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Simple Random Sampling

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Section 7.3 Confidence intervals for a population proportion

Unit 1 Section 1.3 – Day : Sampling Techniques  Sample – a part of a population used in statistical studies.  An unbiased sample is one where.

POPULATION- the entire group of individuals that we want information about SAMPLE- the part of the population that we actually examine in order to gather.

Section 5.1 What is Probability? 5.1 / 1. Probability Probability is a numerical measurement of likelihood of an event. The probability of any event is.

Chapter 2: Sampling and Surveys Section 2.1: Samples, Good and Bad.

1-3 Data Collection and Sampling Techniques Surveys are the most common method of collecting data. Three methods of surveying are: 1) Telephone surveys.

If you have your Parent Letter signed, please return the bottom portion. Scissors are on my desk. Grab the handout at the front. Complete the front. Wait.

Chapter 5 Section 1 – Part 2. Samples: Good & Bad What makes a sample bad? What type of “bad” sampling did we discuss previously?

Confidence Interval Estimation for a Population Proportion Lecture 31 Section 9.4 Wed, Nov 17, 2004.

A simple random sample of n measurements from a population is one selected in such a manner that 1. every sample of size n from the population has equal.

Investigation 1- IB2 Random sampling and Quadrats (G.1.3)

Testing Hypotheses about a Population Proportion Lecture 30 Sections 9.3 Wed, Oct 24, 2007.

Continuous Random Variables Lecture 25 Section Mon, Feb 28, 2005.

Continuous Random Variables Lecture 24 Section Tue, Mar 7, 2006.

Confidence Interval Estimation for a Population Proportion Lecture 33 Section 9.4 Mon, Nov 6, 2006.

Continuous Random Variables Lecture 22 Section Mon, Feb 25, 2008.

Section 5.1.  Three ways to answer this question: 1. Actually carry out an experiment 2. Develop a probability model 3. Start with a model, and then.

Systematic Sampling Lecture 9 – Part 1 Sections 2.8 Wed, Jan 30, 2008.

Copyright © 2013, 2010 and 2007 Pearson Education, Inc. Chapter Data Collection 1.

Copyright © 2016 The McGraw-Hill Companies, Inc. Permission required for reproduction or display. B ASIC C ONCEPTS IN P ROBABILITY Section 5.1.

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Statistics By Jim

Making statistics intuitive

Simple Random Sampling: Definition & Examples

By Jim Frost Leave a Comment

What is Simple Random Sampling?

Simple random sampling (SRS) is a probability sampling method where researchers randomly choose participants from a population . All population members have an equal probability of being selected. This method tends to produce representative, unbiased samples.

For example, if you randomly select 1000 people from a town with a population of 100,000 residents, each person has a 1000/100000 = 0.01 probability. That’s a simple calculation requiring no additional knowledge about the population’s composition. Hence, simple random sampling.

Simple random sampling is a probability sampling method that helps ensure the sample mirrors the population. The process proportionately samples from larger subpopulations more frequently than smaller subpopulations.

Suppose the town contains subpopulation A with 40,000 people and subpopulation B with 10,000. Using SRS with a probability of 0.01, the process will tend to enlist 400 from subpopulation A and 100 from B. Hence, the process tends to produce a proportionate representation in the sample that reflects the entire population. You don’t need to know the details about the subpopulations for this process to work!

Learn more about Types of Sampling Methods in Research .

How to Use Simple Random Sampling

Performing simple random sampling requires that you have a sampling frame that contains a complete list of all population members and the ability to contact and involve them in your study. Learn more about Sampling Frames: Definition, Examples & Uses .

• Define the population.
• Create a list of all population members.
• Assign random numbers to each member.
• Use a random number generator to select participants until you reach your target sample size.

Alternatively, if the population is not too large, you can use a lottery system for drawing the sample. Place all the names in a hat and randomly draw your sample. For large populations, researchers typically use computers to select participants randomly from a database.

Example of Simple Random Sampling

Imagine we are studying the town with 100,000 residents. We want to perform simple random sampling to obtain a sample size of 1000. We first need to define the population. We’ll define it as residents of the town who pay township taxes and are at least 18 years old.

Next, we need to create a complete list of residents who meet those criteria. Perhaps we’ll work with the township tax office to make the list. We’ll add all eligible residents to our list.

Finally, we need to select participants randomly from the list. We can use a computer program to do that. Alternatively, we can print out names on slips of paper and draw them from a basket. We keep drawing from the list until we have 1000 names.

Benefits of Simple Random Sampling

Many statisticians consider simple random sampling to be the gold standard for producing representative samples. Because it is entirely random, it minimizes the potential for researchers biasing the results, even if unintentionally. As you’ll read, there are alternative sampling methods that provide concessions to real-world sampling difficulties. Unfortunately, the alternatives can unwittingly produce a biased sample. Learn more about representative samples .

Procedurally, SRS is the simplest method for obtaining an unbiased sample. While the researchers need a list of the entire population, they don’t need other information about that population, its subpopulations, and its features.

Conversely, other more complex forms of sampling require researchers to understand the population’s characteristics. Then, using that knowledge and a lot of preplanning, they divide the population into strata or clusters and perform other procedures before sampling. With SRS, you just randomly draw from the list until you have enough subjects.

Because simple random sampling tends to produce unbiased samples that mirror the population, it’s excellent for analysts who need to use a sample to infer the properties of a population (i.e., inferential statistics ). In a study, having a representative sample improves both its internal and external validity . After simple random sampling, you can use statistical hypothesis tests to use the sample to draw conclusions about the population.

For more information about inferential statistics, read my articles about Populations, Parameters, and Samples in Inferential Statistics  and Descriptive versus Inferential Statistics .

Drawbacks of Simple Random Sampling

Even though there are great benefits to using this method, simple random sampling has some significant drawbacks.

Population List

First and foremost, this method can be quite cumbersome and require ample resources for large populations. You’ll need a list of all population members, which can be a tremendous hurdle by itself. If that list doesn’t exist, you might need to expend considerable resources to create it. An incomplete list can bias your results. Only a complete list allows the researchers to have an equal probability of selecting all population members.

Attempting to perform SRS with an incomplete population list causes undercoverage bias and a nonrepresentative sample.

Learn more about Undercoverage Bias: Definition & Examples .

Then you’ll need to contact and interact with everyone you randomly select. Depending on the nature of your study, that process can be pretty expensive and time-consuming if your participants span a wide geographic range, particularly when you need a large sample size.

Insufficient Representation of Subpopulations

Despite being entirely random, simple random sampling can miss important subpopulations and features in the population. For example, in our town with 100,000 residents, imagine that we’re particularly interested in surveying those who are at least 90 years old. You plan to obtain a sample size of 1000, which is 1 out of 100 residents. However, there are only 50 people in town who are older than 90. Your sample might not include anyone in this vital group! If it does, it’ll be a tiny number that doesn’t provide a clear picture of this subgroup.

Simple random sampling can fail to provide precise data about particular subgroups and differences between subgroups. Other sampling methods can ensure sufficient numbers from small subgroups that produce a clear picture and increase the ability to compare subgroups.

Simple Random Sampling vs. Other Methods

Because you need a list of the entire population, simple random sampling is most feasible when working with a relatively small population that is already defined. For example, if you’re surveying a company and can easily obtain a list of employees from Human Resources, SRS isn’t too difficult. Large populations can require extensive amounts of time and resources just to create the complete list. Simple random sampling is a great option when you don’t know much about your population other than its membership.

However, other sampling methods can be more efficient when creating the population list is difficult, your population is large and dispersed, or you need to guarantee sufficient data for specific subpopulations. Alternative methods can reduce the need for a complete list and reduce the logistical headaches of a geographically extensive study.

For example, a national opinion poll company might consider an alternative method to assess differences between subpopulations, such as gender, race, and age.

These other methods frequently require you to have a greater understanding of your population than SRS requires. Consider the following alternatives to simple random sampling that can also obtain representative samples:

• Systematic sampling : Uses a random starting point but then samples at a fixed interval. Does not require a complete population list.
• Stratified sampling: Divides the population into dissimilar strata. Ensures that the sample includes specific subpopulations and facilitates comparisons between them.
• Cluster sampling : Divides the population into clusters that mirror the entire population. Then you randomly select from a subset of clusters. Reduces the need for a complete list of the population and eases logistics issues.

For a contrast to representative sampling methods, learn about convenience sampling , which tends to produce biased samples.

Learn about the specialized random sampling process that Political Polls use, allowing a relatively small sample to predict an election.

Sampling in Developmental Science: Situations, Shortcomings, Solutions, and Standards (nih.gov)

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Research Methods for Business

Definition…

Sampling is the process

of selecting a small number of elements

from a larger defined target group (Population)

of elements such that

the information gathered

from the small group will allow judgments

to be made about the larger groups .

Sampling is the act, process, or technique of selecting a suitable sample , or a representative part of a population for the purpose of determining parameters or characteristics of the whole population .

Purpose Of Sampling …

To draw conclusions about populations from samples, which enables us to determine a population`s characteristics by directly observing only a portion (or sample) of the population. We obtain a sample rather than a complete enumeration (a census ) of the population for many reasons.

6 MAIN REASONS FOR SAMPLING…

• . Timeliness
• . The large size of many populations
• . Inaccessibility of some of the population
• . Destructiveness of the observation

REASONS FOR SAMPLING…

• Economy - taking a sample requires fewer resources than a census.
• Time factor - a sample may provide you with needed information quickly.
• The very large populations - many populations about which inferences must be made are quite large
• The partly accessible populations- There are some populations that are so difficult to get access to that only a sample can be used.
• The destructive nature of the observation- sometimes the very act of observing the desired characteristic of a unit of the population destroys it for the intended use.
• Accuracy and sampling- A sample may be more accurate than a census. A sloppily conducted census can provide less reliable information than a carefully obtained sample.

Important terminologies...

• . Population
• . Sampling Unit

The population refers to the entire group of people, events or things of interest that the researcher wishes to investigate.

• If an organizational consultant is interested in studying the effects of a four-day work week on the white-coller workers in a telephone company in Ireland. Then all white-coller workers in that company will make up the population .
• If regulators wants to know how patients in nursing homes run by a company in France, then all the patients in all the nursing homes run by them will form the population. If however, the regulators are interested only in one particular nursing home run by that company, then only the patients in that particular nursing home will make the population.

An element is the

single member of the population.

• If 1000 blue-coller workers in a particular organization are working and an researcher is interested to know the satisfaction level of these workers then each member (blue-coller) of the particular organization will be considered as element.
• Census is a count of all elements in the human population .

A sample is a subset of the population. it comprises some members from it.

• . If 200 members are drawn/selected from a population of 1000 blue-coller workers to

study the desire outcome, then 200 members form the sample for the study .

• . If there are 145 patients in a hospital and 40 of them are to be surveyed by the hospital administrator to assess there level of satisfaction with the treatment received, then these 40 members will be called the sample .

A sample is thus a subgroup or subset of the population. By studying the sample, the researcher should be able to draw conclusions that are generalizable to the population of interest.

Sampling Unit

The sample unit is the element or the set of elements that is available for selection in some stage of the sampling process.

Example of sampling units in a multi stage sample are city blocks, house hold, and individuals with in the households.

A subject is a single member of the sample just as an element is a single member of the population.

• . If 200 members from the total population of 1000 blue-coller workers form the sample for the study. Then each blue-coller worker in the sample is a subject.
• . If there are 145 patients in a hospital and 40 of them are to be surveyed by the hospital administrator to assess there level of satisfaction with the treatment received, then each member from sample of 40 will be called the subject.

Representative of Sampling...�

• Choosing the right sample cannot be overemphasized.
• If we choose the sample in a scientific way, we can be reasonably sure that sample statistics (Mean, Standard Deviation, (S) Variation in the sample ) and population parameters (Mean (u), Standard Deviation, Variation in the sample ) are close to each others. �

Acknowledgments to Uma Sekaran

What is a Good Sample?

• . Accurate: absence of bias
• . Precise estimate: sampling error

Sampling error is any type of bias

that is attributable to mistakes

in either drawing a sample or

determining the sample size.

Sampling Process…

Define the Population

Determine the Sampling Frame

Select Sampling Technique(s)

Determine the Sample Size

Execute the Sampling Process

Defining Population of Interest…

Population of interest is entirely dependent on Management Problem, Research Problems, and Research Design.

Some Bases for Defining Population:

• . Geographic Area (Pakistan, Punjab, Banking sector, Our Institute etc.)
• . Demographics (Gender, Age, Color, Height etc.)
• . Usage/Lifestyle
• . Awareness

Sampling Frame …

A list of population elements (people, companies, houses, cities, etc.) from which units to be sampled can be selected.

• Difficult to get an accurate list.
• Sample frame error occurs when certain elements of the population are accidentally omitted or not included on the list.

Sampling Methods/Techniques

Probability

Nonprobability

Sampling Methods/Techniques/Types

Sampling Techniques

Convenience

Other Sampling

Simple Random

Probability Sampling Designs

A probability sample is one that gives every member of the population a known chance of being selected .

All are selected randomly.

• Simple random sampling - anyone
• Systematic sampling
• Stratified sampling - different groups (ages)
• Proportionate
• Cluster sampling - different areas (cities)

Simple Random Sampling

Simple random sampling is a method of probability sampling in which every unit has an equal nonzero

chance of being selected

• Each element in the population has a known and equal probability of selection.
• This implies that every element is selected independently of every other element.

Systematic Sampling

Systematic Random Sampling is a method of probability sampling in which the defined target population is ordered and the sample is selected according to position using a skip interval.

• The sample is chosen by selecting a random starting point and then picking every i th element in succession from the sampling frame.
• The sampling interval, i, is determined by dividing the population size N by the sample size n and rounding to the nearest integer.
• For example, there are 100,000 elements in the population and a sample of 1,000 is desired. In this case the sampling interval, i, is 100. A random number between 1 and 100 is selected. If, for example, this number is 23, the sample consists of elements 23, 123, 223, 323, 423, 523, and so on.

Stratified Sampling

Stratified Random Sampling is a method of probability sampling in which the population is divided into different subgroups and samples are selected from each

• A two-step process in which the population is partitioned into subpopulations.
• Divide the target population into homogeneous subgroups or strata
• Draw random samples fro each stratum
• Combine the samples from each stratum into a single sample of the target population
• A major objective of stratified sampling is to increase precision without increasing cost.

Cluster Sampling

• The target population is first divided into mutually exclusive and collectively exhaustive subpopulations, or clusters.
• Then a random sample of clusters is selected, based on a probability sampling technique.
• For each selected cluster, either all the elements are included in the sample (one-stage) or a sample of elements is drawn probabilistically (two-stage).
• Elements within a cluster should be as heterogeneous as possible, but clusters themselves should be as homogeneous as possible. Ideally, each cluster should be a small-scale representation of the population.
• In probability proportionate to size sampling, the clusters are sampled with probability proportional to size. In the second stage, the probability of selecting a sampling unit in a selected cluster varies inversely with the size of the cluster.

Nonprobability Sampling

… Nonprobability sample is an arbitrary grouping that limits the use of some statistical tests. It is not selected randomly.

Classifications of Nonprobability Sampling

Convenience Sampling

• Judgment Sampling

Quota Sampling

Snowball Sampling

Convenience sampling attempts to obtain a sample of convenient elements. Often, respondents are selected because they happen to be in the right place at the right time.

• Use of students, and members of social organizations
• Mail intercept interviews without qualifying the

respondents.

• “people on the street” interviews

Judgmental Sampling

Judgmental sampling is a form of convenience sampling in which the population elements are selected based on the judgment of the researcher.

• Test markets
• Engineers selected in industrial marketing research
• Expert witnesses used in court

Quota sampling may be viewed as two-stage restricted judgmental sampling.

• The first stage consists of developing control categories, or quotas, of population elements.
• In the second stage, sample elements are selected based on convenience or judgment.

Population Sample� composition composition� Control �Characteristic Percentage Percentage Number� Sex� Male 48 48 480� Female 52 52 520� ____ ____ ____� 100 100 1000

In Snowball Sampling , an initial group of respondents is selected, usually at random.

• After being interviewed, these respondents are asked to identify others who belong to the target population of interest.
• Subsequent respondents are selected based on the referrals.

Factors to Consider in Sample Design

Research objectives

Degree of accuracy

Knowledge of

target population

Research scope

Determining Sample Size

• How many completed questionnaires do we need to have a representative sample?
• Generally the larger the better, but that takes more time and money.
• Answer depends on :
• How different or dispersed the population is.
• Desired level of confidence.
• Desired degree of accuracy.

In conclusion, it can be said that using a sample in research saves mainly on money and time, if a suitable sampling strategy is used, appropriate sample size selected and necessary precautions taken to reduce on sampling and measurement errors, then a sample should yield valid and reliable information .

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SIMPLE RANDOM SAMPLING - PowerPoint PPT Presentation

SIMPLE RANDOM SAMPLING

Random - each unit in the population has the same ... south east quad. 800 students. bias due to nonresponse. 3,116 total population(nc peach growers) ... – powerpoint ppt presentation.

• PSYCHOLOGY MAJORS
• A separate sample is randomly selected from each stratum (or layer) of the population.
• CMU Residence Hall Students
• Divide population based on meaningful geographic areas.
• 3,116 Total Population(NC peach growers)
• Mean Trees for each group (M329)
• Mail Surveys
• Mail more than one questionnaire
• Call nonrespondents
• Use special delivery or hand stamped
• Make questionnaires clear, easy to read, and attractive layout
• Phone Survey
• Send advanced letter
• Pilot testing
• Call on weekend and/or evenings

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• random — Generate pseudo-random numbers
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random — Generate pseudo-random numbers ¶

Source code: Lib/random.py

This module implements pseudo-random number generators for various distributions.

For integers, there is uniform selection from a range. For sequences, there is uniform selection of a random element, a function to generate a random permutation of a list in-place, and a function for random sampling without replacement.

On the real line, there are functions to compute uniform, normal (Gaussian), lognormal, negative exponential, gamma, and beta distributions. For generating distributions of angles, the von Mises distribution is available.

Almost all module functions depend on the basic function random() , which generates a random float uniformly in the half-open range 0.0 <= X < 1.0 . Python uses the Mersenne Twister as the core generator. It produces 53-bit precision floats and has a period of 2**19937-1. The underlying implementation in C is both fast and threadsafe. The Mersenne Twister is one of the most extensively tested random number generators in existence. However, being completely deterministic, it is not suitable for all purposes, and is completely unsuitable for cryptographic purposes.

The functions supplied by this module are actually bound methods of a hidden instance of the random.Random class. You can instantiate your own instances of Random to get generators that don’t share state.

Class Random can also be subclassed if you want to use a different basic generator of your own devising: see the documentation on that class for more details.

The random module also provides the SystemRandom class which uses the system function os.urandom() to generate random numbers from sources provided by the operating system.

The pseudo-random generators of this module should not be used for security purposes. For security or cryptographic uses, see the secrets module.

M. Matsumoto and T. Nishimura, “Mersenne Twister: A 623-dimensionally equidistributed uniform pseudorandom number generator”, ACM Transactions on Modeling and Computer Simulation Vol. 8, No. 1, January pp.3–30 1998.

Complementary-Multiply-with-Carry recipe for a compatible alternative random number generator with a long period and comparatively simple update operations.

Bookkeeping functions ¶

Initialize the random number generator.

If a is omitted or None , the current system time is used. If randomness sources are provided by the operating system, they are used instead of the system time (see the os.urandom() function for details on availability).

If a is an int, it is used directly.

With version 2 (the default), a str , bytes , or bytearray object gets converted to an int and all of its bits are used.

With version 1 (provided for reproducing random sequences from older versions of Python), the algorithm for str and bytes generates a narrower range of seeds.

Changed in version 3.2: Moved to the version 2 scheme which uses all of the bits in a string seed.

Changed in version 3.11: The seed must be one of the following types: None , int , float , str , bytes , or bytearray .

Return an object capturing the current internal state of the generator. This object can be passed to setstate() to restore the state.

state should have been obtained from a previous call to getstate() , and setstate() restores the internal state of the generator to what it was at the time getstate() was called.

Functions for bytes ¶

Generate n random bytes.

This method should not be used for generating security tokens. Use secrets.token_bytes() instead.

Added in version 3.9.

Functions for integers ¶

Return a randomly selected element from range(start, stop, step) .

This is roughly equivalent to choice(range(start, stop, step)) but supports arbitrarily large ranges and is optimized for common cases.

The positional argument pattern matches the range() function.

Keyword arguments should not be used because they can be interpreted in unexpected ways. For example randrange(start=100) is interpreted as randrange(0, 100, 1) .

Changed in version 3.2: randrange() is more sophisticated about producing equally distributed values. Formerly it used a style like int(random()*n) which could produce slightly uneven distributions.

Changed in version 3.12: Automatic conversion of non-integer types is no longer supported. Calls such as randrange(10.0) and randrange(Fraction(10, 1)) now raise a TypeError .

Return a random integer N such that a <= N <= b . Alias for randrange(a, b+1) .

Returns a non-negative Python integer with k random bits. This method is supplied with the Mersenne Twister generator and some other generators may also provide it as an optional part of the API. When available, getrandbits() enables randrange() to handle arbitrarily large ranges.

Changed in version 3.9: This method now accepts zero for k .

Functions for sequences ¶

Return a random element from the non-empty sequence seq . If seq is empty, raises IndexError .

Return a k sized list of elements chosen from the population with replacement. If the population is empty, raises IndexError .

If a weights sequence is specified, selections are made according to the relative weights. Alternatively, if a cum_weights sequence is given, the selections are made according to the cumulative weights (perhaps computed using itertools.accumulate() ). For example, the relative weights [10, 5, 30, 5] are equivalent to the cumulative weights [10, 15, 45, 50] . Internally, the relative weights are converted to cumulative weights before making selections, so supplying the cumulative weights saves work.

If neither weights nor cum_weights are specified, selections are made with equal probability. If a weights sequence is supplied, it must be the same length as the population sequence. It is a TypeError to specify both weights and cum_weights .

The weights or cum_weights can use any numeric type that interoperates with the float values returned by random() (that includes integers, floats, and fractions but excludes decimals). Weights are assumed to be non-negative and finite. A ValueError is raised if all weights are zero.

For a given seed, the choices() function with equal weighting typically produces a different sequence than repeated calls to choice() . The algorithm used by choices() uses floating point arithmetic for internal consistency and speed. The algorithm used by choice() defaults to integer arithmetic with repeated selections to avoid small biases from round-off error.

Added in version 3.6.

Changed in version 3.9: Raises a ValueError if all weights are zero.

Shuffle the sequence x in place.

To shuffle an immutable sequence and return a new shuffled list, use sample(x, k=len(x)) instead.

Note that even for small len(x) , the total number of permutations of x can quickly grow larger than the period of most random number generators. This implies that most permutations of a long sequence can never be generated. For example, a sequence of length 2080 is the largest that can fit within the period of the Mersenne Twister random number generator.

Changed in version 3.11: Removed the optional parameter random .

Return a k length list of unique elements chosen from the population sequence. Used for random sampling without replacement.

Returns a new list containing elements from the population while leaving the original population unchanged. The resulting list is in selection order so that all sub-slices will also be valid random samples. This allows raffle winners (the sample) to be partitioned into grand prize and second place winners (the subslices).

Members of the population need not be hashable or unique. If the population contains repeats, then each occurrence is a possible selection in the sample.

Repeated elements can be specified one at a time or with the optional keyword-only counts parameter. For example, sample(['red', 'blue'], counts=[4, 2], k=5) is equivalent to sample(['red', 'red', 'red', 'red', 'blue', 'blue'], k=5) .

To choose a sample from a range of integers, use a range() object as an argument. This is especially fast and space efficient for sampling from a large population: sample(range(10000000), k=60) .

If the sample size is larger than the population size, a ValueError is raised.

Changed in version 3.9: Added the counts parameter.

Changed in version 3.11: The population must be a sequence. Automatic conversion of sets to lists is no longer supported.

Discrete distributions ¶

The following function generates a discrete distribution.

Binomial distribution . Return the number of successes for n independent trials with the probability of success in each trial being p :

Mathematically equivalent to:

The number of trials n should be a non-negative integer. The probability of success p should be between 0.0 <= p <= 1.0 . The result is an integer in the range 0 <= X <= n .

Added in version 3.12.

Real-valued distributions ¶

The following functions generate specific real-valued distributions. Function parameters are named after the corresponding variables in the distribution’s equation, as used in common mathematical practice; most of these equations can be found in any statistics text.

Return the next random floating point number in the range 0.0 <= X < 1.0

Return a random floating point number N such that a <= N <= b for a <= b and b <= N <= a for b < a .

The end-point value b may or may not be included in the range depending on floating-point rounding in the expression a + (b-a) * random() .

Return a random floating point number N such that low <= N <= high and with the specified mode between those bounds. The low and high bounds default to zero and one. The mode argument defaults to the midpoint between the bounds, giving a symmetric distribution.

Beta distribution. Conditions on the parameters are alpha > 0 and beta > 0 . Returned values range between 0 and 1.

Exponential distribution. lambd is 1.0 divided by the desired mean. It should be nonzero. (The parameter would be called “lambda”, but that is a reserved word in Python.) Returned values range from 0 to positive infinity if lambd is positive, and from negative infinity to 0 if lambd is negative.

Changed in version 3.12: Added the default value for lambd .

Gamma distribution. ( Not the gamma function!) The shape and scale parameters, alpha and beta , must have positive values. (Calling conventions vary and some sources define ‘beta’ as the inverse of the scale).

The probability distribution function is:

Normal distribution, also called the Gaussian distribution. mu is the mean, and sigma is the standard deviation. This is slightly faster than the normalvariate() function defined below.

Multithreading note: When two threads call this function simultaneously, it is possible that they will receive the same return value. This can be avoided in three ways. 1) Have each thread use a different instance of the random number generator. 2) Put locks around all calls. 3) Use the slower, but thread-safe normalvariate() function instead.

Changed in version 3.11: mu and sigma now have default arguments.

Log normal distribution. If you take the natural logarithm of this distribution, you’ll get a normal distribution with mean mu and standard deviation sigma . mu can have any value, and sigma must be greater than zero.

Normal distribution. mu is the mean, and sigma is the standard deviation.

mu is the mean angle, expressed in radians between 0 and 2* pi , and kappa is the concentration parameter, which must be greater than or equal to zero. If kappa is equal to zero, this distribution reduces to a uniform random angle over the range 0 to 2* pi .

Pareto distribution. alpha is the shape parameter.

Weibull distribution. alpha is the scale parameter and beta is the shape parameter.

Alternative Generator ¶

Class that implements the default pseudo-random number generator used by the random module.

Changed in version 3.11: Formerly the seed could be any hashable object. Now it is limited to: None , int , float , str , bytes , or bytearray .

Subclasses of Random should override the following methods if they wish to make use of a different basic generator:

Override this method in subclasses to customise the seed() behaviour of Random instances.

Override this method in subclasses to customise the getstate() behaviour of Random instances.

Override this method in subclasses to customise the setstate() behaviour of Random instances.

Override this method in subclasses to customise the random() behaviour of Random instances.

Optionally, a custom generator subclass can also supply the following method:

Override this method in subclasses to customise the getrandbits() behaviour of Random instances.

Class that uses the os.urandom() function for generating random numbers from sources provided by the operating system. Not available on all systems. Does not rely on software state, and sequences are not reproducible. Accordingly, the seed() method has no effect and is ignored. The getstate() and setstate() methods raise NotImplementedError if called.

Notes on Reproducibility ¶

Sometimes it is useful to be able to reproduce the sequences given by a pseudo-random number generator. By reusing a seed value, the same sequence should be reproducible from run to run as long as multiple threads are not running.

Most of the random module’s algorithms and seeding functions are subject to change across Python versions, but two aspects are guaranteed not to change:

If a new seeding method is added, then a backward compatible seeder will be offered.

The generator’s random() method will continue to produce the same sequence when the compatible seeder is given the same seed.

Basic examples:

Simulations:

Example of statistical bootstrapping using resampling with replacement to estimate a confidence interval for the mean of a sample:

Example of a resampling permutation test to determine the statistical significance or p-value of an observed difference between the effects of a drug versus a placebo:

Simulation of arrival times and service deliveries for a multiserver queue:

Statistics for Hackers a video tutorial by Jake Vanderplas on statistical analysis using just a few fundamental concepts including simulation, sampling, shuffling, and cross-validation.

Economics Simulation a simulation of a marketplace by Peter Norvig that shows effective use of many of the tools and distributions provided by this module (gauss, uniform, sample, betavariate, choice, triangular, and randrange).

A Concrete Introduction to Probability (using Python) a tutorial by Peter Norvig covering the basics of probability theory, how to write simulations, and how to perform data analysis using Python.

These recipes show how to efficiently make random selections from the combinatoric iterators in the itertools module:

The default random() returns multiples of 2⁻⁵³ in the range 0.0 ≤ x < 1.0 . All such numbers are evenly spaced and are exactly representable as Python floats. However, many other representable floats in that interval are not possible selections. For example, 0.05954861408025609 isn’t an integer multiple of 2⁻⁵³.

The following recipe takes a different approach. All floats in the interval are possible selections. The mantissa comes from a uniform distribution of integers in the range 2⁵² ≤ mantissa < 2⁵³ . The exponent comes from a geometric distribution where exponents smaller than -53 occur half as often as the next larger exponent.

All real valued distributions in the class will use the new method:

The recipe is conceptually equivalent to an algorithm that chooses from all the multiples of 2⁻¹⁰⁷⁴ in the range 0.0 ≤ x < 1.0 . All such numbers are evenly spaced, but most have to be rounded down to the nearest representable Python float. (The value 2⁻¹⁰⁷⁴ is the smallest positive unnormalized float and is equal to math.ulp(0.0) .)

Generating Pseudo-random Floating-Point Values a paper by Allen B. Downey describing ways to generate more fine-grained floats than normally generated by random() .

Table of Contents

• Bookkeeping functions
• Functions for bytes
• Functions for integers
• Functions for sequences
• Discrete distributions
• Real-valued distributions
• Alternative Generator
• Notes on Reproducibility

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statistics — Mathematical statistics functions

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Types of Sampling

Jan 05, 2020

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Types of Sampling. Simple Random Cluster Stratified Systematic (every kth element is sampled). Assumptions. Members of the Population can be numbered from 1 to N where N is the population size

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Types of Sampling • Simple Random • Cluster • Stratified • Systematic (every kth element is sampled)

Assumptions • Members of the Population can be numbered from 1 to N where N is the population size • You have a mechanism for randomly picking n numbers (usually without replacement) from the collection of numbers 1,2,3,…,N • If number k is selected, the population element with that number is in the sample

Simple Random Sample 1 12 17 10 6 9 • Population above right • In a Simple Random Sample each population member is equally likely to be selected • Click to sample 2 13 8 18 16 20 3 14 4 15 19 7 11 21 5

1 2 13 14 12 4 17 3 5 18 15 16 8 24 7 22 9 21 6 19 23 20 11 10 Cluster Sampling • The Population (right) can be divided into strata • A stratum consists of population elements that are similar in some way • All of one or more strata are randomly selected and used in the sample—see next slide

24 22 21 19 23 20 Population Above—Click to take a Cluster Sample 1 2 13 14 12 4 17 3 5 18 15 16 8 7 9 6 11 10

1 2 13 14 12 4 17 3 5 18 15 16 8 24 7 22 9 21 6 19 23 20 11 10 Stratified Sampling • The Population (right) can be divided into strata • A stratum consists of population elements that are similar in some way • A random sample from each stratum is randomly selected and used in the sample—see next slide

Population Above—Click to take a Stratified Sample 1 2 13 14 12 4 17 3 5 18 15 16 8 24 7 22 9 21 6 19 23 20 11 10

Systematic (every kth) • A starting number is randomly chosen • The population element corresponding to the starting number and every kth population element after that are selected for the sample • ‘Circular’ counting is used

Every 7th Element Sample 1 12 17 10 6 9 • Population above right • Click once to get a random starting element • Click again to get every 7th element after that 2 13 8 18 16 20 3 14 4 15 19 7 11 21 5

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