AI Grounds
Open AI Grounds on a desktop
These interactive lessons need a larger screen. Please continue on a desktop or laptop computer.
AI Grounds
These interactive lessons need a larger screen. Please continue on a desktop or laptop computer.
Guided discovery
Read a value in standard-deviation units.
These are assumed normal models, not fitted observations. Current mean 0, standard deviation 1. Presets replace Mean, Standard deviation and Value; they keep the selected tail.
The normal model’s center and reference for signed distance. Use arrow keys on the slider. Press Enter or leave the number field to apply an exact edit.
Positive spread in the same raw units as Value. Use arrow keys on the slider. Press Enter or leave the number field to apply an exact edit.
Compare this value with the current mean and spread. Use arrow keys on the slider. Press Enter or leave the number field to apply an exact edit.
Solid curve · density. Shading · selected tail within the window. Dashed line · mean. Solid vertical line · selected value.
Window: -8 to 8. It follows the mean and covers at least four SD units each side; it widens for SD above 2. The density scale stays 0–0.85. Shading shows only the visible part of the selected tail; the probability includes its entire infinite extent.
Raw units relative to the model mean.
0.00
0 − 0
Signed distance in SD units.
0.00
0 / 1
All values at or below Value.
0.5000
P(X ≤ 0)
z = (0 − 0) / 1 = 0.00.
| Event | Approx. probability |
|---|---|
| At or below 0 | 0.5000 |
| Above 0 | 0.5000 |
| Single point: X = 0 | 0 (exact) |
Density at Value = 0.398942 per raw unit. This height is not a point or tail probability. A z-score is a distance, and can be negative or greater than 1; probabilities stay within 0–1.
Tails are calculated numerically and rounded to four decimals. A tiny positive normal tail may display 0.0000; that does not mean an off-window value is impossible.
The normal model is symmetric about its mean, has positive standard deviation and extends over all real values. Each tail at the mean is 0.5. The standardized variable Z = (X − mean) / SD has mean 0 and SD 1 under this normal model. A value’s z-score describes signed relative distance; standardization does not make an arbitrary distribution normal.
Normal density is exp(−z²/2) / (SD × √(2π)). Tail probabilities use the Abramowitz–Stegun 26.2.17 numerical approximation, with the smaller tail computed directly. Finite precision and rounding affect very small tails. The plotted curve and shading use finite sampled line segments; probability is calculated independently over the entire selected tail.
This lesson assumes a distribution. It does not infer normality from data, estimate parameters, perform a hypothesis test or attach a probability to a single point.
NIST · Normal distribution