Math With AmarA C A D E M Y
All math animations

High school · Statistics

Turn area under a bell curve into probability

Widen an interval around the mean and watch its probability grow while the density curve stays fixed.

Predict what will happen, press Play, then pause and explain what changed. Every control also works without playback.

Watch the relationship

Paused
Turn area under a bell curve into probability. Distance from mean: 0σ. Shaded interval: [0, 0]. Shaded probability (approx.): 0%Probability is the shaded areaDensity-6-3036Shading μ ± 0σ · Point probability is 0
This is an ideal normal distribution, not a fit to measured data. The horizontal axis is in arbitrary measurement units and the vertical axis is density per unit. The curve extends beyond the plotted window. Probabilities are numerical integrals; an exact single value has probability zero.

Starts paused. Play once, pause anywhere, or use Step to inspect the mathematics. Playback stops when this panel leaves the screen.

Make it your experiment

Change one value. Notice what follows.

The controls adjust the model. Numbers below describe the current frame. Decimals are rounded.

Distance from mean
0σ
Shaded interval
[0, 0]
Shaded probability (approx.)
0%

HD animation studio

From experiment to screen.

Present a crisp Canvas scene, save a full-HD image, or capture your model as a silent video.

Understand what you are seeing

The idea behind the motion.

A continuous density gives probability through area over an interval. The height at one point is not that point's probability. Moving the mean shifts the curve, while increasing the standard deviation spreads the same total area over a wider range.

A relationship to keep

f(x) = exp(−(x−μ)²/(2σ²)) / (σ√(2π))

Read the symbols alongside the explanation. A diagram shows the relationship; the assumptions tell you when it applies.

  1. STEP 1

    Locate and spread

    The center is μ. The positive number σ controls the spread. A narrow curve is taller because its total area must still equal one.

  2. STEP 2

    Sweep out an interval

    Playback shades from μ − zσ to μ + zσ as z grows from zero to three. The boundaries move together, staying equally far from the mean.

  3. STEP 3

    Read area, not height

    The probability readout integrates the shaded density numerically. At one standard deviation it is about 68.27%; at two, about 95.45%; at three, about 99.73%.

Your turn to explain

Make a prediction. Test your reasoning.

If μ = 2 and σ = 1.5, which interval lies within two standard deviations?

Use the values specified in the question. Reset restores the initial values for this investigation.

Compare your explanation

[−1, 5], because 2 ± 2(1.5) = 2 ± 3. Its normal-model probability is approximately 95.45%.

Work through a full lesson

Connect the animation to a worked example and practice questions.

Intermediate · Statistics

What does two standard deviations mean for package weights?

Standardize a normal measurement, read an interval probability, and distinguish individual variation from variation in an average.