Normal Distribution Calculator
Compute PDF, CDF, and inverse CDF for the normal (Gaussian) distribution with given μ and σ.
Inputs
P(X ≤ x)
0.97500217
PDF f(x)
0.05844094
P(X > x)
0.02499783
Mean
0.0000
Variance
1.0000
Step by step
Values used
Mean (μ) = 0; Standard Deviation (σ) = 1; x value = 1.96
PDF
f(x) = (1/(σ√(2π))) × exp(−(x−μ)²/(2σ²))
Standard Z
z = (x − μ) / σ
P(X ≤ x)
= 0.97500217
PDF f(x)
= 0.05844094
P(X > x)
= 0.02499783
Mean
= 0.0000
Variance
= 1.0000
How it works
The normal (Gaussian) distribution is the most important continuous distribution in statistics. It is completely defined by its mean μ and standard deviation σ. About 68% of values lie within 1σ of the mean, 95% within 2σ, and 99.7% within 3σ.
Formulas
f(x) = (1/(σ√(2π))) × exp(−(x−μ)²/(2σ²))
- μ
- Mean
- σ
- Standard deviation
- x
- Point of evaluation
Standard Z
z = (x − μ) / σ
- x
- Value
- μ
- Mean
- σ
- Standard deviation
Frequently Asked Questions
What is the standard normal distribution?
It is the normal distribution with μ = 0 and σ = 1. Any normal variable can be converted to standard normal using z = (x − μ)/σ.
Why is the normal distribution so important?
The Central Limit Theorem states that the sum (or mean) of many independent random variables tends toward a normal distribution regardless of the underlying distribution.
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