4.5 Hypergeometric Distribution
3 min read•june 25, 2024
The models experiments where items are drawn without replacement from a finite population. It's crucial for scenarios like quality control sampling or analyzing election results, where each selection impacts subsequent probabilities.
Understanding this distribution helps in calculating probabilities for specific outcomes in sampling situations. It differs from the binomial distribution in key ways, making it essential for accurately analyzing experiments with changing probabilities between trials.
Hypergeometric Distribution
Characteristics of hypergeometric experiments
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- Involves a population divided into two distinct groups
- Group of interest (successes) (red marbles in a jar)
- Group of non-interest (failures) (blue marbles in a jar)
- Sampling occurs without replacement
- Selected items are not replaced back into the population before the next selection
- Leads to non-independence of picks as the probability of success changes with each draw (probability of drawing a red marble changes after each draw)
- is fixed and predetermined
- Number of items selected from the population is set in advance (drawing 5 marbles from the jar)
- Probability of success changes with each draw
- As items are not replaced, the probability of selecting a success or failure varies with each subsequent draw (probability of drawing a red marble decreases after each red marble is drawn)
- Focuses on the in the sample
- Goal is to determine the probability of obtaining a specific number of successes in the sample (probability of drawing 3 red marbles out of 5 draws)
Hypergeometric distribution calculations
- Hypergeometric distribution formula:
- : total (total number of marbles in the jar)
- : number of successes in the population (number of red marbles in the jar)
- : sample size (number of marbles drawn from the jar)
- : number of successes in the sample (number of red marbles drawn)
- Formula calculates the probability of obtaining exactly successes in a sample of size drawn from a population of size containing successes
- To calculate probabilities:
- Identify the values of , , , and based on the given information
- Substitute these values into the formula
- Simplify the expression to obtain the probability (use calculator or statistical software for large values)
- This formula represents the of the hypergeometric distribution
Hypergeometric vs binomial distributions
- Hypergeometric distribution:
-
- Selected items are not replaced back into the population (marbles drawn from a jar are not put back)
- Leads to non-independence of picks as probability of success changes with each draw
- Finite, known population size (number of marbles in the jar is fixed and known)
- Appropriate when:
- Sampling from a relatively small population without replacement (drawing marbles from a jar)
- Probability of success changes with each draw
-
- Binomial distribution:
- Sampling with replacement or from an infinite population
- Selected items are replaced back into the population or population is assumed to be infinite (flipping a coin multiple times)
- Leads to independence of picks as probability of success remains constant
- Fixed number of trials predetermined (flipping a coin 10 times)
- Constant probability of success for each trial (probability of getting heads on a coin flip is always 0.5)
- Appropriate when:
- Sampling with replacement or from an infinite population (flipping a coin, rolling a die)
- Probability of success remains constant with each draw
- Number of trials is fixed
- Sampling with replacement or from an infinite population
Statistical Measures and Functions
- : The average number of successes expected in a hypergeometric experiment
- : Measures the spread of the distribution around the expected value
- : Square root of the variance, indicating the typical deviation from the expected value
- : Gives the probability of obtaining up to a certain number of successes in the sample
Key Terms to Review (16)
Cumulative Distribution Function: The cumulative distribution function (CDF) is a function that describes the probability that a random variable takes on a value less than or equal to a specific value. It provides a complete picture of the distribution of probabilities for both discrete and continuous random variables, enabling comparisons and insights across different types of distributions.
Expected Value: Expected value is a fundamental concept in probability that represents the long-term average or mean of a random variable's outcomes, weighted by their probabilities. It provides a way to quantify the center of a probability distribution and is crucial in decision-making processes involving risk and uncertainty.
Fisher's exact test: Fisher's exact test is a statistical significance test used to determine if there are nonrandom associations between two categorical variables in a contingency table. It is particularly useful when sample sizes are small, allowing researchers to evaluate the significance of the observed frequencies in relation to the expected frequencies under the null hypothesis, which states that there is no association between the variables. This test provides an exact p-value rather than an approximation, making it valuable in situations where traditional chi-square tests may not be applicable.
Hypergeometric distribution: The hypergeometric distribution is a probability distribution that describes the number of successes in a fixed number of draws from a finite population without replacement. It is particularly useful when dealing with scenarios where the sampling is done from distinct groups, such as drawing cards from a deck. This distribution helps in understanding situations where we want to determine the likelihood of certain outcomes when the population is divided into two categories.
Hypergeometric probability: Hypergeometric probability is the probability of $k$ successes in $n$ draws from a finite population without replacement. It is used when the sample size and the population size are both known, and each draw changes the composition of the population.
Hypergeometric Probability Formula: The hypergeometric probability formula is a discrete probability distribution that calculates the probability of a specific number of successes in a given number of trials, without replacement, from a finite population. It is particularly useful in situations where the population size is relatively small, and the sampling is done without replacement.
K: In probability and statistics, 'k' typically represents the number of successes in a given number of trials. This concept is crucial in various types of distributions, as it helps to determine the probability of achieving a specific number of successful outcomes. In these contexts, 'k' can vary based on the scenario being analyzed, allowing for calculations related to success rates in independent trials or draws from finite populations.
N: The variable 'n' is a fundamental concept in probability and statistics, representing the number of trials or observations in a given experiment or sample. It is a crucial parameter that appears in various statistical distributions and theorems, providing crucial information about the size and structure of the data being analyzed.
Number of successes: The number of successes refers to the count of favorable outcomes in a given experiment or sample space, particularly in situations where events can be classified as successes or failures. In the context of the hypergeometric distribution, this term is critical as it specifically denotes the exact number of times a desired outcome occurs when drawing from a finite population without replacement. Understanding this term helps to clarify the calculations involving probabilities when considering both the size of the population and the sample drawn.
Population Size: Population size refers to the total number of individuals in a given group or population from which samples are drawn for statistical analysis. It plays a crucial role in determining the outcomes and characteristics of various statistical methods, particularly in calculating probabilities and making inferences. Understanding population size is essential when using models like the hypergeometric distribution, as it affects the likelihood of selecting specific types of individuals from a finite population without replacement.
Probability Mass Function: A probability mass function (PMF) is a mathematical function that gives the probability of a discrete random variable taking on a specific value. This function summarizes the distribution of probabilities for all possible outcomes, ensuring that the total probability across all values equals one. The PMF provides essential insights into the likelihood of various outcomes occurring in situations modeled by discrete distributions.
R.A. Fisher: R.A. Fisher was a prominent British statistician and geneticist known for his significant contributions to statistical methods and the foundations of modern statistical science. His work laid the groundwork for many statistical concepts, including the design of experiments and the development of the hypergeometric distribution, which is essential for understanding sampling without replacement and is crucial in fields such as genetics and quality control.
Sample Size: Sample size refers to the number of observations or data points collected in a statistical study or experiment. It is a crucial factor in determining the reliability and precision of the results, as well as the ability to make inferences about the larger population from the sample data.
Sampling without replacement: Sampling without replacement refers to a method where each member of a population can be selected only once for the sample. This technique ensures that the same individual or item is not chosen more than once, which is crucial in certain statistical analyses and experiments, especially when dealing with limited populations. This concept is essential when working with the hypergeometric distribution and has implications for the central limit theorem.
Standard Deviation: Standard deviation is a statistic that measures the dispersion or spread of a set of values around the mean. It helps quantify how much individual data points differ from the average, indicating the extent to which values deviate from the central tendency in a dataset.
Variance: Variance is a statistical measurement that describes the spread or dispersion of a set of data points in relation to their mean. It quantifies how far each data point in the set is from the mean and thus from every other data point. A higher variance indicates that the data points are more spread out from the mean, while a lower variance shows that they are closer to the mean.