The normal or Gauss distribution (after Carl Friedrich Gauss) is an important type of continuous probability distribution in stochastics. Its probability density function is also called the Gaussian function, Gaussian normal distribution, Gaussian distribution curve, Gaussian curve, Gaussian bell curve, Gaussian bell function, Gaussian bell or simply bell curve.

The normal or Gauss distribution is defined as:

$f\left(x\right)=\frac{1}{\sigma \sqrt{2\pi}}\phantom{\rule{0.3em}{0ex}}{e}^{-\frac{1}{2}\frac{{\left(x-\mu \right)}^{2}}{{\sigma}^{2}}}$

The graph of this density function has a "bell-shaped" form and is symmetrical around parameter μ as centre of symmetry, which also represents the expected value, the median and the mode of the distribution.

Using the sliders in the lower part of the graph, the parameters of the Gauss distribution can be varied. The adjustable parameter range can be specified in the numeric fields. The red points on the bell curve can be moved. The integral of the bell curve is calculated for the range between the points. As the total area of the Gauss distribution is normalized to one, the integral corresponds to the area fraction. This means, for example, if the points are set to ±σ, the area is 0.68 or 68% of the total area.

μ and σ are the parameters of the normal distribution. In μ is the center of the distribution and the bell curve takes its maximum there. The inflection points of the function are located at a distance ±σ from the center of symmetry.

For random variables that are normally distributed, the following applies:

- In the interval of the deviation ±σ from the mean value μ 68.27 % of all measured values are to be found
- In the interval of the deviation ±2σ from the mean value μ 95.45 % of all measured values can be found
- In the interval of the deviation ±3σ from the mean value μ 99.73 % of all measured values can be found

- 50 % of all measured values have a deviation of no more than 0.675σ from the mean value μ
- 90 % of all measured values have a deviation of no more than 1.645σ from the mean value μ
- 95 % of all measured values have a deviation of no more than 1.960σ from the mean value μ
- 99 % of all measured values have a deviation of no more than 2.576σ from the mean value μ

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The curve fitting of the Gaussian distribution to the measured values is done by calculation of the weighted average of the measured values. The weighted average corresponds to the μ in the Gaussian distribution. The standard deviation of the measured values from the mean μ is the σ in the normal distribution formula.

$$\mu =\frac{\sum _{i=1}^{n}{x}_{i}{y}_{i}}{\sum _{i=1}^{n}{y}_{i}}$$

$$\sigma =\sqrt{\frac{\sum _{i=1}^{n}{\left({x}_{i}-\mu \right)}^{2}{y}_{i}}{\sum _{i=1}^{n}{y}_{i}}}$$

The displayed bell curve is the fitted Gaussian distribution multiplied by the area A of the measured values.

$f\left(x\right)=\frac{A}{\sigma \sqrt{2\pi}}\phantom{\rule{0.3em}{0ex}}{e}^{-\frac{1}{2}\frac{{\left(x-\mu \right)}^{2}}{{\sigma}^{2}}}$

The area A is calculated by the trapeze formula.

$$A=\sum _{i=1}^{n-1}\frac{\left({x}_{i+1}-{x}_{i}\right)\left({y}_{i+1}+{y}_{i}\right)}{2}$$

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Here is a list of of further useful sites:

Index Circle Triangle Mean Value Calculator Trigonometry Curve fit calculator Damped Vibration Plot Sine (sin) Plot Cosine (cos) Plot Tangent (tan) Plot Beat frequencies Plot Normal distribution at Wikipedia