5 min read•december 31, 2022

B

Brianna Bukowski

Josh Argo

Jed Quiaoit

A **sampling distribution** is a distribution of all possible samples of a given size drawn from a population. It represents the distribution of statistics (such as the mean or proportion) calculated from these samples. 💠

For example, suppose you're interested in estimating the mean income of a population of workers. You can take a sample of workers and calculate the mean income of the sample. However, this sample mean is likely to be different from the population mean due to sampling error. To account for this, you can take multiple samples of the same size and calculate the mean income for each sample. The distribution of these sample means is called the sampling distribution of the mean.

In the previous units, every distribution consisted of one sample, such as a class of students grade in a class. With a sampling distribution, you take the average of *all* means (quantitative) or proportions (categorical) of each possible sample size (n) and use these averages as your data points. The normal model now also represents the distribution of all possible samples of a given sample size.

To find the sampling distribution for differences in a sample proportion or mean, remember that variances **always **add to find the new variance. If one needs the standard deviation, you should take the square root of the variance. However, for means you can just subtract. ➖

There are two major types of random variables in AP Statistics: **discrete** and **continuous**. **Discrete random variables **are variables that have a certain and definite set of values that the variable could be. Usually, these are whole numbers in real world situations (1, 2, 3, 4, 5…, 100, etc.). For discrete random variables, to calculate the mean, you use the expected variable formula:

For discrete random variables, to calculate the standard deviation, you use a formula similar (in a way) to the expected value formula, but with a square root:.

The other type of random variable, **continuous random variables**, can take on any value at any point along an interval. Generally, continuous random variables can be measured while discrete random variables are counted. A histogram is used to display continuous data, while a bar graph displays discrete data! 📊

A **population parameter **is a measure of a characteristic of a population, such as the mean or proportion of a certain attribute. It is a fixed value that represents the true value of the population. 🌎

A **sample statistic** is a measure of a characteristic of a sample, such as the mean or proportion of a certain attribute. It is a calculated value that estimates the value of the population parameter. 🏠

In AP Statistics, you will be asked to compare **S**tatistics from a **S**ample to the **P**arameters of a **P**opulation. Here is a chart to help you remember which symbols are from sample statistics and from population parameters:

Measurement | Population Parameter | Sample Statistic |

Mean | 𝝁 | x̅ |

Standard Deviation | σ | s |

Proportions | 𝝆 | p̂ |

Browse Study Guides By Unit

👆Unit 1 – Exploring One-Variable Data

✌️Unit 2 – Exploring Two-Variable Data

🔎Unit 3 – Collecting Data

🎲Unit 4 – Probability, Random Variables, & Probability Distributions

📊Unit 5 – Sampling Distributions

⚖️Unit 6 – Proportions

😼Unit 7 – Means

✳️Unit 8 – Chi-Squares

📈Unit 9 – Slopes

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