Standard Deviation
A free Statistics and Data Analysis lesson from the “Data Foundations” unit, with a worked example and practice problems including step-by-step solutions.
Standard deviation measures how far values typically sit from the mean. Small standard deviation means values cluster near the mean; large standard deviation means values are more spread out. 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
- Interpret standard deviation as typical distance from the mean
- Compare variability using standard deviation
- Recognize when standard deviation is sensitive to outliers
Worked example
Problem. A larger standard deviation means the data values are generally:
- Worked Example: First identify exactly what the question is asking: A larger standard deviation means the data values are generally:
- For data questions, identify what each statistic measures before calculating so the result matches the question.
- Standard deviation measures typical distance from the mean.
- Larger standard deviation means more spread.
Answer: farther from the mean
Practice problems
1. Practice case A: A larger standard deviation means the data values are generally:
Choices: categorical labels · farther from the mean · closer to the median · all equal
Show solution
- Warm-up: First identify exactly what the question is asking: A larger standard deviation means the data values are generally:
- For data questions, identify what each statistic measures before calculating so the result matches the question.
- Standard deviation measures typical distance from the mean.
- Larger standard deviation means more spread.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
Answer: farther from the mean
2. Practice case B: Which display shows median, quartiles, and range at once?
Choices: Two-way table · Pie chart · Box plot · Scatter plot
Show solution
- Warm-up: First identify exactly what the question is asking: Which display shows median, quartiles, and range at once?
- For range questions, identify the possible output values after the input restrictions and graph shape are considered.
- A box plot is built from the five-number summary.
- It shows median, quartiles, and overall range.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
Answer: Box plot
3. Practice case C: When comparing two distributions, a complete comparison should mention:
Choices: only the largest number · only the sample size · only the graph title · shape, center, spread, and unusual values
Show solution
- Warm-up: First identify exactly what the question is asking: When comparing two distributions, a complete comparison should mention:
- Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
- Comparisons should address the full distribution.
- Shape, center, spread, and unusual values give the clearest comparison.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
Answer: shape, center, spread, and unusual values
4. Practice case D: One group has range 8 and another has range 13. How much larger is the second range?
Show solution
- Warm-up: First identify exactly what the question is asking: One group has range 8 and another has range 13. How much larger is the second range?
- For range questions, identify the possible output values after the input restrictions and graph shape are considered.
- Compare the two spreads by subtracting.
- 13 - 8 = 5.
- Check the result by substituting or estimating: the response should match 5 and make sense in the original problem.
Answer: 5
5. Practice case E: An outlier is best described as:
Choices: a required part of every sample · a value far from the overall pattern · the middle value · the total number of values
Show solution
- Warm-up: First identify exactly what the question is asking: An outlier is best described as:
- Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
- Outliers are unusually far from the rest of the data.
- They can affect center and spread.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
Answer: a value far from the overall pattern
Practice this interactively with instant feedback and an AI tutor.
Practice Standard Deviation Take the free placement check