Standard Deviation Examples in Math
Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Standard Deviation.
This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.
Concept Recap
The standard deviation measures the average distance of data values from the mean, giving a typical spread around the center.
The typical distance from the average. Low SD = clustered. High SD = spread out.
Read the full concept explanation →How to Use These Examples
- Read the first worked example with the solution open so the structure is clear.
- Try the practice problems before revealing each solution.
- Use the related concepts and background knowledge badges if you feel stuck.
What to Focus On
Core idea: About 68\% of data falls within 1 SD of the mean (for normal distributions).
Common stuck point: SD uses squared differences, so negative distances become positive.
Sense of Study hint: Build a table: one column for each value, one for the deviation from the mean, one for the squared deviation. Then average and take the square root.
Worked Examples
Example 1
mediumSolution
- 1 Compute the mean: \bar{x} = \frac{2+4+4+4+5+5+7+9}{8} = \frac{40}{8} = 5.
- 2 Find each squared deviation: (2-5)^2 = 9, (4-5)^2 = 1 (three times), (5-5)^2 = 0 (twice), (7-5)^2 = 4, (9-5)^2 = 16.
- 3 Sum of squared deviations: 9 + 1 + 1 + 1 + 0 + 0 + 4 + 16 = 32.
- 4 Variance: \sigma^2 = \frac{32}{8} = 4.
- 5 Standard deviation: \sigma = \sqrt{4} = 2.
Answer
Example 2
hardPractice Problems
Try these problems on your own first, then open the solution to compare your method.
Example 1
mediumExample 2
mediumRelated Concepts
Background Knowledge
These ideas may be useful before you work through the harder examples.