Topic 5 · Statistics · Lesson 5.5

Normal Distribution Explorer

Drag the mean (μ), the standard deviation (σ), and a cutoff value (X). The bell curve shifts and resizes, the shaded region updates, and the empirical-rule bands recolor in real time. Watch what "two standard deviations from the mean" actually means as you change the inputs.

Normal distribution · empirical rule

68% within μ ± 1σ · 95% within μ ± 2σ · 99.7% within μ ± 3σ

With μ = 100 and σ = 15, about 15.87% of values fall below X = 85. The middle 95% falls between 70 and 130.

P(X < x)

15.87%

area to the left of X

P(X > x)

84.13%

area to the right of X

Z-score of X

−1.000

how many σ from μ

Empirical rule

68 / 95 / 99.7

% within ±1σ / ±2σ / ±3σ

Mean (μ) 100

−50250

Standard deviation (σ) 15

150

Cutoff X 85

−100300

Scenarios