Center vs Spread Math Example 2

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

Example 2

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Three data sets all have mean = 10: Set A = {10,10,10,10}\{10, 10, 10, 10\}, Set B = {8,9,11,12}\{8, 9, 11, 12\}, Set C = {1,5,15,19}\{1, 5, 15, 19\}. Calculate the SD of each and describe what the spread reveals.

Solution

  1. 1
    Set A: all values = mean, so all deviations = 0; ฯƒA=0\sigma_A = 0
  2. 2
    Set B: deviations โˆ’2,โˆ’1,+1,+2-2,-1,+1,+2; ฯƒB=4+1+1+44=2.5โ‰ˆ1.58\sigma_B = \sqrt{\frac{4+1+1+4}{4}} = \sqrt{2.5} \approx 1.58
  3. 3
    Set C: deviations โˆ’9,โˆ’5,+5,+9-9,-5,+5,+9; ฯƒC=81+25+25+814=53โ‰ˆ7.28\sigma_C = \sqrt{\frac{81+25+25+81}{4}} = \sqrt{53} \approx 7.28
  4. 4
    Interpretation: same center (10), but A is perfectly consistent, B has minor variability, C is widely scattered

Answer

ฯƒA=0,ฯƒBโ‰ˆ1.58,ฯƒCโ‰ˆ7.28\sigma_A=0, \sigma_B\approx1.58, \sigma_C\approx7.28. Same mean, very different spreads.
Spread quantifies consistency. Zero spread means no variability; high spread means chaotic data. Without spread measures, identical means are falsely treated as equivalent distributions. Always pair mean with standard deviation.

About Center vs Spread

Center and spread are two complementary ways to describe a data distribution. Center (mean, median, mode) tells you where values cluster; spread (range, interquartile range, standard deviation) tells you how far values are from that center. Together they give a complete picture of any dataset.

Learn more about Center vs Spread โ†’

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