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Proof Edit. 1 2 π ∫ − n n e − u 2 2 d u , >>\int _^e^>>>du\end>,> Pr ( μ − 1 σ ≤ X ≤ μ + 1 σ ) = 1 2 π ∫ − 1 1 e − u 2 2 d u ≈ 0.6827 Pr ( μ − 2 σ ≤ X ≤ μ + 2 σ ) = 1 2 π ∫ − 2 2 e − u 2 2 d u ≈ 0.9545 Pr ( μ − 3 σ ≤ X ≤ μ + 3 σ ) = 1 2 π ∫ − 3 3 e − u 2 2 d u ≈ 0.9973. &\Pr(\mu -1\sigma \leq X\leq \mu +1\sigma )=>>\int _^e^>>>du\approx 0.6827\\&\Pr(\mu -2\sigma \leq X\leq \mu +2\sigma )=>>\int _^e^>>>du\approx 0.9545\\&\Pr(\mu -3\sigma \leq X\leq \mu +3\sigma )=>>\int _^e^>>>du\approx 0.9973.\end>> Diagram showing the cumulative distribution function for the normal distribution with mean ( μ ) 0 and variance ( σ 2 ) 1. For example, Φ (2) ≈ 0.9772 , or Pr( X ≤ μ + 2 σ ) ≈ 0.9772 , corresponding to a prediction interval of (1 − (1 − 0.97725)·2) = 0.9545 = 95.45% . This is not a symmetrical interval – this is merely the probability that an observation is less than μ + 2 σ . To compute the probability that an observation is within two standard deviations of the mean (small differences due to rounding): This is related to confidence interval as used in statistics: X ¯ ± 2 σ n >\pm 2>>> is approximately a 95% confidence interval when X ¯ >> is the average of a sample of size n .
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