Normal Distribution Calculator

Enter X Enter μ or X Enter σ or s Enter n
Click here if you already know your z-score

Given a normal distribution with μ = 100 and σ = 15, calculate the 68-95-99.7 rule, or three-sigma rule, or empirical rule ranges

Calculate Range 1:
Range 1, or the 68% range, states that 68% of the normal distribution values lie within 1 standard deviation of the mean
68% of values are within μ ± σ
μ ± σ = 100 ± 15
100 - 15 <= 68% of values <= 100 + 15
85 <= 68% of values <= 115

Calculate Range 2:
Range 2, or the 95% range, states that 95% of the normal distribution values lie within 2 standard deviations of the mean
95% of values are within μ ± 2σ
μ ± 2σ = 100 ± 2(15)
100 - 2 x 15 <= 95% of values <= 100 + 2 x 15
100 - 30 <= 95% of values <= 100 + 30
70 <= 95% of values <= 130

Calculate Range 3:
Range 3, or the 99.7% range, states that 99.7% (virtually ALL) of the normal distribution values lie within 3 standard deviations of the mean
99.7% of values are within μ ± 3σ
μ ± 3σ = 100 ± 3(15)
100 - 3 x 15 <= 99.7% of values <= 100 + 3 x 15
100 - 45 <= 99.7% of values <= 100 + 45
55 <= 99.7% of values <= 145