To obtain samples that are unbiased—i.e., that give each subject in the population an equally likely chance of being selected—statisticians use four basic methods of sampling: random, systematic, stratified, and cluster sampling.

### Random Sampling

**Random samples **are selected by using chance methods or random numbers. One such method is to number each subject in the population. Then place numbered cards in a bowl, mix them thoroughly, and select as many cards as needed. The subjects whose numbers are selected constitute the sample. Since it is difficult to mix the cards thoroughly, there is a chance of obtaining a

biased sample. For this reason, statisticians use another method of obtaining

numbers. They generate random numbers with a computer or calculator. Before the

invention of computers, random numbers were obtained from tables.

Some two-digit random numbers are shown in

Table 1–3. To select a random sample of, say, 15 subjects out of 85 subjects,

it is necessary to number each subject from 01 to 85. Then select a starting

number by closing your eyes and placing your finger on a number in the table.

(Although this may sound somewhat unusual, it enables us to find a starting

number at random.) In this case suppose your finger landed on the number 12 in

the second column. (It is the sixth number down from the top.) Then proceed downward

until you have selected 15 different numbers between 01 and 85. When you reach

the bottom of the column, go to the top of the next column. If you select a

number greater than 85 or the number 00 or a duplicate number, just omit it. In

our example, we will use the subjects numbered 12, 27, 75, 62, 57, 13, 31, 06,

16, 49, 46, 71, 53, 41, and 02.

### Systematic Sampling

Researchers obtain **systematic samples **by numbering each subject of the population and then selecting every *k*th subject. For example, suppose there were 2000 subjects in the population and a sample of 50 subjects were needed. Since 2000 4 50 5 40, then *k *5 40, and every 40th subject would be selected; however, the first subject (numbered between 1 and 40) would be selected at random. Suppose subject 12 were the first subject selected; then the sample would consist of the subjects whose numbers were 12, 52, 92, etc., until 50 subjects were obtained. When using systematic sampling, you must be careful about how the subjects in the population are numbered. If subjects were arranged in a manner

### Stratified Sampling

Researchers obtain **stratified samples **by dividing the population into groups (called strata) according to some characteristic that is important to the study, then sampling from each group. Samples within the strata should be randomly selected. For example, suppose the president of a two-year college wants to learn how students feel about a certain issue. Furthermore, the president wishes to see if the opinions of the first-year students differ from those of the second-year students. The president will randomly select students from each group to use in the sample.

### Cluster Sampling

Researchers also use **cluster samples. **Here the population is divided into groups called clusters by some means such as geographic area or schools in a large school district, etc. Then the researcher randomly selects some of these clusters and uses all members of the selected clusters as the subjects of the samples. Suppose a researcher wishes to survey apartment dwellers in a large city. If there are 10 apartment buildings in the city, the researcher can select at random 2 buildings from the 10 and interview all the residents of these buildings. Cluster sampling is used when the population is large or when it involves subjects residing in a large geographic area. For example, if one wanted to do a study involving the patients in the hospitals in New York City, it would be very costly and time-consuming to try to obtain a random sample of patients since they would be spread over a large area. Instead, a few hospitals could be selected at random, and the patients in these hospitals would be interviewed in a cluster.