7.4 Finite Population Correction Factor
2 min read•june 25, 2024
When sampling from small populations, the becomes crucial. It adjusts standard errors and variances when exceeds 5% of the population, improving estimate accuracy for means and proportions.
This factor accounts for reduced variability in larger samples relative to . It's essential in scenarios like election polls, quality control, and market research, where impacts independence and estimation precision.
Finite Population Correction Factor
Finite population correction factor calculation
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- Adjusts of when sample size exceeds 5% of population size
- Formula: where = population size and = sample size
- Apply to of sampling distribution of means by multiplying where = standard error of sampling distribution of means
- Apply to standard error of sampling distribution of proportions by multiplying where = standard error of sampling distribution of proportions
- Reduces standard error as it accounts for the reduction in variability when a larger proportion of the population is sampled (election polls, small town surveys)
- Used when from a finite population
Adjustment of sampling variances
- Adjusts of sampling distribution when sample size exceeds 5% of population size
- Apply to variance of sampling distribution of means by multiplying where = variance of sampling distribution of means
- Apply to variance of sampling distribution of proportions by multiplying where = variance of sampling distribution of proportions
- Reduces variance as it accounts for the reduction in variability when a larger proportion of the population is sampled (quality control testing, market research on niche segments)
Independence in population sampling
- Assumes each observation is independent of all other observations in the sample
- Violated when sample size exceeds 5% of population size as probability of selecting an individual changes after each selection due to decreasing population size (drawing cards without replacement, selecting marbles from a jar)
- Population proportion near 0 or 1 also affects independence as probability of selecting an individual with the characteristic of interest changes significantly after each selection, especially with larger sample sizes relative to population (rare disease testing, defect rates in manufacturing)
- Lack of independence can lead to biased estimates and incorrect conclusions about the population (overestimating support for an unpopular policy, underestimating failure rates of a product)
Statistical Inference and Estimation
- allows for inference about population parameters from sample statistics
- quantifies the precision of estimates derived from samples
- Population parameters are estimated using corresponding sample statistics
- Finite population correction factor improves the accuracy of these estimates for small populations
Key Terms to Review (30)
√((N-n)/(N-1)): The square root of the ratio of the difference between the population size (N) and the sample size (n), divided by the difference between the population size (N) and 1. This term is known as the Finite Population Correction Factor and is used to adjust the standard error of a sample statistic when the sample size is a significant proportion of the total population.
Central Limit Theorem: The central limit theorem is a fundamental concept in probability and statistics that states that the sampling distribution of the mean of a random variable will tend to a normal distribution as the sample size increases, regardless of the underlying distribution of the variable.
Cluster Sampling: Cluster sampling is a probability sampling technique where the entire population is divided into groups or clusters, and a random sample of these clusters is selected to represent the whole population. This method is often used when the population is geographically dispersed or when a complete list of all individual members is not available.
Confidence interval: A confidence interval is a range of values, derived from sample data, that is likely to contain the value of an unknown population parameter. It provides an estimated range that is believed to contain the parameter with a certain level of confidence.
Confidence Interval: A confidence interval is a range of values that is likely to contain an unknown population parameter, such as a mean or proportion, with a specified level of confidence. It provides a measure of the precision of an estimate and allows researchers to make inferences about the population based on a sample.
Estimate of the error variance: Estimate of the error variance is a measure of the variability in the observed values that cannot be explained by the regression model. It is often denoted as $\hat{\sigma}^2$ and calculated as the sum of squared residuals divided by the degrees of freedom.
Finite Population Correction Factor: The finite population correction factor (FPC) is a statistical adjustment used to account for the impact of sampling from a finite population, rather than an infinite population, when estimating population parameters. It is particularly relevant when the sample size is a significant proportion of the total population size.
Heterogeneity: Heterogeneity refers to the state of being diverse, varied, or composed of different elements. In the context of statistical analysis, heterogeneity describes a situation where the data being studied exhibits significant differences or variability within the sample or population.
Homogeneity: Homogeneity refers to the uniformity or consistency of a population or dataset. It describes the degree to which the elements within a group share similar characteristics or properties, particularly in the context of statistical analysis and sampling.
Margin of Error: The margin of error is a statistical measure that quantifies the amount of uncertainty or imprecision in a sample statistic, such as the sample mean or sample proportion. It represents the range of values around the sample statistic within which the true population parameter is expected to fall with a given level of confidence.
Population Parameter: A population parameter is a numerical characteristic or measurement that describes the entire population being studied. It is a fixed, unknown value that represents the true state of the population, and is contrasted with a sample statistic, which is a numerical characteristic or measurement calculated from a sample drawn from the population.
Population Size: Population size refers to the total number of individuals or units that make up a given population. It is a fundamental concept in statistics and is particularly relevant in the context of statistical distributions and sampling methods.
Response Rate: The response rate is a measure of the proportion of the target population that participates in a survey or study. It is an important metric in statistical analysis and research, as it can impact the reliability and representativeness of the data collected.
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 that determines the reliability and precision of the conclusions drawn from the data.
Sample Statistic: A sample statistic is a numerical value calculated from a sample of data that is used to estimate or describe a characteristic of the larger population. It serves as a representation of the population parameter and is crucial in statistical inference and decision-making.
Sampling Distribution: The sampling distribution is a probability distribution that describes the possible values of a statistic, such as the sample mean or sample proportion, obtained from all possible samples of the same size drawn from a population. It represents the distribution of a statistic across all possible samples, rather than the distribution of the population itself.
Sampling Fraction: The sampling fraction is the ratio of the sample size to the size of the population being studied. It represents the proportion of the population that is included in the sample and is a crucial concept in understanding the Finite Population Correction Factor.
Sampling Frame: The sampling frame is the list or set of all the elements or units in the population from which a sample is to be drawn. It serves as the foundation for selecting a representative sample for statistical analysis.
Sampling without replacement: Sampling without replacement is a method of sample selection where each selected unit is not returned to the population before the next draw. This ensures that no unit can be chosen more than once.
Sampling Without Replacement: Sampling without replacement is a statistical technique where items or individuals are selected from a finite population, and once an item is selected, it is not returned to the population before the next selection. This method ensures that each item in the population has a unique chance of being chosen and prevents the same item from being selected multiple times within a single sample.
SRS: SRS, or Simple Random Sampling, is a fundamental sampling technique in statistics where each member of the population has an equal chance of being selected for the sample. This method ensures that the sample is representative of the overall population, making it a crucial component in understanding finite population correction factors.
Standard error: Standard error measures the accuracy with which a sample distribution represents a population by using standard deviation. It is crucial for estimating population parameters and conducting hypothesis tests.
Standard Error: The standard error is a measure of the variability or spread of a sample statistic, such as the sample mean. It represents the standard deviation of the sampling distribution of a statistic, indicating how much the statistic is expected to vary from one sample to another drawn from the same population.
Stratified Sampling: Stratified sampling is a probability sampling technique where the population is divided into distinct subgroups or strata, and samples are randomly selected from each stratum in proportion to the stratum's size. This method ensures that the sample is representative of the overall population, allowing for more precise estimates and inferences.
The Central Limit Theorem: The Central Limit Theorem (CLT) states that the distribution of the sample mean approaches a normal distribution as the sample size grows, regardless of the original population's distribution. This theorem is fundamental in inferential statistics because it allows for making predictions about population parameters.
Variance: Variance is a measure of the spread or dispersion of a dataset, indicating how far each data point deviates from the mean or average value. It is a fundamental statistical concept that quantifies the variability within a distribution and plays a crucial role in various statistical analyses and probability distributions.
σ_p̂: The standard deviation of the sampling distribution of the sample proportion, denoted as σ_p̂, is a measure of the variability or spread of the sample proportions around the true population proportion. It is a crucial concept in the context of finite population correction factor, as it helps quantify the uncertainty associated with estimating the population proportion from a sample.
σ_x̄: The standard deviation of the sampling distribution of the sample mean, denoted as σ_x̄, is a measure of the variability or spread of the sample means around the population mean. It represents the average amount that the sample means are expected to deviate from the true population mean.
σ²_p̂: σ²_p̂ represents the variance of the sample proportion, which is a measure of the spread or dispersion of the sample proportion around the true population proportion. It is a key concept in the context of the Finite Population Correction Factor, as it helps quantify the uncertainty associated with estimating the population proportion from a sample.
σ²_x̄: σ²_x̄ represents the variance of the sampling distribution of the sample mean, which is the expected variance of the sample means drawn from a population. It is a crucial concept in understanding the precision and reliability of statistical inferences made from sample data.