A-level · Beta
The normal distribution
Move the mean, spread and bounds; connect shaded area to probability.
Move the bounds to shade a probability. The mean and spread also change the curve.
Shaded probability ≈ 68.269%.
- Mean μ
- 0
- Standard deviation σ
- 1
- Variance σ²
- 1
- Standardised bounds
- -1 to 1
Probability is area, not curve height. Bounds are ordered from smallest to largest. Equal bounds give zero probability. CDF values are computed numerically; the curve has tails beyond the display.
Predict, test and explain
Step 1 of 3
Predict
For a continuous normal variable, what is P(X = μ)?
Explore and compare
Choose the investigation above. Predict what will change before you move a control.
Try it in the simulation. Change one thing at a time.
Notes and saved values stay in this tab. They disappear when you reload.
Hints, self-check and connections
Try a self-check
Fixed values, separate from the diagram controls.
For a continuous normal variable, what is P(X = μ)?
Hint
For N(μ, σ²), σ is standard deviation and σ² is variance.
Reveal answer
One exact point has no width and therefore no probability area.
Build an explanation
Hint 1
For N(μ, σ²), σ is standard deviation and σ² is variance.
Hint 2
One exact point has no width and therefore no probability area.
Hint 3
Compare the result with the model, then explain why it occurs.
Worked example
This example uses fixed values, separate from the diagram controls.
For X ~ N(10, 4), standardise X = 12.
- Read variance 4, so standard deviation is 2.
- Use z = (x − μ)/σ.
- Calculate (12 − 10)/2.
Answer: z = 1.
Connect this idea
Watch out: Probability is area under the density, not the height at a point.
Specification and learning route
AQA N2 · Edexcel Statistics 4.2
A-level extension. This model illustrates selected content; references are not a claim of full coverage or exam-board endorsement.
Useful starting knowledge: Probability as area; Mean and standard deviation; Standardisation; Algebra.