How this calculator works
The mean locates the centre and the standard deviation sets the scale. Standardising a value gives its z-score. Cumulative probability is area to its left; interval probability is the area between two bounds. Point density has inverse measurement units and is not the probability of one exact value.
Formula
z = (x - μ)/σ; f(x) = exp(-z²/2)/(σ√(2π)); F(x) = Φ(z); x_p = μ + σΦ⁻¹(p).
Worked example
With mean 100 and SD 15, x = 100 has cumulative probability 50%. The interval 85 to 115 contains about 68.2689%.
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Frequently asked questions
Can a density exceed one?
Yes. Density is area per unit of x. Only an integrated probability must lie between zero and one.
Is the model appropriate for any data?
No. Strong skew, truncation or several clusters can make a normal model misleading.
What is the probability at one exact value?
It is zero for this continuous model. Use an interval to describe measurement tolerance.