Variability Math Example 1

Follow the full solution, then compare it with the other examples linked below.

Example 1

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

Solution

  1. 1
    Range: 25โˆ’5=2025 - 5 = 20 โ€” captures total spread including extremes
  2. 2
    Q1=7.5Q_1 = 7.5, Q3=22.5Q_3 = 22.5; IQR=22.5โˆ’7.5=15IQR = 22.5 - 7.5 = 15 โ€” captures middle 50% spread
  3. 3
    Mean: ฮผ=15\mu = 15; deviations: โˆ’10,โˆ’5,0,5,10-10,-5,0,5,10; ฯƒ2=100+25+0+25+1005=50\sigma^2 = \frac{100+25+0+25+100}{5} = 50; ฯƒ=50โ‰ˆ7.07\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, SDโ‰ˆ7.07. 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.

About Variability

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

Learn more about Variability โ†’

More Variability Examples

Example 2 medium

Two data sets have the same mean of 50 but different standard deviations: Set A has [formula], Set B

Example 3 easy

Which data set has greater variability? Set A: [formula] or Set B: [formula]? Use range and explain.

Example 4 hard

A factory produces bolts. Machine A produces bolts with diameter mean 10 mm, SD = 0.1 mm. Machine B

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