# is simple random sampling usually done with or without replacement?

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## Simple random sample

## Simple random sample

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In statistics, a **simple random sample** (or **srs**) is a subset of individuals (a sample) chosen from a larger set (a population) in which a subset of individuals are chosen randomly, all with the same probability. In srs, each subset of individuals has the same probability of being chosen for the sample as any other subset of individuals.[1] A simple random sample is an unbiased sampling technique. Simple random sampling is a basic type of sampling and can be a component of other more complex sampling methods.

## Contents

1 Introduction

2 Relationship between simple random sample and other methods

2.1 Equal probability sampling (epsem)

2.2 Distinction between a systematic random sample and a simple random sample

3 Sampling a dichotomous population

4 Algorithms 5 See also 6 References 7 External links

## Introduction[edit]

The principle of simple random sampling is that every set of items has the same probability of being chosen. For example, suppose college students want to get a ticket for a basketball game, but there are only < tickets for them, so they decide to have a fair way to see who gets to go. Then, everybody is given a number in the range from 0 to -1, and random numbers are generated, either electronically or from a table of random numbers. Numbers outside the range from 0 to -1 are ignored, as are any numbers previously selected. The first numbers would identify the lucky ticket winners.

In small populations and often in large ones, such sampling is typically done "**without replacement**", i.e., one deliberately avoids choosing any member of the population more than once. Although simple random sampling can be conducted with replacement instead, this is less common and would normally be described more fully as simple random sampling **with replacement**. Sampling done without replacement is no longer independent, but still satisfies exchangeability, hence many results still hold. Further, for a small sample from a large population, sampling without replacement is approximately the same as sampling with replacement, since the probability of choosing the same individual twice is low.

An unbiased random selection of individuals is important so that if many samples were drawn, the average sample would accurately represent the population. However, this does not guarantee that a particular sample is a perfect representation of the population. Simple random sampling merely allows one to draw externally valid conclusions about the entire population based on the sample.

Conceptually, simple random sampling is the simplest of the probability sampling techniques. It requires a complete sampling frame, which may not be available or feasible to construct for large populations. Even if a complete frame is available, more efficient approaches may be possible if other useful information is available about the units in the population.

Advantages are that it is free of classification error, and it requires minimum advance knowledge of the population other than the frame. Its simplicity also makes it relatively easy to interpret data collected in this manner. For these reasons, simple random sampling best suits situations where not much information is available about the population and data collection can be efficiently conducted on randomly distributed items, or where the cost of sampling is small enough to make efficiency less important than simplicity. If these conditions do not hold, stratified sampling or cluster sampling may be a better choice.

## Relationship between simple random sample and other methods[edit]

### Equal probability sampling (epsem)[edit]

A sampling method for which each individual unit has the same chance of being selected is called **equal probability sampling** (epsem for short).

Using a simple random sample will always lead to an epsem, but not all epsem samples are SRS. For example, if a teacher has a class arranged in 5 rows of 6 columns and she wants to take a random sample of 5 students she might pick one of the 6 columns at random. This would be an epsem sample but not all subsets of 5 pupils are equally likely here, as only the subsets that are arranged as a single column are eligible for selection. There are also ways of constructing multistage sampling, that are not srs, while the final sample will be epsem.[2] For example, systematic random sampling produces a sample for which each individual unit has the same probability of inclusion, but different sets of units have different probabilities of being selected.

Samples that are epsem are **self weighting**, meaning that the inverse of selection probability for each sample is equal.

### Distinction between a systematic random sample and a simple random sample[edit]

Consider a school with 1000 students, and suppose that a researcher wants to select 100 of them for further study. All their names might be put in a bucket and then 100 names might be pulled out. Not only does each person have an equal chance of being selected, we can also easily calculate the probability () of a given person being chosen, since we know the sample size () and the population ():

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Explain the difference between a parameter and a statistic.

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A parameter is a measure of the population, and a statistic is a measure of a sample.

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Explain the difference between a sample and a census. Every 10 years, the U.S. Census Bureau takes a census. What does that mean?

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A sample is a collection of people or objects taken from the population of interest. A census is a survey in which every member of the population is measured. When the U.S. Census Bureau takes a census, it conducts a survey of all people living in the U.S.

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Discrete Mathematics and Its Applications

8th Edition Kenneth Rosen 4,399 explanations Trigonometry 8th Edition Charles P. McKeague 3,733 explanations

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### Terms in this set (18)

Explain the difference between a parameter and a statistic.

A parameter is a measure of the population, and a statistic is a measure of a sample.

Explain the difference between a sample and a census. Every 10 years, the U.S. Census Bureau takes a census. What does that mean?

A sample is a collection of people or objects taken from the population of interest. A census is a survey in which every member of the population is measured. When the U.S. Census Bureau takes a census, it conducts a survey of all people living in the U.S.

The mean GPA of all 7000 students at a college is 2.83. A sample of 100 GPAs from this school has a mean of 3.06. Which number is mu and which is x overbar?

The population mean is muequals2.83, and the sample mean is x overbarequals3.06.

Suppose you find all the heights of the members of the men's basketball team at your school. Could you use those data to make inferences about heights of all men at your school? Why or why not?

One should not use these data to make inferences about heights of all men at the school because the sample is not random and is not representative of the population.

You are receiving a large shipment of batteries and want to test their lifetimes. Explain why you would want to test a sample of batteries rather than the entire population.

If you test all the batteries to failure you would have no batteries to sell.

Suppose you want to estimate the mean GPA of all students at your school. You set up a table in the library asking for volunteers to tell you their GPAs. Do you think you would get a representative sample? Why or why not?

One would probably not get a representative sample because of response bias (students who volunteer will probably have higher GPAs than students who don't volunteer) and measurement bias (students may inflate their GPAs).

Explain the difference between sampling with replacement and sampling without replacement. Suppose you had the names of 10 students, each written on a 3 by 5 notecard, and want to select two names. Describe both procedures.

Describe sampling with replacement. Choose the correct answer below.

Draw a notecard, note the name, replace the notecard and draw again. It is possible the same student could be picked twice.

Explain the difference between sampling with replacement and sampling without replacement. Suppose you had the names of 10 students, each written on a 3 by 5 notecard, and want to select two names. Describe both procedures.

Describe sampling without replacement. Choose the correct answer below.

Draw a notecard, note the name, do not replace the notecard and draw again. It is not possible the same student could be picked twice.

Is simple random sampling usually done with or without replacement?

Simple random sampling is usually done without replacement, which means that a subject cannot be selected for a sample more than once.

You need to select a simple random sample of two from six friends who will participate in a survey. Assume the friends are numbered 1, 2, 3, 4, 5, and 6.

Use the line from a random number table shown below to select your sample. Start from the left.

0 5 8 5 7 8 1 4 9 9 7 2 4 3 5 2 1 1 0 6 7 5 5 1

Friend 5 and friend 1

Assume your class has 30 students and you want a random sample of 10 of them. A student suggests asking each student to flip a coin, and if the coin comes up heads, then he or she is in your sample. Explain why this is not a good method.

This method is not good because it is unlikely to result in a sample size of 10.

A teacher at a community college sent out questionnaires to evaluate how well the administrators were doing their jobs. All teachers received questionnaires, but only 10% returned them. Most of the returned questionnaires contained negative comments about the administrators. Explain how an administrator could dismiss the negative findings of the report.

There is nonresponse bias. The results could be biased because the small percentage who chose to return the survey might be very different from the majority who did not return the survey.

A phone survey asked whether Social Security should be continued or abandoned immediately. Only landlines (not cell phones) were called. Do you think this would introduce bias? Explain.

## Sampling With Replacement and Sampling Without Replacement

Sampling With Replacement and Sampling Without Replacement

**Sampling with replacement:**

Consider a population of potato sacks, each of which has either 12, 13, 14, 15, 16, 17, or 18 potatoes, and all the values are equally likely. Suppose that, in this population, there is exactly one sack with each number. So the whole population has seven sacks. If I sample two with replacement, then I first pick one (say 14). I had a 1/7 probability of choosing that one. Then I replace it. Then I pick another. Every one of them still has 1/7 probability of being chosen. And there are exactly 49 different possibilities here (assuming we distinguish between the first and second.) They are: (12,12), (12,13), (12, 14), (12,15), (12,16), (12,17), (12,18), (13,12), (13,13), (13,14), etc.

**Sampling without replacement:**

Consider the same population of potato sacks, each of which has either 12, 13, 14, 15, 16, 17, or 18 potatoes, and all the values are equally likely. Suppose that, in this population, there is exactly one sack with each number. So the whole population has seven sacks. If I sample two without replacement, then I first pick one (say 14). I had a 1/7 probability of choosing that one. Then I pick another. At this point, there are only six possibilities: 12, 13, 15, 16, 17, and 18. So there are only 42 different possibilities here (again assuming that we distinguish between the first and the second.) They are: (12,13), (12,14), (12,15), (12,16), (12,17), (12,18), (13,12), (13,14), (13,15), etc.

**What's the Difference?**

When we sample with replacement, the two sample values are independent. Practically, this means that what we get on the first one doesn't affect what we get on the second. Mathematically, this means that the covariance between the two is zero.

In sampling without replacement, the two sample values aren't independent. Practically, this means that what we got on the for the first one affects what we can get for the second one. Mathematically, this means that the covariance between the two isn't zero. That complicates the computations. In particular, if we have a SRS (simple random sample) without replacement, from a population with variance , then the covariance of two of the different sample values is , where is the population size. (A brief summary of some formulas is provided here. For a discussion of this in a textbook for a course at the level of M378K, see the chapter on Survey Sampling in by John A. Rice, published by Wadsworth & Brooks/Cole Publishers. There is an outline of an slick, simple, interesting, but indirect, proof in the problems at the end of the chapter.)

**Population size -- Leading to a discussion of "infinite" populations.**

When we sample without replacement, and get a non-zero covariance, the covariance depends on the population size. If the population is very large, this covariance is very close to zero. In that case, sampling with replacement isn't much different from sampling without replacement. In some discussions, people describe this difference as sampling from an infinite population (sampling with replacement) versus sampling from a finite population (without replacement).

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