Variability Math Example 2

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

Example 2

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

Solution

  1. 1
    Set A (ฯƒ=2\sigma=2): values cluster tightly around 50 โ€” most values fall within [46,54][46,54]; distribution is narrow/peaked
  2. 2
    Set B (ฯƒ=15\sigma=15): values spread widely around 50 โ€” most values fall within [20,80][20,80]; distribution is wide/flat
  3. 3
    Same mean: both are centered at 50, but the "typical" distance from center differs dramatically
  4. 4
    Visual: Set A looks like a tall, thin bell; Set B looks like a short, wide bell

Answer

Set A is narrowly concentrated around 50; Set B is widely spread. Same center, very different spread.
Standard deviation directly describes the typical distance of data points from the mean. Two distributions with the same center can look completely different based on spread. Never summarize data with only the mean โ€” always report a spread measure.

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 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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