Chapter 5 · Standardizing Values

Section 5.2The Normal Model and the 68-95-99.7 Rule

When a distribution is symmetric, unimodal, and bell-shaped, the normal model gives us a powerful framework for understanding it. In this section, you'll explore the key features of the normal distribution, apply z-scores within the normal model, and use the 68-95-99.7 rule to determine how unusual any data value really is.

Summary

Normal modelA smooth, bell-shaped curve that describes how data is distributed when a histogram is symmetric, unimodal, and roughly bell-shaped. Defined entirely by its mean (μ) and standard deviation (σ).

68-95-99.7 ruleIn a normal distribution, approximately 68% of values fall within one standard deviation of the mean, 95% within two, and 99.7% within three. This rule lets you quickly assess how unusual any value is.

Z-score in contextA z-score tells you how many standard deviations a value sits from the mean. Combined with the 68-95-99.7 rule, it lets you estimate the proportion of values that are more or less extreme than a given data point.

Mean (μ)The center of the normal distribution — the peak of the bell curve. Most values cluster near the mean, with fewer values appearing as you move farther away in either direction.

Standard deviation (σ)Controls the spread of the bell curve. A small standard deviation produces a narrow, tall curve; a large standard deviation produces a wide, flat curve.

The normal model is the foundation for many statistical procedures — including confidence intervals and hypothesis testing — developed in later chapters.

The Real World Statistics Learning Progression

Each section follows the same learning progression: understand the concept, practice the skill, think like a statistician, then apply your learning to authentic data.

Learn This Section

Whether learning independently or teaching a class, the learning path below is designed to help students master the concepts in this section. Page references correspond to the Real World Statistics instructional system.

Read the sectionRead pages 151–154 to reinforce the concepts introduced in the lesson.

Practice what you've learnedComplete the practice questions on page 155. Check your work using the Answer Key, then review and correct any questions you missed before moving on.

Think like a statisticianExplore Enrichment: Role-Based Statistical Analysis Tasks and the Chapter Dataset to apply statistical thinking in real-world contexts.

Need Extra Support? If you'd like additional guidance while working through the practice questions, explore the Worked Solutions Manual for step-by-step solutions and explanations.

From Learners to Statisticians

Examples of authentic statistical work supported by this section include:

Using the 68-95-99.7 rule to interpret standardized test score distributions, identifying what proportion of students scored within one or two standard deviations of the mean and flagging unusually high or low performers (e.g., an assessment coordinator reviewing district-wide exam results).
Applying z-scores and the normal model to patient health data — such as cholesterol levels or blood pressure readings — to determine which patients fall outside the typical range and may warrant further clinical attention (e.g., a healthcare analyst screening population health data).
Evaluating product quality in a manufacturing process by modeling measurements as a normal distribution and using the 68-95-99.7 rule to estimate the proportion of items that fall outside acceptable tolerances (e.g., a quality control engineer assessing production consistency).

You may use these ideas to design your own investigations or explore the Real World Statistics System.

Enrichment: Role-Based Statistical Analysis Tasks

Enrichment tasks place students in the role of a statistician working on real research problems, where they evaluate data, justify decisions, and communicate findings clearly. Each section includes multiple enrichment tasks, including one with a model response that demonstrates statistical reasoning in context, showing how evidence is evaluated, decisions are justified, and conclusions are communicated. This serves as a bridge to working with authentic datasets and independent statistical investigation within each chapter.

Available in the Interactive eBook at the end of the section (page 155) and in the Appendix of the Print Book (page 401).

Teaching with Real World Statistics?

These structured learning paths — including Enrichment tasks and Chapter Datasets — can be adapted or copied directly into Canvas, Google Classroom, Schoology, and other learning management systems to simplify lesson planning and student assignments.

Part of the Real World Statistics instructional system — bringing together print, interactive learning, videos, datasets, enrichment, worked solutions, and teacher resources.