conduct a small
pre-sample survey. A company will often have conducted a prior survey on a subject that relates
closely to the issue you are researching. If this is the case, the standard deviation from this
survey can be used as an estimate of the population standard deviation. If such a survey does not
exist then the best thing to do is take a small sample survey and use the standard deviation from
this survey as your estimate.
Another way to develop an estimate of population standard deviation is to use secondary data. It
is possible that another firm conducted a similar survey. The results of this survey could be used
to develop an estimate. If all else fails, you can always use judgment to estimate the standard
deviation. This can be done by obtaining a potential range of answers from management that is
in a position to make an educated guess regarding these figures. Ask management to estimate
what the highest and lowest response will be. The range is the difference between the high and
low responses. Using ¼ of this estimated range can be used as an estimate for the population
variance. The population standard deviation can then be estimated as the square root of the
population variance (or you can put the variance estimate directly into the above equation).
Once all of these numbers have been obtained, the sample size can be calculated by using the
Excel spreadsheet titled, “sample size calculation,” located in the class folder on the Marriott
School web site and by following these steps:
1. Enter the estimated population variance (standard deviation squared) into cell B12 of the
spreadsheet.
2. Enter the acceptable amount of sampling error obtained from management into cell E12
of the spreadsheet.
3. Enter the Z-score into cell D14 of the spreadsheet.
The required sample size is returned in cell H12 of the spreadsheet. You may also want to enter
the level of significance associated with the Z-score into cell B14. However, this is only to help
you keep track of the significance level that the Z-score is based on. It will not have any effect
on the calculation used to determine the required sample size.
Example
Suppose a fast food restaurant has hired you. As part of your research you are looking to find
out the average number of times per month each customer purchases a particular menu item from
the restaurant. In discussing the research with the management, they decide they would like to
know the average number within .25 of the actual value. You conduct a brief survey in which
you ask 25 customers how often they purchase the menu item and find an estimated standard
deviation of 1.5 (estimated variance of 2.25). Use a 90% level of confidence (.10 level of
significance).
Enter the values obtained into the spreadsheet. Enter 2.25 into cell B12. Enter .25 into cell E12.
Enter 1.645 into cell D14. The calculator returns a sample size of 97 into cell H12. This means
you would need to have a total sample size of just fewer than 100 people in order to obtain the
average number of times per month each customer purchases the menu item and be 90%
confident you are within plus or minus .25 of the actual value.