Variability Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Variability.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

Variability is the degree to which data points in a set differ from each other and from the center of the distribution.

How spread out or bunched up the data is. No variability = everyone is the same.

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: Variability is natural and expected—understanding it is key to statistics.

Common stuck point: Mean alone doesn't tell the story—you need variability measures too.

Sense of Study hint: Compare two small data sets with the same mean but different spreads. Which set's mean feels more trustworthy?

Worked Examples

Example 1

easy
Three measures of spread exist for the data \{5, 10, 15, 20, 25\}: range, IQR, and standard deviation. Calculate all three and compare what each captures.

Solution

  1. 1
    Range: 25 - 5 = 20 — captures total spread including extremes
  2. 2
    Q_1 = 7.5, Q_3 = 22.5; IQR = 22.5 - 7.5 = 15 — captures middle 50% spread
  3. 3
    Mean: \mu = 15; deviations: -10,-5,0,5,10; \sigma^2 = \frac{100+25+0+25+100}{5} = 50; \sigma = \sqrt{50} \approx 7.07
  4. 4
    Each captures different aspects: range is simple but sensitive to outliers; IQR is resistant; SD accounts for all deviations from mean

Answer

Range=20, IQR=15, SD≈7.07. Each measures spread differently.
Variability can be measured in multiple ways depending on the context and the presence of outliers. Range is simplest; IQR is most resistant; standard deviation is used in most statistical inference procedures.

Example 2

medium
Two data sets have the same mean of 50 but different standard deviations: Set A has \sigma=2, Set B has \sigma=15. Describe what this means and sketch what their distributions would look like.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
Which data set has greater variability? Set A: \{48, 49, 50, 51, 52\} or Set B: \{10, 30, 50, 70, 90\}? Use range and explain.

Example 2

hard
A factory produces bolts. Machine A produces bolts with diameter mean 10 mm, SD = 0.1 mm. Machine B produces bolts with mean 10 mm, SD = 0.5 mm. The specification requires bolts between 9.8 mm and 10.2 mm. Explain which machine is preferable and why variability matters here.
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Background Knowledge

These ideas may be useful before you work through the harder examples.

data abstract