Two datasets can share the exact same average while representing totally different worlds. Standard deviation measures the spread around that center.
The tale of two classrooms
Consider two high school physics classes taking a test. Both classes achieve an identical average score of 75%.
• Class A scores: 73, 74, 75, 76, 77. The scores are closely clustered. Sample standard deviation is about 1.58 points.
• Class B scores: 40, 50, 75, 100, 110. The scores are more spread out (this hypothetical test permits extra credit). Sample standard deviation is about 30.41 points.
Without standard deviation, an administrator would conclude both classrooms are performing identically. Standard deviation reveals the crucial truth.
Sample vs Population Standard Deviation
• Population Standard Deviation (σ): Used when you have recorded every single member of the entire group (e.g. all 30 employees in a company). Divides by N.
• Sample Standard Deviation (s): Used when your data is a sample of a larger population (e.g. 500 surveyed citizens out of a country of millions). Divides by (n – 1) known as Bessel’s correction, which makes sample variance an unbiased estimator of population variance under standard sampling assumptions.
Sample s = √[ Σ(xᵢ - x̄)² ÷ (n - 1) ]
The Empirical Rule (68-95-99.7)
In a standard bell curve (normal distribution):
• ~68% of all data points fall within 1 standard deviation of the mean (x̄ ± 1s).
• ~95% fall within 2 standard deviations (x̄ ± 2s).
• ~99.7% fall within 3 standard deviations (x̄ ± 3s).
Try it yourself
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