Variability Math Example 4

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

Example 4

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.

Solution

  1. 1
    Machine A: within ยฑ3ฯƒ\pm 3\sigma, bolts range from 10โˆ’3(0.1)=9.710 - 3(0.1) = 9.7 to 10.310.3 mm; ยฑ2ฯƒ\pm 2\sigma range is [9.8,10.2][9.8, 10.2] โ€” fits specification
  2. 2
    Machine B: within ยฑ1ฯƒ\pm 1\sigma, bolts range from 9.59.5 to 10.510.5 mm; bolts outside [9.8,10.2][9.8,10.2] will occur frequently
  3. 3
    With normal distribution, Machine A produces ~95% of bolts in spec; Machine B produces far fewer
  4. 4
    Machine A is preferred โ€” lower SD means tighter quality control even with same mean

Answer

Machine A (SD=0.1) is preferable; its lower variability keeps nearly all production within specification.
In manufacturing, low variability is critical for quality control. Two machines with identical means can have dramatically different defect rates if their standard deviations differ. This is why variability, not just the mean, determines product quality.

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 1 easy

Three measures of spread exist for the data [formula]: range, IQR, and standard deviation. Calculate

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.

โ† All Variability Examples Variability Concept