The normal distribution
Move the mean μ and the standard deviation σ, then ask a question. The curve is the density; the shaded area under it is the probability the question asks for.
Parameters
Ask a question
Compare with
P(X ≤ 1.96) = 0.9750
P(X ≤ 1.96) = F(1.96) = 0.9750
Summary
| Measure | N(μ = 0, σ = 1) |
|---|---|
| mean | 0.0000 |
| variance | 1.0000 |
| standard deviation | 1.0000 |
| skewness | 0.0000 |
The same in code
- Python (scipy.stats)
norm.cdf(1.96, loc=0, scale=1)- Excel
=NORM.DIST(1.96, 0, 1, TRUE)
Formulas
Try this
The 68-95-99.7 rule
On the standard normal, ask for P(−1 ≤ X ≤ 1), then widen the ends to ±2 and ±3.
Look for: The areas are 0.6827, 0.9545 and 0.9973. Any normal gives the same areas between μ ± 1σ, 2σ and 3σ.
Density is not probability
Set σ = 0.2.
Look for: The peak rises to about 2, above 1. The height is a density; only areas are probabilities, and the total area is still 1.
Exact, not from a table
Open the preset for an effect within 11.5 hours.
Look for: The exact area is 0.3415. The course quotes 0.3412, read from a z-table with z rounded to two decimals; the lab computes the area directly.
Heavier tails
Compare the standard normal with Student's t at ν = 3 and ask for P(X > 2).
Look for: The normal leaves 0.0228 beyond 2, the t 0.0697, about three times as much. Raise ν and the difference shrinks; by ν = 30 it has nearly gone.