Gaussian Distribution as function plot f(x) = 1/(√2π⋅σ) ⋅ e-(x-μ)2/(2σ2)
By means of the slider in the lower region of the graph, the parameters of the Gaussian distribution can be varied. The parameter range of the slider can be adjusted in the numeric field 'Range of the parameter'. The red dots on the bell curve can be shifted. For the area between the points, the integral of the bell curve is calculated. Since the total area of the Gaussian distribution is normalized to one so corresponds the integral to the area. That e.g. if the points be placed on ±σ then is is the area 0.68 or 68% of the total. With the 'lines Style' can be switched on the display of the derivative.
μ and σ are the parameters of the normal distribution. In μ is the center of the distribution and the bell curve, where it holds at its maximum. In the distance ±σ from symetric center are the turning points of the function.
For random variables are normally distributed applies:
In the interval the deviation ±σ from the mean μ are 68.27% of all measurements to find
In the interval the deviation ±2σ from the mean μ are 95.45% of all measurements to find
In the interval the deviation ±3σ from the mean μ are 99.73% of all measurements to find
50% of all measured values have a deviation of 0.675 σ from mean μ
90% of all measured values have a deviation of 1.645 σ from mean μ
95% of all measured values have a deviation of 1.960 σ from mean μ
99% of all measured values have a deviation of more than 2,576 σ from mean μ
Fitting of the Gaussian distribution to measured values
The 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.