## sample

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# Sample

Samples and populations

Random samples

Sample size and accuracy

Resources

In statistics, a sample is a subset of actual observations taken from any larger set of possible observations. The larger set of observations is known as a population. Samples are collected and the results analyzed so the inferences (or, conclusions) can be made to the population. Such a procedure is called sampling. For example, suppose that a researcher would like to know how many hours the average 11th grade student in the United States spends studying English literature every night. One way to answer that question would be to interview a select number (say 50, 500, or 5,000) of 11th grade students and ask them how many hours they spend on English literature each evening. The researcher could then draw some conclusions about the time spent studying English literature by all 11th grade students based on what he or she learned from the sample that was studied.

## Samples and populations

Sampling is a crucial technique in the science of statistical analysis. It represents a compromise between a researcher collecting all possible information on some topic and the amount of information that he or she can realistically collect. For example, in the example used above, the ideal situation might be for a researcher to collect data from every single 11th grade student in the United States. However, the cost, time, and effort required to do this kind of study would be enormous. No one could possibly do such a study.

The alternative is to select a smaller subset of 11th grade students and collect data from them. If the sample that is chosen is typical of all 11th grade students throughout the United States, the data obtained could also be considered to be typical. That is, if the average 11th grade student in the sample studies English literature two hours every evening, then the researcher might be justified in saying that the average 11th grade student in the United States also studies English literature two hours a night.

## Random samples

The key to using samples in statistical analysis is to be sure that they are random. A random sample is one in which every member of the population has an equal chance of being selected for the sample. For example, a researcher could not choose 11th grade students for a sample if they all came from the same city, from the same school, were of the same sex, or had the same last name. In such cases, the sample chosen for study would not be representative of the total population.

Many systems have been developed for selecting random samples. One approach is simply to put the name of every member of the population on a piece of paper, put the pieces of paper into a large fishbowl, mix them up, and then draw names at random for the sample. Although this idea sounds reasonable, it has a number of drawbacks. One is that complete mixing of pieces of paper is very difficult. Pieces may stick to each other, they may be of different sizes or weight, or they may differ from each other in some other respect. Still, this method is often used for statistical studies in which precision is not crucial.

### KEY TERMS

Extrapolation The process of using some limited set of data to make predictions about a broader set of data on the same subject.

Population Any set of observations that could potentially be made.

Random sample A sample in which every member of the population has an equal chance of being selected for the sample.

Today, researchers use computer programs to obtain random samples for their studies. When the U.S. federal government collects statistics on the number of hours that people work, the kinds of jobs they do, the wages they earn, and so on, they tell a computer to sift through the names of every citizen for whom they have records and choose every hundredth name, every five-hundredth name, or to make selections at some other interval. Only the individuals actually chosen by the computer are used for the sample. From the results of that sample, extrapolations are made for the total population of all working Americans.

## Sample size and accuracy

The choice a researcher always has to make is how large a sample to choose. It stands to reason that the larger the sample, the more accurate will be the results of the study. Consequently, a smaller sample that is used, results in less accurate results. Statisticians have developed mathematical formulas that allow them to estimate how accurate their results are for any given sample size. The sample size used depends on how much money they have to spend, how accurate the final results need to be, how much variability among data are they willing to accept, and so on.

Interestingly enough, the sample size needed to produce accurate results in a study is often surprisingly small. For example, the Gallup Poll (which is a public opinion poll that is performed by The Gallup Organization) regularly chooses samples of people of whom they ask a wide variety of questions. The organization is perhaps best known for its predictions of presidential and other elections. For its presidential election polls, the Gallup organization interviews no more than a few thousand people out of the tens of millions who actually vote. Yet, their results are often accurate within a percentage point or so of the actual votes cast in an election. The secret of success for Gallupand for other successful polling organizationsis to be sure that the sample they select is truly random; that is, that the people interviewed are completely typical of everyone who belongs to the general population. When invalid populations are used, erroneous predictions, such as those that took place relative to the 2004 U.S. presidential election, often occur.

## Resources

### BOOKS

McCollough, Celeste, and Loche Van Atta. Statistical Concepts: A Program for Self-Instruction. New York: McGraw Hill, 1963.

Montgomery, Douglas C. Applied Statistics and Probability for Engineers. Hoboken, NJ: Wiley, 2007.

Newman, Isadore, et al. Conceptual Statistics for Beginners. Lanham, MD: University Press of America, 2006.

Rice, John A. Mathematical Statistics and Data Analysis. Belmont, CA: Thompson/Brooks/Cole, 2007.

Walpole, Ronald, and Raymond Myers, et al. Probability and Statistics for Engineers and Scientists. Englewood Cliffs, NJ: Prentice Hall, 2002.

David E. Newton

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# Sample

A sample is a subset of actual observations taken from any larger set of possible observations. The larger set of observations is known as a population. For example, suppose that a researcher would like to know how many hours the average 11th grade student in the United States spends studying English literature every night. One way to answer that question would be to interview a select number (say 50, 500, or 5,000) of 11th grade students and ask them how many hours they spend on English literature each evening. The researcher could then draw some conclusions about the time spent studying English literature by all 11th grade students based on what he or she learned from the sample that was studied.

## Samples and populations

Sampling is a crucial technique in the science of statistical analysis. It represents a compromise between a researcher collecting all possible information on some topic and the amount of information that he or she can realistically collect. For example, in the example used above, the ideal situation might be for a researcher to collect data from every single 11th grade student in the United States. But the cost, time, and effort required to do this kind of study would be enormous. No one could possibly do such a study.

The alternative is to select a smaller subset of 11th grade students and collect data from them. If the sample that is chosen is typical of all 11th grade students throughout the United States, the data obtained could also be considered to be typical. That is, if the average 11th grade student in the sample studies English literature two hours every evening, then the researcher might be justified in saying that the average 11th grade student in the United States also studies English literature two hours a night.

## Random samples

The key to using samples in statistical analysis is to be sure that they are random . A random sample is one in which every member of the population has an equal chance of being selected for the sample. For example, a researcher could not choose 11th grade students for a sample if they all came from the same city, from the same school, were of the same sex, or had the same last name. In such cases, the sample chosen for study would not be representative of the total population.

Many systems have been developed for selecting random samples. One approach is simply to put the name of every member of the population on a piece of paper , put the pieces of paper into a large fishbowl, mix them up, and then draw names at random for the sample. Although this idea sounds reasonable, it has a number of drawbacks. One is that complete mixing of pieces of paper is very difficult. Pieces may stick to each other, they may be of different sizes or weight, or they may differ from each other in some other respect. Still, this method is often used for statistical studies in which precision is not crucial.

Today, researchers use computer programs to obtain random samples for their studies. When the United States government collects statistics on the number of hours people work, the kinds of jobs they do, the wages they earn, and so on, they ask a computer to sift through the names of every citizen for whom they have records and choose every hundredth name, every five-hundredth name, or to make selections at some other interval . Only the individuals actually chosen by the computer are used for the sample. From the results of that sample, extrapolations are made for the total population of all working Americans.

## Sample size and accuracy

The choice a researcher always has to make is how large a sample to choose. It stands to reason that the larger the sample, the more accurate will be the results of the study. The smaller the sample, the less accurate the results. Statisticians have developed mathematical formulas that allow them to estimate how accurate their results are for any given sample size. The sample size used depends on how much money they have to spend, how accurate the final results need to be, how much variability among data are they willing to accept, and so on.

Interestingly enough, the sample size needed to produce accurate results in a study is often surprisingly small. For example, the Gallup Poll regularly chooses samples of people of whom they ask a wide variety of questions. The organization is perhaps best known for its predictions of presidential and other elections. For its presidential election polls, the Gallup organization interviews no more than a few thousand people out of the tens of millions who actually vote. Yet, their results are often accurate within a percentage point or so of the actual votes cast in an election. The secret of success for Gallup—and for other successful polling organizations—is to be sure that the sample they select is truly random, that is, that the people interviewed are completely typical of everyone who belongs to the general population. When invalid populations are used, erroneous predictions, such as those that took place relative to the 2000 U.S. presidential election, often occur.

## Resources

### books

McCollough, Celeste, and Loche Van Atta. Statistical Concepts: A Program for Self-Instruction. New York: McGraw Hill, 1963.

Walpole, Ronald, and Raymond Myers, et al. Probability and Statistics for Engineers and Scientists. Englewood Cliffs, NJ: Prentice Hall, 2002.

David E. Newton

## KEY TERMS

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Extrapolation

—The process of using some limited set of data to make predictions about a broader set of data on the same subject.

Population

—Any set of observations that could potentially be made.

Random sample

—A sample in which every member of the population has an equal chance of being selected for the sample.

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sam·ple / ˈsampəl/ • n. a small part or quantity intended to show what the whole is like: investigations involved analyzing samples of handwriting. ∎  a specimen taken for scientific testing or analysis: a urine sample. ∎  Statistics a portion drawn from a population, the study of which is intended to lead to statistical estimates of the attributes of the whole population. ∎  a small amount of a food or other commodity, esp. one given to a prospective customer. ∎  a sound created by sampling. • v. [tr.] take a sample or samples of (something) for analysis: bone marrow cells were sampled. ∎  try the qualities of (food or drink) by tasting it. ∎  get a representative experience of: sample the pleasures of Saint Maarten. ∎  Electr. ascertain the momentary value of (an analog signal) many times a second so as to convert the signal to digital form. ∎  record or extract a small piece of music or sound digitally for reuse as part of a composition or song.

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sample (sahm-pŭl) n. a subgroup of a population that is selected for study because it is regarded as representative of the population as a whole. random s. a sample selected by a random process ensuring that each member of the population has an equal chance of being included in it.

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sample †illustrative or confirmatory fact, etc.; †example, warning XIII; specimen of material or goods XV. Aphetic — AN. assample, var. of OF. essample EXAMPLE.
Hence vb. XVI.

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# Sample

a small quantity; an example.

Examples : sample of ingenuity, 1706; sample of salesmenLipton, 1970.