# 4: Sampling Distributions

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### Introduction

#### Learning objectives for this lesson

Upon completion of this lesson, you should be able to:

• determine the standard error for the sample proportion
• determine the standard error for the sample mean
• apply the Central Limit Theorem properly to a set of continuous data

#### Sampling Distributions of Sample Statistics

Two common statistics are the sample proportion,  $\hat{p}$, (read as “pi-hat”) and sample mean, $\bar{x}$, (read as “x-bar”). Sample statistics are random variables and therefore vary from sample to sample. For instance, consider taking two random samples, each sample consisting of 5 students, from a class and calculating the mean height of the students in each sample. Would you expect both sample means to be exactly the same? As a result, sample statistics also have a distribution called the sampling distribution. These sampling distributions, similar to distributions discussed previuosly, have a mean and standard deviation. However, we refer to the standard deviation of a sampling distribution as the standard error. Thus, the standard error is simply the standard deviation of a sampling distribituion. Often times people will interchange these two terms. This is okay as long as you understand the distinction between the two: standard error refers to sampling distributions and standard deviation refes to probability distributions.
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