Sampling

The following video introduces the different methods that statisticians use collect samples of data.

Gathering information about an entire population often costs too much or is virtually impossible. Instead, we use a sample of the population. A sample should have the same characteristics as the population it is representing. Most statisticians use various methods of random sampling in an attempt to achieve this goal. This section will describe a few of the most common methods. There are several different methods of random sampling. In each form of random sampling, each member of a population initially has an equal chance of being selected for the sample. Each method has pros and cons. The easiest method to describe is called a simple random sample. Any group of [latex]n[/latex] individuals is equally likely to be chosen by any other group of [latex]n[/latex] individuals if the simple random sampling technique is used. In other words, each sample of the same size has an equal chance of being selected. For example, suppose Lisa wants to form a four-person study group (herself and three other people) from her pre-calculus class, which has [latex]31[/latex] members not including Lisa. To choose a simple random sample of size three from the other members of her class, Lisa could put all [latex]31[/latex] names in a hat, shake the hat, close her eyes, and pick out three names. A more technological way is for Lisa to first list the last names of the members of her class together with a two-digit number, as in the following table.

Lisa can use a table of random numbers (found in many statistics books and mathematical handbooks), a calculator, or a computer to generate random numbers. For this example, suppose Lisa chooses to generate random numbers from a calculator. The numbers generated are as follows:

[latex]0.94360[/latex]; [latex]0.99832[/latex]; [latex]0.14669[/latex]; [latex]0.51470[/latex]; [latex]0.40581[/latex]; [latex]0.73381[/latex]; [latex]0.04399[/latex]

Lisa reads two-digit groups until she has chosen three class members (that is, she reads [latex]0.94360[/latex] as the groups [latex]94[/latex], [latex]43[/latex], [latex]36[/latex], [latex]60[/latex]). Each random number may only contribute one class member. If she needed to, Lisa could have generated more random numbers.

The random numbers [latex]0.94360[/latex] and [latex]0.99832[/latex] do not contain appropriate two digit numbers. However the third random number, [latex]0.14669[/latex], contains [latex]14[/latex] (the fourth random number also contains [latex]14[/latex]), the fifth random number contains [latex]05[/latex], and the seventh random number contains [latex]04[/latex]. The two-digit number [latex]14[/latex] corresponds to Macierz, [latex]05[/latex] corresponds to Cuningham, and [latex]04[/latex] corresponds to Cuarismo. Besides herself, Lisa’s group will consist of Marcierz, Cuningham, and Cuarismo.

USING THE TI-83, 83+, 84, 84+ CALCULATOR

To generating Random Numbers:

• Press MATH.
• Arrow over to PRB.
• Press [latex]5[/latex]:randInt(. Enter [latex]0[/latex], [latex]30[/latex]).
• Press ENTER for the first random number.
• Press ENTER two more times for the other [latex]2[/latex] random numbers. If there is a repeat press ENTER again.

Note: randInt([latex]0, 30, 3[/latex]) will generate [latex]3[/latex] random numbers.

Besides simple random sampling, there are other forms of sampling that involve a chance process for getting the sample. Other well-known random sampling methods are the stratified sample, the cluster sample, and the systematic sample.

To choose a stratified sample, divide the population into groups called strata and then take a proportionate number from each stratum. For example, you could stratify (group) your college population by department and then choose a proportionate simple random sample from each stratum (each department) to get a stratified random sample. To choose a simple random sample from each department, number each member of the first department, number each member of the second department, and do the same for the remaining departments. Then use simple random sampling to choose proportionate numbers from the first department and do the same for each of the remaining departments. Those numbers picked from the first department, picked from the second department, and so on represent the members who make up the stratified sample.

To choose a cluster sample, divide the population into clusters (groups) and then randomly select some of the clusters. All the members from these clusters are in the cluster sample. For example, if you randomly sample four departments from your college population, the four departments make up the cluster sample. Divide your college faculty by department. The departments are the clusters. Number each department, and then choose four different numbers using simple random sampling. All members of the four departments with those numbers are the cluster sample.

To choose a systematic sample, randomly select a starting point and take every [latex]n[/latex]^th piece of data from a listing of the population. For example, suppose you have to do a phone survey. Your phone book contains [latex]20,000[/latex] residence listings. You must choose [latex]400[/latex] names for the sample. Number the population [latex]1[/latex]–[latex]20,000[/latex] and then use a simple random sample to pick a number that represents the first name in the sample. Then choose every fiftieth name thereafter until you have a total of [latex]400[/latex] names (you might have to go back to the beginning of your phone list). Systematic sampling is frequently chosen because it is a simple method.

A type of sampling that is non-random is convenience sampling. Convenience sampling involves using results that are readily available. For example, a computer software store conducts a marketing study by interviewing potential customers who happen to be in the store browsing through the available software. The results of convenience sampling may be very good in some cases and highly biased (favor certain outcomes) in others.

Sampling data should be done very carefully. Collecting data carelessly can have devastating results. Surveys mailed to households and then returned may be very biased (they may favor a certain group). It is better for the person conducting the survey to select the sample respondents.

True random sampling is done with replacement. That is, once a member is picked, that member goes back into the population and thus may be chosen more than once. However for practical reasons, in most populations, simple random sampling is done without replacement. Surveys are typically done without replacement. That is, a member of the population may be chosen only once. Most samples are taken from large populations and the sample tends to be small in comparison to the population. Since this is the case, sampling without replacement is approximately the same as sampling with replacement because the chance of picking the same individual more than once with replacement is very low.

In a college population of [latex]10,000[/latex] people, suppose you want to pick a sample of [latex]1,000[/latex] randomly for a survey. For any particular sample of [latex]1,000[/latex], if you are sampling with replacement,

• the chance of picking the first person is [latex]1,000[/latex] out of [latex]10,000[/latex] ([latex]0.1000[/latex]);
• the chance of picking a different second person for this sample is [latex]999[/latex] out of [latex]10,000[/latex] ([latex]0.0999[/latex]);
• the chance of picking the same person again is [latex]1[/latex] out of [latex]10,000[/latex] (very low).

If you are sampling without replacement,

• the chance of picking the first person for any particular sample is [latex]1000[/latex] out of [latex]10,000[/latex] ([latex]0.1000[/latex]);
• the chance of picking a different second person is [latex]999[/latex] out of [latex]9,999[/latex] ([latex]0.0999[/latex]);
• you do not replace the first person before picking the next person.

Compare the fractions [latex]\displaystyle\frac{{999}}{{10,000}}[/latex] and [latex]\displaystyle\frac{{999}}{{9,999}}[/latex]. For accuracy, carry the decimal answers to four decimal places. To four decimal places, these numbers are equivalent ([latex]0.0999[/latex]).

Sampling without replacement instead of sampling with replacement becomes a mathematical issue only when the population is small. For example, if the population is [latex]25[/latex] people, the sample is ten, and you are sampling with replacement for any particular sample, then the chance of picking the first person is ten out of [latex]25[/latex], and the chance of picking a different second person is nine out of [latex]25[/latex] (you replace the first person).

If you sample without replacement, then the chance of picking the first person is ten out of [latex]25[/latex], and then the chance of picking the second person (who is different) is nine out of [latex]24[/latex] (you do not replace the first person).

Compare the fractions [latex]\displaystyle\frac{{9}}{{25}}[/latex] and [latex]\displaystyle\frac{{9}}{{24}}[/latex]. To four decimal places, [latex]\displaystyle\frac{{9}}{{25}}[/latex] = [latex]0.3600[/latex] and [latex]\displaystyle\frac{{9}}{{24}}[/latex] = [latex]0.3750[/latex]. To four decimal places, these numbers are not equivalent.

When you analyze data, it is important to be aware of sampling errors and nonsampling errors. The actual process of sampling causes sampling errors. For example, the sample may not be large enough. Factors not related to the sampling process cause nonsampling errors. A defective counting device can cause a nonsampling error.

In reality, a sample will never be exactly representative of the population so there will always be some sampling error. As a rule, the larger the sample, the smaller the sampling error.

In statistics, a sampling bias is created when a sample is collected from a population and some members of the population are not as likely to be chosen as others (remember, each member of the population should have an equally likely chance of being chosen). When a sampling bias happens, there can be incorrect conclusions drawn about the population that is being studied.

Watch the following video to learn more about sources of sampling bias.