Sample Size: Simple Random Samples

This lesson describes how to find the smallest sample size that provides the desired precision, when the sampling method is simple random sampling.

Factors That Influence Sample Size

The "right" sample size for a particular application depends on many factors, including the following:

• Cost considerations (e.g., maximum budget, desire to minimize cost).
• Administrative concerns (e.g., complexity of the design, research deadlines).
• Minimum acceptable level of precision.
• Confidence level.
• Variability within the population or subpopulation (e.g., stratum, cluster) of interest.
• Sampling method.

These factors interact in complex ways. Although a consideration of all the variations is beyond the scope of this tutorial, the remainder of this lesson covers a situation that commonly occurs with simple random samples: How to find the smallest sample size that provides the required precision.

For other computations not covered in this lesson, consider using the Sample Size Calculator (described below).

How to Choose Sample Size for a Simple Random Sample

Consider the following problem. You are conducting a survey to estimate a population mean or proportion. The sampling method is simple random sampling, without replacement. You want your survey to provide a specified level of precision.

To choose the right sample size for a simple random sample, you need to define the following inputs.

• Specify the desired margin of error ME. This is your measure of precision.
• Specify the desired confidence level. This is necessary to find the critical standard score (see below).
• Find the critical standard score z. For an estimation problem, the critical standard score (z) is the value for which the z-score cumulative probability is:

  P = [ CL + (100 - CL)/2 ] / 100

  where P is the z-score cumulative probability, and CL is the confidence level. Use a normal probability table (found in most introductory statistics texts) or an online calculator to find the z-score for the cumulative probability. On this website, we use Stat Trek's Normal Distribution Calculator to find z-scores.

• Unless the population size is very large relative to sample size (e.g., 20 times larger), you need to specify the size of the population (N).
• And finally, you need to know the variance of the population, σ². Since the population variance is seldom known, researchers typically estimate population variance based on judgment and/or previous research.

Given these inputs, the following formulas find the smallest sample size that provides the desired level of precision.

Sample statistic	Population size	Sample size
Mean	Known	n = { z2 * σ2 * [ N / (N - 1) ] } / { ME2 + [ z2 * σ2 / (N - 1) ] }
Mean	Unknown	n = ( z2 * σ2 ) / ME2
Proportion	Known	n = [ ( z2 * p * q ) + ME2 ] / [ ME2 + z2 * p * q / N ]
Proportion	Unknown	n = [ ( z2 * p * q ) + ME2 ] / ( ME2 )

This approach works when the sample size is relatively large (greater than or equal to 30). Use the first or third formulas when the population size is known. When the population size is large but unknown, use the second or fourth formulas.

For proportions, the sample size requirements vary, based on the value of the proportion. If you are unsure of the right value to use, set p equal to 0.5. This will produce a conservative sample size estimate; that is, the sample size will produce at least the precision called for and may produce better precision.

Sample Problem

At the end of every school year, the state administers a reading test to a simple random sample drawn without replacement from a population of 100,000 third graders. Over the last five years, students who took the test correctly answered 75% of the test questions.

What sample size should you use to achieve a margin of error equal to plus or minus 4%, with a confidence level of 95%?

Solution: To solve this problem, we follow the steps outlined above.

• Specify the margin of error. This was given in the problem definition. The margin of error is plus or minus 4% or 0.04.
• Specify the confidence level (CL). This was also given. The confidence level is 95% or 0.95.
• Determine the critical standard score (z). The critical standard score is the value for which the cumulative probability is [ CL + (100 - CL)/2 ] / 100 = [ 95 + (100 - 95)/2 ] / 100 = 0.975. To find that value, we use the Normal Distribution Calculator. Recall that the distribution of standard scores has a mean of 0 and a standard deviation of 1. Therefore, we plug the following entries into the normal calculator: Probability = 0.975; Mean = 0; and Standard deviation = 1. The calculator tells us that the value of the standard score is about 1.96.

• And finally, we assume that the population proportion p is equal to its past value over the previous 5 years. That value is 0.75. Given these inputs, we can find the smallest sample size n that will provide the required margin of error.

  n = [ (z² * p * q ) + ME² ] / [ ME² + z² * p * q / N ]

  n = [ (1.96)² * 0.75 * 0.25 + 0.0016] / [ 0.0016 + (1.96)² * 0.75 * 0.25 / 100,000 ]

  n = (0.7203 + 0.0016) / ( 0.0016 + 0.0000072)

  n = 449.2

Assuming a 95% confidence level, we need to survey a simple random sample of 450 students to achieve a margin of error of plus or minus 4 percent.