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Indiana Institute of Technology
MTH101 Mathematics Indiana Institute of Technology
In statistics, a simple random sample is a subset of individuals (a sample) chosen from a
larger set (a population). Each individual is chosen randomly and entirely by chance, such
that each individual has the same probability of being chosen at any stage during the
sampling process, and each subset of k individuals has the same probability of being chosen
for the sample as any other subset of k individuals.[1] This process and technique is known
as simple random sampling, and should not be confused with systematic random
sampling. A simple random sample is an unbiased surveying technique.
Simple random sampling is a basic type of sampling, since it can be a component of other
more complex sampling methods. The principle of simple random sampling is that every
object has the same probability of being chosen. For example, suppose N college students
want to get a ticket for a basketball game, but there are only X < N 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 N-1, and random numbers are generated, either electronically or from a
table of random numbers. Numbers outside the range from 0 to N-1 are ignored, as are any
numbers previously selected. The first X 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 odds of choosing the same individual twice is low.
An unbiased random selection of individuals is important so that if a large number of
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.
Random samples, sampling distributions of estimators, Methods of Moments and
Maximum Likelihood.
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.
Algorithms[edit]
Several efficient algorithms for simple random sampling have been developed.[2][3] A naive
algorithm is the draw-by-draw algorithm where at each step we remove the item at that step
from the set with equal probability and put the item in the sample. We continue until we have
sample of desired size k. The drawback of this method is that it requires random access in
the set.
The selection-rejection algorithm developed by Fan et al. in 1962[4] requires single pass over
data; however, it is a sequential algorithm and requires knowledge of total count of items n,
which is not available in streaming scenarios.
A very simple random sort algorithm was proved by Sunter in 1977[5] which simply assigns a
random number drawn from uniform distribution (0, 1) as key to each item, sorts all items
using the key and selects the smallest k items.
J. Vitter in 1985[6] proposed reservoir sampling algorithm which is often widely used. This
algorithm does not require advance knowledge of n and uses constant space.
Random sampling can also be accelerated by sampling from the distribution of gaps
between samples,[7] and skipping over the gaps.
Random sampling is a technique used in selecting people or or items for research. There are
many techniques that can be used; but, each technique makes sure that each person or item
considered for the research has an equal opportunity to be chosen as part of the group.
Common Random Sampling Techniques
Random Number Table
Random number tables are created when every person or every item receives a number.
The numbers are entered into a table with digits, starting with the number one and including
a number for every person or item. The numbers are added onto the table in a random
order.
To use this method for random sampling, each person in the population receives a unique
number that is included on the table. Numbers are chosen at random from the table. The
choice of one digit is unaffected by the choice of any other given digit.
An example of using a random number table is to assign a number to each of 100 people
who have expressed interest in attending a special event. Individual numbers are chosen
from the random number table. When a person’s number is chosen, they are approved to
attend the event.
This is random sampling because any number can be chosen from the table. The
numbers don’t have to be chosen in numerical order.
Each number has an equal opportunity to be chosen from the table.
Replacement Sampling
With Replacement
Sampling with replacement is the act of choosing an individual once and then replacing their
number or name into the original group of potential people such that the same person has
the ability to be chosen more than once.
Without Replacement
This procedure is the same as sampling with replacement, except that the name or number
of the individual is not replaced into the original group. This results in only one opportunity to
be chosen, rather than multiple.
Lottery
All of the names or assigned numbers of individuals are entered into a given group and then
chosen at random.
Using a Computer
Software and other programs are available for the purpose of making random samples.
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