Chapter 6: The Normal Distribution
Student Resources
I use the 4 “P’s” framework to help you learn the material in this chapter: Prepare, Practice, Participate, and Perform. To increase the chances to succeed in this course, I strongly encourage you to complete all four “P’s” for each chapter.
1 Prepare
1.1 Chapter Overview
This chapter introduces the normal distribution—one of the most important concepts in statistics. You’ll learn about the properties of the bell curve, how to use z-scores and the standard normal distribution, and when normality matters (and when it doesn’t) in movement science research.
1.2 Multimedia Resources
The following table provides access to video and slide resources for this chapter. Click the links to open them in an overlay for better viewing on all devices.
| Resource | Description | Link |
|---|---|---|
| Long Video Overview | A detailed video explaining the key concepts of the normal distribution, z-scores, and when normality matters in movement science research. | 🔗 Watch Video |
| Slide Overview PDF | PDF slides that serve as an overview of this chapter. Read these before the textbox to introduce the main concepts and vocabulary. | 🔗 View Slides |
| Slide Deck HTML | Interactive HTML slides for class. During class, the instructor controls the presentation; after class, review at your own pace. | 🔗 Open Slides |
| Slide Deck PDF | PDF version of the slide deck for download and offline viewing. | 🔗 Download PDF |
1.3 Read the Chapter
Read (Weir & Vincent, 2021, p. Ch6) and (Furtado, 2026, p. Ch7) - optional but recommended - to understand the bell curve, probability, and when normality matters.
To succeed in this course, you must read the textbook chapters assigned for each topic. This is the only way to learn the material in depth.
Once done, proceed to the next section to practice what you learned.
2 Practice
Practicing what you learned in the chapter is essential to mastering the material. Below are some resources to help you practice the material in this chapter.
2.1 Frequently Asked Questions
The normal distribution is a symmetric, bell-shaped probability distribution that is completely defined by its mean (μ) and standard deviation (σ). It’s also called the Gaussian distribution or bell curve. Many natural phenomena approximate this distribution, and it’s fundamental to statistical inference.
Key properties include: - Symmetric (mirror image on both sides of the mean) - Bell-shaped (highest frequency at the mean, tapering off at the tails) - Mean = Median = Mode (all at the center) - About 68% of values within ±1 SD, 95% within ±2 SD, 99.7% within ±3 SD - Total area under the curve equals 1.0 (100%)
A z-score is a standardized score that indicates how many standard deviations a raw score is from the mean: z = (X − μ)/σ. Positive z-scores are above the mean; negative z-scores are below the mean. A z-score of 0 means the score equals the mean.
Z-scores allow us to:
- Compare scores from different distributions or scales
- Determine the relative standing of a score
- Calculate probabilities and percentiles
- Standardize data for analysis They put different measurements on the same scale.
The standard normal distribution is a normal distribution with mean = 0 and standard deviation = 1. Any normal distribution can be converted to the standard normal distribution using z-scores. We use this to look up probabilities in the standard normal table.
The 68-95-99.7 rule (empirical rule) states that in a normal distribution:
- About 68% of values fall within ±1 standard deviation of the mean
- About 95% fall within ±2 standard deviations
- About 99.7% fall within ±3 standard deviations This rule helps us quickly estimate probabilities.
To use the z-table:
- Calculate the z-score: z = (X − μ)/σ
- Find the z-score in the table (row for first two digits, column for second decimal)
- The table value gives the proportion of scores below that z-score
- For scores above, subtract from 1.0
Data are normally distributed if they follow the pattern of the normal distribution—symmetric, bell-shaped, with most values near the mean and fewer values at the extremes. In practice, we usually look for “approximately normal” rather than perfectly normal.
Many statistical tests (t-tests, ANOVA, regression) assume normally distributed data or sampling distributions. However, many of these tests are “robust” to violations of normality, especially with larger sample sizes. It’s important to check, but small departures from normality are usually not a problem.
You can check normality by:
- Creating a histogram or density plot (look for bell shape)
- Making a Q-Q plot (points should fall on diagonal line)
- Calculating skewness and kurtosis
- Conducting formal tests (Shapiro-Wilk, Kolmogorov-Smirnov) Visual inspection is often most useful.
If data aren’t normal:
- Use nonparametric tests that don’t assume normality
- Transform the data (log, square root) to make them more normal
- Use robust statistics (median, IQR instead of mean, SD)
- Recognize that many tests are robust to violations with adequate sample size
These terms are often used interchangeably. The normal curve is the graphical representation (the bell-shaped curve), while normal distribution refers to the probability distribution itself. Both describe the same concept.
Technically, the normal distribution is continuous. However, discrete variables with many possible values can approximate a normal distribution. For example, the number of successful free throws out of 100 attempts might look approximately normal even though it’s discrete.
2.2 Test your Knowledge
Take this low-stakes quiz to test your knowledge of the material in this chapter. This quiz is for practice only and will help you identify areas where you may need additional review.
3 Participate
This section includes activities and discussions that will be completed during class time. Your active participation is essential for deepening your understanding of the material.
During class, we will:
- Calculate and interpret z-scores for movement science data
- Use the z-table to find probabilities and percentiles
- Assess normality of real datasets
- Discuss when normality assumptions matter (and when they don’t)
4 Perform
4.1 Apply Your Learning
Now that you’ve prepared, practiced, and participated, it’s time to demonstrate your mastery of the material through assignments and assessments.
I strongly encourage you to complete the previous “Ps” (Prepare, Practice, Participate) before attempting any assignments or assessments associated with this chapter.