68-95-99.7 Rule - Derivation

Derivation

These numerical values come from the cumulative distribution function of the normal distribution. For example, Φ(2) ≈ 0.9772, or Pr(x ≤ μ + 2σ) ≈ 0.9772. Note that this is not a symmetrical interval - this is merely the probability that an observation is less than μ + 2σ. To compute the probability that an observation is within 2 standard deviations of the mean (small differences due to rounding):

$\Pr(\mu-2\sigma \le x \le \mu+2\sigma) = \Phi(2) - \Phi(-2) \approx 0.9772 - (1 - 0.9772) \approx 0.9545$

Statisticians might express these intervals as confidence intervals: is approximately a 95% confidence interval.

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