# Sample Size Calculator

The Sample Size Calculator helps you find the right sample design for your research, in just a few steps. In each section, provide the data requested. The Calculator does the rest. It documents inputs, analyzes data, reports key findings, and describes the analysis - all in clear, easy-to-understand language.

If anything is unclear, refer to the Frequently-Asked Questions.

## Problem Definition

The first step in sample planning is to describe the research you are conducting.

Sampling method

Parameter of interest

Purpose of research

Instructions: To find the answer to a frequently-asked question, simply click on the question.

### What is a sampling method?

Sampling method refers to the way that sample elements are selected from the total population. The Sample Size Calculator offers a choice of three sampling methods.

• Simple random sampling
• Cluster sampling
• Random sampling

All three of these methods are examples of probability sampling. With a probability sample, each element from the population has a known probability of being selected. A random method (e.g., a coin flip, a table of random numbers) is used to choose which elements are included in the sample.

### What is simple random sampling?

Simple random sampling is a type of sampling method. With simple random sampling, each member of the population has an equal chance of being selected. Further, each possible subset of n elements is equally likely to be chosen.

A random method (e.g., a coin flip, a table of random numbers) is used to choose which elements are included in the sample.

### What is stratified sampling?

Stratified sampling is a type of sampling method. With stratified sampling, each element of the population is assigned to a subgroup or stratum. Then, separate samples are selected from each stratum.

A random method (e.g., a coin flip, a table of random numbers) is used to determine which elements from each stratum are included in the sample.

### What is cluster sampling?

Cluster sampling is a type of sampling method. With cluster sampling, each element of the population is assigned to a subgroup or cluster.

A random method (e.g., a coin flip, a table of random numbers) is used to determine which clusters are included in the sample.

### Which sampling method is best?

It all depends. Each method has strengths and weaknesses. Here's a quick overview of pros and cons.

• Simple random sampling. Simple random sampling is sometimes, but not always, easiest to administer. It may or may not provide the most precision or the lowest cost.
• Stratified sampling. Stratified samples are often more difficult to administer than simple random samples. Often, they provide better precision and/or lower cost.
• Cluster sampling. Given equal sample sizes, cluster samples provide less precision than simple random samples or stratified samples. Sometimes, though, cluster samples are less costly to administer. If cost savings can be used to increase sample size and improve precision, cluster samples may be a good choice.

As you see, the "best" method is not obvious. Use the Sample Size Calculator to compare different options. Choose the approach that provides the best combination of research precision, cost, and administrative complexity.

### Should I sample with replacement or without replacement?

Suppose we have an urn full of marbles. To estimate the percentage of red marbles, we draw a sample of marbles, one by one.

After each drawing, if we put the marble back in the urn, it might be chosen again; if we do not put it back, it can be chosen only once. Sampling methods that allow elements from the population to be chosen more than once are examples of sampling with replacement. Methods that allow each element to be chosen only once are examples of sampling without replacement.

Sampling without replacement always provides more precision than sampling with replacement. Therefore, unless you have a good reason to sample with replacement, sampling without replacement is a better choice.

### What is the parameter of interest?

With survey sampling, the researcher is typically interested in estimating the value of a population parameter, based on sample data. The Sample Size Calculator offers a choice of three parameters.

• A mean score.
• A proportion.
• A total score.

A mean score is an average score. It is calculated by dividing the sum of scores by the number of observations. For example, if four students received scores of 70, 80, 80, and 90 on a test, their mean test score would be 80.

A proportion refers to the fraction of the group that has a given characteristic. In the previous example, we might ask what proportion of students received a test score greater than 80. Since only 1 of 4 students received a score greater than 80, the fraction would be one quarter and the proportion would be 0.25.

A total score refers to the sum of values for all of the observations. For example, to determine the total weight of passengers in an airplane, we would weigh each passenger and add the weights.

### How does an estimation problem differ from a hypothesis test?

With an estimation problem, the purpose of the study is to estimate the value of a population parameter, such as a mean score or a proportion. The researcher uses sample data make that estimate.

With a hypothesis test, the purpose of the study is to confirm or disconfirm a hypothesis. For example, a researcher might want to test the hypothesis that a population mean is equal to 10. He/she uses sample data to accept or reject that hypothesis.

If you are using survey data to estimate the value of a parameter (e.g., a mean, a proportion, or a total score), you are working on an estimation problem. If you are trying to confirm or disconfirm a hypothesis about a population parameter, you are conducting a hypothesis test.