Probability Sampling

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Probability sampling is not just a statistical nicety; it is the foundational step that determines whether your analysis has any claim to truth about the world beyond your dataset.

By thoughtfully choosing and implementing these methods, you move from making educated guesses to providing mathematically sound estimates with known uncertainty. And that is the true mark of a rigorous data scientist or statistician.

As data scientists and statisticians, we live in a world of inference. We take a small, manageable piece of the world (called a sample: a small representative portion of the whole) and use it to make powerful claims about a much larger whole (called the population in statistics). But the bridge between that sample and the population is only as strong as the method we use to build it. That bridge is a probability sampling.

Probability Sampling

Forget black-box models and “garbage in, garbage out” for a moment. The most fundamental step in ensuring the statistical validity of your analysis starts right here, at the sampling stage. Today, we are diving deep into the why and how of probability sampling, the gold standard for any analysis that aims to be truly representative.

Probability Sampling

Probability Sampling is the procedure in which the sample is selected in such a way that every element of a population has a known or non-zero probability of being included in the sample. Simple random sampling, stratified Random Sampling, systematic sampling, and Cluster Sampling are some Probability sampling designs.

Why Probability Sampling is Non-Negotiable in Data Science

In a probability sampling design, every single member of the population has a known, non-zero chance of being selected. This one simple rule is what unlocks everything we hold dear in statistical inference:

  • Unbiased Estimation: It allows us to calculate estimates (like the sample mean) that are unbiased with respect to the population parameter.
  • Quantifiable Uncertainty: This is the big one. Knowing the probability of selection allows us to calculate the sampling error and construct confidence intervals. We can finally say, “We are 95% confident the true population mean lies within this range.” This is the bedrock of statistical confidence.
  • Generalizability: The results from your sample can be legitimately generalized to the entire population you drew it.

Without probability sampling, you are left with non-probability sampling (like convenience sampling or voluntary response surveys), which is riddled with selection bias and offers no way to measure the error of your estimates.

The Core Methods of Probability Sampling: A Practical Guide

Let us break down the four workhorse techniques. Choosing the right one depends on your population structure, budget, and desired precision.

1. Simple Random Sampling (SRS)

  • The purest form of probability sampling. Every possible subset of $n$ individuals has an equal chance of being the selected sample, and each possible sample with or without replacement has equal probability of being selected as a sample. It is like a lottery for your population.
  • How: Assign every member a number and use a random number generator to select the sample.
  • When to use: Ideal when your population is homogeneous (very similar). It’s simple to understand, but it can be inefficient or expensive if the population is large and spread out.
  • Data Scientist’s Note: In Python, you can use numpy.random.choice() or pandas.DataFrame.sample() to implement this easily on a frame.

2. Systematic Sampling

  • In the sampling method, the sample is obtained by selecting every $k$-th element from a list of the population. You choose a random starting point between $1$ and $k$ and then select every $k$-th element after that. The $k$ is the sampling interval and stands for the integer nearest to $\frac{N}{n} = \frac{Population\, Size}{Sample\, Size}$.
  • How: Calculate k (the sampling interval) by dividing the population size (N) by your desired sample size (n). Start randomly and select every $k$-th item.
  • When to use: A simpler and more practical alternative to SRS when you have a complete sampling frame (e.g., a customer list, a phone book).
  • Warning: Be cautious of hidden periodicities in the list that could introduce bias (e.g., sampling every 7th day might always land on a Tuesday).

3. Stratified Sampling

  • Sometimes population units are not homogeneous according to certain characteristics. Dividing the population into distinct, homogeneous subgroups called strata (e.g., age groups, income brackets, product types), then performing a random sample within each stratum. The combined sample from each stratum is called a stratified random sample, and this whole procedure is called stratified random sampling.
  • How: You can proportionally allocate the sample size to each stratum (e.g., if a stratum is 20% of the population, it gets 20% of the sample) or use optimal allocation to allocate more samples to strata with higher variability.
  • When to use: Perfect when you want to ensure representation of key subgroups or when you know certain strata are more variable than others. It often provides greater statistical precision and reduces sampling error compared to SRS.
  • Data Scientist’s Note: This is crucial for building training sets for machine learning models on imbalanced data (e.g., fraud detection) to ensure the model sees enough rare cases.

4. Cluster Sampling

  • Dividing the population into naturally occurring, heterogeneous groups called clusters (e.g., cities, schools, factories). You then randomly select a subset of clusters and survey every individual within the chosen clusters.
  • How: This is often done in multiple stages (multi-stage sampling), e.g., randomly select states, then cities within those states, then schools within those cities.
  • When to use: Primarily for cost and logistics. It’s far cheaper to visit a few random clusters (e.g., five cities) than to sample individuals randomly spread across an entire country. The trade-off is that clusters are often similar internally, which can lead to higher sampling error compared to SRS.

Choosing the Right Sampling Design: A Quick Guide

MethodBest ForKey AdvantageKey Consideration
Simple Random SimplingSmall, homogeneous populationsConceptual simplicity, unbiasedCan be inefficient/logistically hard
Systematic SamplingHaving a complete list/frameEasy to implementVulnerable to hidden patterns
Stratified SamplingPopulations with important subgroupsIncreases precision, ensures subgroup representationRequires prior knowledge to form strata
Cluster SamplingLarge, geographically dispersed populationsMajor cost and logistical savingsHigher sampling error for same sample size

Sampling in R Language

WHERE and IF Statements in SAS

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Master the difference between WHERE and IF Statements in SAS for efficient data filtering. Learn when to use WHERE statements for speed on existing variables and IF statements for new variables. Our guide includes syntax examples, performance tips, and helps you avoid common subsetting errors to clean your datasets faster.

Differentiate between WHERE and IF Statements in SAS

There is a fundamental difference between WHERE and IF Statements in SAS. The breakdown of differences between WHERE and IF statements in SAS is as follows:

Key Differences Where and If Statements in SAS

Where Statement in SAS

The WHERE statement in SAS acts as a filter at the source. It tells SAS only to read observations that meet a specific condition from the input dataset(s). This happens very early in the process, often before the Program Data Vector (PDV) is fully constructed.

When to use WHERE Statement in SAS: Almost always for simple filtering, especially when working with large datasets, as it significantly improves performance by reducing I/O.

Example 1: Basic Filtering in a DATA Step

data high_earners;
    set sashelp.class;
    where age > 13; /* Only reads observations where Age is >13 */
run;

In this case, observations where age <= 13 are never even loaded into the PDV for processing.

Example 2: Using a WHERE Dataset Option

This is very powerful in procedures or when merging.

proc print data=sashelp.class(where=(sex='F')); /* Prints only females */
run;

data combined;
    merge ds1(where=(valid=1)) /* Merge only valid records from ds1 */
          ds2;
    by id;
run;

IF Statement in SAS

The IF statement in SAS is a processing-time filter. The entire observation is read into the PDV, all variables are calculated, and then the IF condition is evaluated. If the condition is false, the OUTPUT statement is bypassed (for that observation), and SAS returns to the beginning of the Data Step to process the next observation.

When to use IF Statement in SAS: When your filtering condition involves variables created within the Data Step or requires complex logic that must be executed row-by-row.

Example 1: Filtering on a New Variable

data tall_people;
    set sashelp.class;
    height_inches = height * 2.54;      /* Create a new variable */
    if height_inches > 64;                     /* Subsetting IF statement;*/
                                                              /* Equivalent to: if height_inches <= 64 then delete; */
run;

This works perfectly because the new variable height_inches exists in the PDV by the time the IF statement is executed.

Example 2: Complex Row-by-Row Logic

data flagged_records;
    set mydata;
    if some_var = . then do;        /* Check for missing value */
        error_flag = 'M';                   /* Set a flag */
        error_count + 1;                  /* Increment a counter */
    end;
    if error_flag = 'M';                    /* Output only the records with errors */
run;

This kind of multi-step logic is not possible with a WHERE statement.

What is the Special Case of Subsetting IF vs DELETE Statements?

A common use of an IF statement is the subsetting IF (if condition;). This outputs an observation only if the condition is true. Its logical opposite is if condition then delete; which deletes an observation if the condition is true.

data adults;
    set people;
    if age >= 18; * Output if true;
run;

/* Is logically equivalent to: */

data adults;
    set people;
    if age < 18 then delete; * Delete if true;
run;

WHERE or IF Statement: Which One to Use?

  • Use WHERE when:
    • You are filtering based on variables that exist in the input dataset.
    • You want the most efficient processing, especially for large data.
    • You are working in a PROC step (like PROC PRINT, PROC SORT).
    • You want to use special operators like CONTAINS or LIKE.
  • Use IF when:
    • You need to filter based on a variable created within the same Data Step.
    • Your filtering logic is complex and requires other SAS statements (like DO loops or ARRAY processing).
    • You are already reading every observation into the PDV for other necessary calculations, and the efficiency gain of WHERE is negligible.

The WHERE statement can be used …. IF statement cannot be used

  • WHERE statement can be used in procedures to subset data, while the IF statement cannot be used in procedures.
  • WHERE can be used as a data set option, while IF cannot be used as a data set option.
  • WHERE statement is more efficient than the IF statement. It tells SAS not to read all observations from the data set
  • WHERE statement can be used to search for all similar character values that sound alike, while the IF statement cannot be used.
  • WHERE statement can not be used when reading data using the INPUT statement, whereas the IF statement can be used.
  • Multiple IF statements can be used to execute multiple conditional statements
  • When it is required to use newly created variables, use an IF statement, as it doesn’t require variables to exist in the READIN data set.  

What is the one statement to set the criteria of data that can be coded in any step?

A WHERE statement can set the criteria for any data set in a data step or a proc step.

General Knowledge Quizzes

Online Quiz Sampling Distribution 16

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Master the fundamentals of Online Quiz Sampling Distribution with this 20-question MCQ quiz. Test your knowledge on bias, standard deviation of proportions, sampling methods (like quota, judgment, and stratified), and key statistical measures—essential exam prep for statistics students, data analysts, and data scientists. Let us start with the Online Quiz Sampoing Distribution now.

Online Quiz Sampling Distribution with Answers

Online Quiz Sampling Distribution with Answers

  • Bias, which occurs when randomly drawn samples from a population fail to represent the whole population, is classified as
  • If the proportion of the population is 10.5, then the proportional mean of the sampling distribution is
  • The value of the estimator is subtracted from the mean and then divided by the standard deviation to calculate
  • In statistical analysis, a sample size is considered small if
  • Sample statistics are denoted by the
  • A procedure in which the number of elements in the stratum is not proportional to the number of elements in the population is classified as
  • Method of sampling in which random sampling will not be possible because the population is widely spread is classified as
  • In the sampling distribution, the standard deviation must be equal to
  • Elements in the sample with specific characteristics are divided into the sample size to calculate
  • The method of random sampling, which is also called the  area sampling method, is classified as
  • In sampling, measures such as variance, mean, and standard deviation are considered as
  • If the value of $p$ is 0.70 and the sample size is 28, then the value of the standard deviation of the sample proportion is
  • If the value of $p$ is 0.70 and the sample size is 28, then the value of the standard deviation of the sample proportion is
  • Quota sampling, judgment sampling, and convenience sampling are classified as types of
  • Bias occurred in the collection of the sample because of confusing questions in the questionnaire, which is classified as
  • Bias in which a few respondents respond to the offered questionnaire is classified as
  • The difference between the corresponding population and the unbiased estimate in terms of absolute value is classified as
  • Bias, which occurs when a randomly drawn sample from a population fails to represent the whole population, is classified as
  • A border patrol checkpoint that stops every passenger van is using
  • Under equal allocation in stratified sampling, the sample from each stratum is

General Knowledge Quiz Tests