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?
- Worked Example: First identify exactly what the question is asking: A z-score is 2, the mean is 55, and the standard deviation is 6. What is the data value?
- For data questions, identify what each statistic measures before calculating so the result matches the question.
- A z-score of 2 is two standard deviations above the mean.
- 55 + 2(6) = 67.
Answer: 67
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
- Warm-up: First identify exactly what the question is asking: A z-score is 2, the mean is 55, and the standard deviation is 6. What is the data value?
- For data questions, identify what each statistic measures before calculating so the result matches the question.
- A z-score of 2 is two standard deviations above the mean.
- 55 + 2(6) = 67.
- Check the result by substituting or estimating: the response should match 67 and make sense in the original problem.
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
- Warm-up: First identify exactly what the question is asking: A z-score of -1 means the value is:
- For signed numbers, track both distance from zero and direction so the sign of the answer makes sense.
- Negative z-scores are below the mean.
- The size tells the number of standard deviations.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
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
- Warm-up: First identify exactly what the question is asking: A normal distribution is usually described as:
- Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
- Normal curves are symmetric around the mean.
- Their shape is bell-like.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
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
- Warm-up: First identify exactly what the question is asking: Using the 68% rule, about what percent of normal data is within 1 standard deviation of the mean?
- For percents, convert the percent to a decimal or fraction and connect it to the base amount in the problem.
- The empirical rule says about 68% is within 1 standard deviation.
- Enter 68 as the percent.
- Check the result by substituting or estimating: the response should match 68 and make sense in the original problem.
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
- Warm-up: First identify exactly what the question is asking: A value at the 90th percentile is:
- For percents, convert the percent to a decimal or fraction and connect it to the base amount in the problem.
- A percentile describes relative position.
- The 90th percentile is above about 90% of values.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
Answer: greater than about 90% of the data
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