Now you’ve seen that by knowing the mean and standard deviation of a normally distributed population, we can describe how a value compares to others in the population in terms of its percentile. And, we can also deduce the percent of values in-between, less than, or greater than any values.

In this lesson, we’ll apply this same logic to *samples*. How can we decide if a *sample* of values is typical or atypical? Let’s start with a simple example.

We have this population (for simplicity, N = 5):

21 24 65 42 79

Now let’s say we randomly select a sample of size n = 2, and we get 21 and 24.

21 24 65 42 79

How can we decide if this sample is typical of the population? You may guess that we can compare the mean of this sample to the mean of the population. That’s a start.

x̅ = 22.5

μ = 46.2

The mean of the sample looks very different from the mean of the population, but *how different*? Well, let’s look at the means we could get from all other possible samples of size n=2.

21, 21 = 21 | 24, 21 = 22.5 | 65, 21 = 43 | 42, 21 = 31.5 | 79, 21 = 50 |

21, 24 = 22.5 | 24, 24 = 24 | 65, 24 = 44.5 | 42, 24 = 33 | 79, 24 = 51.5 |

21, 65 = 43 | 24, 65 = 44.5 | 65, 65 = 65 | 42, 65 = 53.5 | 79, 65 = 72 |

21, 42 = 31.5 | 24, 42 = 33 | 65, 42 = 53.5 | 42, 42 = 42 | 79, 42 = 60.5 |

21, 79 = 50 | 24, 79 = 51.5 | 65, 79 = 72 | 42, 79 = 60.5 | 79, 79 = 79 |

Note that if we take the mean of means (the average of all the purple numbers), we get the population mean (=46.2).

Now we can compare a mean of 22.5 with the means of other samples of the same size.

**This is a preview of Lesson 7. To access the full book, please purchase a hard copy or a digital version. If you opt for the digital version, you will receive a link via email within 1 business day.**

Continue to Lesson 8, or select a lesson below.

Lesson 1: Introduction to Statistical Research Methods

Lesson 2: Visualizing Data

Lesson 3: Central Tendency

Lesson 4: Variability

Lesson 5: Standardizing

Lesson 6: Normal Distribution

Lesson 7: Sampling Distributions

Lesson 8: Estimation

Lesson 9: Hypothesis Testing

Lesson 10: t-Tests for Dependent Samples

Lesson 11: t-Tests for Independent Samples

Lesson 12: Intro to One-Way ANOVA

Lesson 13: One-Way ANOVA: Test significance of differences

Lesson 14: Correlation

Lesson 15: Linear Regression

Lesson 16: Chi-Squared Tests

Afterward

Index