🔔 Normal Distribution Explorer
Skill goal: connect z-values to area — and therefore to probability — under the standard normal curve.
Observe
The curve is the standard normal (mean μ = 0, standard deviation σ = 1). The two blue handles mark z₁ and z₂, and the teal region between them is the probability P(z₁ < Z < z₂).
Manipulate
👆 Drag the z₁ and z₂ handles left or right along the axis.
Predict
Before you change it, what do you think will happen? Example: “If I move z₂ farther from 0 while z₁ stays at −1.96, will the probability go up or down?”
See
Probability = Φ(z₂) − Φ(z₁), where Φ is the standard normal CDF computed with the Abramowitz–Stegun error-function approximation.
Explain — Why it works
For a standard normal variable Z, the probability of landing in an interval equals the area under the bell curve over that interval. The total area under the curve is exactly 1, so every shaded region is a probability between 0 and 1. The CDF Φ(z) = P(Z ≤ z) is computed from the error function: Φ(z) = ½(1 + erf(z/√2)). Stretching the bounds outward always adds area; shrinking them removes it — that’s why moving z₂ away from the center increases P.
Challenge
1. Drag the bounds to shade exactly the middle 68% (the empirical-rule region). Then press Check.
2. Now shade a left tail with probability ≈ 0.50 (a region touching the far-left edge of the graph). Press Check.