What You Will Learn
- What a sampling distribution is
- Why different samples produce different statistics
- How sample means form a distribution
- The difference between a sample distribution and a sampling distribution
- How sample size affects sampling distributions
- Why sampling distributions are important for inference
Start With a Simple Idea
Imagine a population containing thousands of observations. Instead of taking one sample, imagine repeatedly taking many samples of the same size.
Each sample will usually produce a slightly different statistic. For example, if we calculate the mean of every sample, we will get many different sample means.
Population
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Sample 1
Mean = 48
Sample 2
Mean = 52
Sample 3
Mean = 50
The important point is that the sample means are not necessarily identical. Their variation is what gives us a sampling distribution.
What Is a Sampling Distribution?
A sampling distribution is the probability distribution of a statistic obtained from repeated samples of the same size drawn from a population.
In simple terms:
Take many samples → calculate the same statistic from each sample → look at the distribution of those statistics.
Sample Distribution vs Sampling Distribution
These terms sound similar, but they describe different things.
Sampling Distribution of the Sample Mean
One of the most important sampling distributions in statistics is the sampling distribution of the sample mean.
Suppose we repeatedly take samples of the same size and calculate the mean of each sample. Those sample means form a distribution.
Example: Sample means
Each number represents the mean from one sample.
Where Does the Sampling Distribution Center?
If samples are selected appropriately, the sample mean tends to center around the population mean.
This does not mean every sample mean equals the population mean. Instead, the collection of sample means tends to be centered around the population mean.
Population Mean = μ
Sampling Distribution of X̄ tends to center around μ
Why Do Sample Means Vary?
Different samples contain different observations. Because the observations differ, the calculated sample means can also differ.
Sample A
Mean = 47
Sample B
Mean = 51
Sample C
Mean = 49
This variation is expected. Inferential statistics provides tools for understanding and quantifying this sampling variation.
What Happens When Sample Size Increases?
Sample size has an important effect on the variability of sample statistics.
Smaller Samples
Sample statistics tend to show more sampling variability.
Larger Samples
Sample statistics generally show less sampling variability.
This idea will become especially important when we study the Standard Error and the Central Limit Theorem.
The Sampling Distribution Workflow
Start with a population
Begin with the population you want to understand.
Take a sample
Select observations from the population.
Calculate a statistic
For example, calculate the sample mean.
Repeat
Imagine repeatedly taking samples of the same size.
Study the statistics
The collection of those statistics forms a sampling distribution.
Real-World Data Analytics Example
An e-commerce company wants to estimate the average order value of its customers.
Instead of taking one sample of customers, imagine taking 1,000 different samples, with each sample containing 100 orders.
Population
All customer orders
Sample Size
100 orders
Repeated Samples
1,000 samples
The 1,000 resulting sample means would form a sampling distribution of the sample mean.
A Common Mistake
A sampling distribution is not the distribution of the raw observations from one sample.
It is the distribution of a statistic calculated across repeated samples.
One Sample
Customer ages: 21, 25, 29, 31, 34...
Sampling Distribution
Sample means: 27.4, 28.1, 26.9, 27.8...
Interactive Practice
Test Your Understanding
CHECK YOUR UNDERSTANDING
What is a sampling distribution?
CHECK YOUR UNDERSTANDING
Suppose we repeatedly take samples of 50 customers and calculate the mean spending for every sample. What forms the sampling distribution?
CHECK YOUR UNDERSTANDING
Why can two samples from the same population have different sample means?
CHECK YOUR UNDERSTANDING
The sampling distribution of the sample mean tends to be centered around what quantity?
CHECK YOUR UNDERSTANDING
What generally happens to sampling variability when sample size increases?
CHECK YOUR UNDERSTANDING
Which statement is correct?
Fill in the Blanks
A distribution of a statistic calculated from repeated samples is called a ______ distribution.
A statistic calculated from each sample can vary because different samples contain different ______.
The sampling distribution of the sample mean tends to be centered around the population ______.
Increasing sample size generally reduces sampling ______.
Data Analytics Scenario
Think Like a Data Analyst
A retail company wants to estimate average customer spending. Analysts repeatedly take samples of 100 transactions and calculate the mean transaction value for each sample.
CHECK YOUR UNDERSTANDING
What collection of values would form the sampling distribution?
CHECK YOUR UNDERSTANDING
If the sample size increases from 100 to 400, what would generally happen to the variability of the sample means?
Key Takeaways
A sampling distribution describes the behavior of a statistic across repeated samples.
Different samples can produce different sample statistics.
The sampling distribution of the sample mean tends to center around the population mean.
Sampling variability is expected and is an important part of statistical inference.
Larger samples generally produce less sampling variability.
Sampling distributions are the foundation for concepts such as standard error, confidence intervals, and hypothesis testing.