Gaussian (Normal) Distribution Calculator

What is a Gaussian (Normal) Distribution?

The Gaussian or Normal distribution is a continuous probability distribution that is symmetrical about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean. The graph of the normal distribution depends on two factors: the mean and the standard deviation.

Formulas and Their Meanings

1. Probability Density Function (PDF): \[f(x) = \frac{1}{\sigma\sqrt{2\pi}} e^{-\frac{1}{2}(\frac{x-\mu}{\sigma})^2}\] Where \(\mu\) is the mean and \(\sigma\) is the standard deviation.

2. Cumulative Distribution Function (CDF): \[F(x) = P(X \leq x) = \int_{-\infty}^x f(t) dt = \frac{1}{2}[1 + erf(\frac{x-\mu}{\sigma\sqrt{2}})]\] This gives the probability that a value falls below x.

3. Z-score: \[Z = \frac{X - \mu}{\sigma}\] This standardizes a normal distribution to have a mean of 0 and a standard deviation of 1.

Calculation Steps

1. Calculate the Z-score for the given x-value.
2. Use a standard normal table or a computational method to find the area under the curve up to the Z-score.
3. This area represents the probability P(X < x).
4. For P(X > x), subtract P(X < x) from 1.
5. For P(x1 < X < x2), calculate P(X < x2) - P(X < x1).

Example Calculation

Let's calculate probabilities for a normal distribution with μ = 10 and σ = 2:

1. For x = 12: \[Z = \frac{12 - 10}{2} = 1\] \[P(X < 12) \approx 0.8413\]
2. For x = 8: \[Z = \frac{8 - 10}{2} = -1\] \[P(X < 8) \approx 0.1587\]
3. P(8 < X < 12) = P(X < 12) - P(X < 8) ≈ 0.8413 - 0.1587 = 0.6826

Visual Representation

This graph illustrates a standard normal distribution. The red dashed line indicates the mean (μ), and the green dashed lines show one standard deviation (σ) on either side of the mean.