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