Sampling Distribution
A sampling distribution is a probability distribution of a statistic obtained through a large number of samples drawn from a specific population. Understanding this concept is pivotal in statistical inference, allowing statisticians to make predictions about the population parameters based on sample statistics.
Definition and Purpose
The sampling distribution shows how the sample statistic, such as the mean, median, or proportion, varies from sample to sample. Here's how it works:
- Each sample drawn from the population has its own statistic, like a sample mean or variance.
- By plotting these statistics, we form a distribution which, under certain conditions, tends to follow a known probability distribution (e.g., normal distribution for the sample mean).
Key Properties
- Expected Value: The mean of the sampling distribution of a statistic is equal to the population parameter.
- Variance: The variance of the sampling distribution depends on the sample size and the population variance.
- Central Limit Theorem (CLT): For large sample sizes, the sampling distribution of the sample mean will be approximately normal, regardless of the population's distribution.
History
The concept of sampling distributions can be traced back to the early 20th century:
- Jerzy Neyman and Egon Pearson in the 1930s formalized many concepts related to statistical inference, including sampling distributions.
- However, the groundwork was laid by earlier statisticians like Karl Pearson and Ronald A. Fisher, who worked on the theory of errors and probability distributions.
Context and Application
Sampling distributions are crucial for:
- Hypothesis Testing: To determine how likely it is to observe a sample statistic under the null hypothesis.
- Confidence Intervals: To estimate the range within which the true population parameter is likely to fall.
- Power Analysis: To understand the probability of rejecting the null hypothesis when it is false.
Examples
Here are some common examples of sampling distributions:
- Sample Mean: The distribution of sample means tends to be normally distributed for large sample sizes.
- Sample Proportion: When dealing with proportions, the sampling distribution often follows a binomial distribution, which approximates to normal for large samples.
- Sample Variance: This distribution is related to the chi-square distribution.
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