Normal Distributions and Z-Scores
A free Statistics and Data Analysis lesson from the “Probability and Distributions” unit, with a worked example and practice problems including step-by-step solutions.
A normal distribution is symmetric and bell-shaped. A z-score tells how many standard deviations a value is from the mean, allowing values from different scales to be compared. This lesson builds the habit of reading the context first, choosing the right statistical tool, calculating carefully, and then writing what the result means. By the end, students should be able to do the computation and explain why that computation answers the question.
What you'll learn
- Use z-scores to standardize values
- Read distance from the mean in standard deviations
- Connect normal models to percentiles
Worked example
Problem. A z-score is 2, the mean is 55, and the standard deviation is 6. What is the data value?
- A z-score of 2 is two standard deviations above the mean.
- 55 + 2(6) = 67.
Answer: 67
Lesson preview: this page shows 5 practice items. The interactive activity contains 28 practice items and 15 quiz items (typed response and multiple choice). Counts include repeated prompts and variants.
Practice problems
1. Practice case A: A z-score is 2, the mean is 55, and the standard deviation is 6. What is the data value?
Show solution
- A z-score of 2 is two standard deviations above the mean.
- 55 + 2(6) = 67.
Answer: 67
2. Practice case B: A z-score of -1 means the value is:
Choices: 1 standard deviation below the mean · 1 standard deviation above the mean · equal to the mean · below the median by 1 unit
Show solution
- Negative z-scores are below the mean.
- The size tells the number of standard deviations.
Answer: 1 standard deviation below the mean
3. Practice case C: A normal distribution is usually described as:
Choices: a two-way table · symmetric and bell-shaped · strongly right-skewed · categorical
Show solution
- Normal curves are symmetric around the mean.
- Their shape is bell-like.
Answer: symmetric and bell-shaped
4. Practice case D: Using the 68% rule, about what percent of normal data is within 1 standard deviation of the mean?
Show solution
- The empirical rule says about 68% is within 1 standard deviation.
- Enter 68 as the percent.
Answer: 68
5. Practice case E: A value at the 90th percentile is:
Choices: equal to 90 · below every other value · the sample size · greater than about 90% of the data
Show solution
- A percentile describes relative position.
- The 90th percentile is above about 90% of values.
Answer: greater than about 90% of the data
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