Normal Distribution & Z-Score Explorer

Set the mean and standard deviation to see the bell curve update in real time. Choose a probability mode to shade the area under the curve, convert between raw scores and z-scores, and follow step-by-step calculations. All probabilities are computed using numerical integration with no external libraries.

Bell Curve

Parameters

Mean (μ)

Standard deviation (σ)

Probability mode

Value a

Z-Score Calculator

Raw score (x)

Presets

Probability

P(X < 1)

84.1345%

Z-score for a = 1

Z-Score Conversion

Z-score for x = 1

Step-by-Step Calculation

1. Z-score formula

z = \frac{x - \mu}{\sigma}

2. Z-score for a = 1

z_a = \frac{1 - 0}{1} = 1

3. Probability

P(X < 1) = \Phi(1) = 84.1345\%

4. Z-score conversion

z = \frac{1 - 0}{1} = 1

Reference Guide

The Normal Distribution

The normal (Gaussian) distribution is the most important probability distribution in statistics. Its bell-shaped curve is defined by two parameters.

f(x) = \frac{1}{\sigma\sqrt{2\pi}} \, e^{-\frac{(x-\mu)^2}{2\sigma^2}}

$\mu$ (mu) is the mean, which determines where the peak sits. $\sigma$ (sigma) is the standard deviation, which controls how spread out the curve is.

The Z-Score

A z-score tells you how many standard deviations a value is from the mean. It standardizes any normal distribution to the standard normal (mean 0, SD 1).

z = \frac{x - \mu}{\sigma}

A z-score of 2 means the value is 2 standard deviations above the mean. A z-score of -1.5 means 1.5 standard deviations below. This lets you compare values from different distributions.

The 68-95-99.7 Rule

For any normal distribution, the percentage of data within 1, 2, and 3 standard deviations of the mean follows a fixed pattern.

Within ±1σ68.27%

Within ±2σ95.45%

Within ±3σ99.73%

This means that about 95% of values fall within two standard deviations of the mean. Values beyond ±3σ are extremely rare.

Cumulative Probability

The cumulative distribution function (CDF) gives $P(X \le x)$ , the probability of observing a value at or below $x$ . It is the area under the bell curve from negative infinity to $x$ .

\Phi(z) = P(Z \le z) = \int_{-\infty}^{z} \frac{1}{\sqrt{2\pi}} e^{-t^2/2} \, dt

For the standard normal, $\Phi(0) = 0.5$ , $\Phi(1.96) \approx 0.975$ , and $\Phi(-1.96) \approx 0.025$ .