SAMPLING DISTRIBUTION

Sampling Distribution
sampling distribution

Sampling Distribution

A sampling distribution is the probability distribution of a statistic (such as the mean, proportion, or standard deviation) obtained from multiple samples drawn from the same population.

In simpler terms, it represents how a sample statistic (like the sample mean) varies when we take multiple samples from the population.

Key Points:

  • It is formed by repeatedly selecting samples from a population and calculating a statistic for each sample.
  • The shape of the sampling distribution depends on the sample size and the population distribution.
  • As the sample size increases, the sampling distribution tends to become more normal due to the Central Limit Theorem (CLT).

Step-by-Step Methods for Sampling Distribution

The process of creating a sampling distribution involves multiple steps, from selecting samples to analyzing their distribution. Here’s a structured step-by-step guide:

Step 1: Define the Population

  • Identify the entire group of individuals or data points you want to study.
  • Example: A university wants to analyze the average height of all its students.

Step 2: Select a Statistic for Analysis

  • Choose a statistic to study, such as: Mean (average), Proportion, Variance
  • Example: If we are studying students’ heights, we focus on the mean height.

Step 3: Take Multiple Random Samples

  • Randomly select multiple samples from the population, ensuring each sample has the same size (n).
  • Example: Take 100 different samples, each containing 50 students.

Step 4: Compute the Sample Statistic

  • Calculate the chosen statistic for each sample.
  • Example: Compute the average height for each sample of 50 students.

Step 5: Create the Sampling Distribution

  • Plot the frequency distribution of the sample statistics (e.g., sample means).
  • This forms the sampling distribution of the mean (if studying averages).

Step 6: Analyze the Shape of the Distribution

  • The shape of the sampling distribution depends on: Sample size (n), Population distribution, Number of samples
  • Key Concept: Central Limit Theorem (CLT)
  • If sample size n is large (n ≥ 30), the sampling distribution will be approximately normal (bell-shaped) even if the population is not normally distributed.

Step 7: Calculate the Mean and Standard Error

  • The mean of the sampling distribution (μₓ̄) is equal to the population mean (μ).
  • The standard deviation of the sampling distribution, called Standard Error (SE) 

Step 8: Apply Statistical Inference

  • Use the sampling distribution to estimate population parameters and make hypothesis tests.
  • Example: If the average sample height is 5.7 feet, we infer the true population mean is around 5.7 feet +- margin of error.
Sampling Distribution