## Sampling Distribution

### Introduction

In this lession, the sampling distribution will be introduced.

### Definition of Sampling Distributions

Suppose we have a population of interest and we have done the following works:

*Step 1:*We take a**random sample**from it.*Step 2:*Based on that sample, we calculate a**sample statistic**, e.g. the**mean**of that sample.*Step 3:*Then, we take another random sample and also calculate and record its mean.- ......
*Step n:*Then, we do the step 1 and 2 again and again, many more times.

Each one of the samples will have their own distribution, which we call **sample distributions**. Each observation in these distributions is a randomly sampled unit from the population. The values we recorded from example, the sample statistic (In this case, a **sample mean**), also make a new distribution, where each observation is not a unit from the population, but a sample statistic.

The distribution of these sample statistic is called the **sampling distribution**.

Note: the two terms, **sample distributions** and **sampling distribution** are different concepts.

### Example of Sampling Distribution

Suppose we are interested in the average height of the US women.

- Our population of interest is US women, denoted by
**N**. - The average height of the US women is denoted by
**μ**.

Then suppose we take random samples of 1000 women from each state, represented by AL, NC ... WY. For each state, we calculate the **state mean**, denoted by **x̄** (x bar). So, there is a dataset consisting of a bunch of **state means**. We call this distribution the **sampling distribution**.

- The mean of the sample means (In this case, the state mean) will probably be around the true population,
**mean(x̄) ≈ μ**; - The standard deviation (SD) of the sample means will probably be much lower than the population SD,
**SD(x̄) < σ**. This is because we would expect the average height for each state to be pretty close to one another. We call the SD of the sample means the**standard error**. - In fact, as the sample size
**n increases**, the**standard error will decrease**. The fewer women we sample from each state, the more variable we would expect the sample means to be.

### References & Resources

- N/A

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