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STATISTICS • PROBABILITY DISTRIBUTIONS
Normal Distribution
Connect raw values, z-scores, density and cumulative probability while comparing two distributions and solving directly for a target percentile.
● LIVE PROBABILITY CURVEz-score and percentile
z-score—
Density—
Percentile—
Reference percentile—
Variance—
Target error—
Read density and probability separately
The bell curve shows density, while the shaded area and CDF graph show accumulated probability. The same x marker links raw units, standardized z-score and percentile.
Fast investigations
- Keep x fixed and widen σ to see density flatten.
- Shift μ while tracking the same raw x.
- Compare two distributions at one observed value.
- Use the percentile challenge to find x for common percentiles.
LEARN WITH THE SIMULATION
Understand Normal Distribution by experimenting.
START HERE
The normal distribution is a continuous probability model shaped by its mean and standard deviation. Standardizing a value into a z-score lets you compare positions across differently centered or scaled normal distributions.
BUILD THE IDEA
What you should understand first
- See how μ shifts the curve and σ changes spread.
- Convert an observed value into a z-score.
- Connect cumulative probability to percentiles.
TRY IT YOURSELF
What happens to the peak when standard deviation increases?
- Keep the mean fixed.
- Increase σ gradually.
- Compare both the width and maximum height of the density curve.
Read the visual:
As σ increases, the curve spreads out and its peak falls so the total probability remains 1.
WHY IT WORKS 01
See how μ shifts the curve and σ changes spread.
The normal distribution is a continuous probability model shaped by its mean and standard deviation. Standardizing a value into a z-score lets you compare positions across differently centered or scaled normal distributions. Focus on the relationship between the controlled variable and the measured response rather than treating the visual change as the explanation.
WHY IT WORKS 02
Convert an observed value into a z-score.
As σ increases, the curve spreads out and its peak falls so the total probability remains 1. Use an extreme or boundary setting after the baseline trial to check whether the same reasoning still holds.
DO NOT MIX THESE UP
Common misconception
Probability density is not the same as probability at a single point. A z-score is measured in standard deviations, not raw units. Normal models are approximations and are not appropriate for every dataset.
PRACTICE CHALLENGE
What happens to the peak when standard deviation increases?
- Keep the mean fixed.
- Increase σ gradually.
- Compare both the width and maximum height of the density curve.
Reveal the reasoning
As σ increases, the curve spreads out and its peak falls so the total probability remains 1. Probability density is not the same as probability at a single point. A z-score is measured in standard deviations, not raw units. Normal models are approximations and are not appropriate for every dataset.
MODEL BOUNDARIES
What this simulation assumes
Every simulation simplifies reality or a mathematical system. These are the assumptions and limits to keep in mind while interpreting the results.
Model assumptions
- The distribution is an ideal continuous Gaussian model defined by a mean and standard deviation.
- Probability is represented by area under the theoretical density curve.
- The displayed domain truncates tails that mathematically extend to infinity.
Model limitations
- Real data need not be normally distributed.
- Extreme-tail probabilities may be visually negligible even though they are mathematically nonzero.
CONNECT IT
Where this matters
- Measurement error
- Standardized testing
- Statistical modeling
WORDS THAT MATTER
Key vocabulary
- Mean
- Center parameter of the normal distribution.
- Standard deviation
- Scale parameter controlling spread.
- z-score
- Standardized distance from the mean.
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