# Normal Distribution Calculator

Cumulative distribution function (CDF), Percentile, Probability between two values and Probability density function (PDF)

The normal distribution calculator computes the cumulative distribution function (CDF): p or the percentile: 𝑥₁.
One is the normal CDF calculator and the other is the inverse normal distribution calculator

Choose 𝑥1 to calculate the cumulative probability based on the percentile, p(X ≤ 𝑥1) to calculate the percentile based on the cumulative probability, 𝑥1, 𝑥2 to calculate p(𝑥1 ≤ X ≤ 𝑥2) or p(X ≤ 𝑥1), p(X ≤ 𝑥2) to calculate x1, x2, p(𝑥1 ≤ X ≤ 𝑥2)

### What is the normal distribution?

The normal distribution (also known as the Gaussian distribution), is the most widely used in statistical analyses.
This is generally because many natural processes are naturally distributed or have a very similar spread.
Some examples of normally distributed data include height, weight, and error in measurements.

The Normal distribution has a symmetric "Bell Curve" structure. more data exist around the center, which is the average, and as further the value is from the center the less likely it occurs.

Usually, when adding independent random variables, the result tends toward the normal distribution (CLT - The Central Limit Theorem)
You can calculate the values of any normal distribution based on the standard normal distribution (a normal distribution with mean equals zero and standard deviation equals one)
when X distributes normally, μ mean and σ standard deviation, Z=(x-μ)/σ distributes as the standard normal distribution, so you can calculate any normal distribution based on the standard normal distribution.

### When to use the normal distribution

1. Normal population
• Bell shape histogram
• Approximate line in the Q-Q plot.
• Pass the normality test, like QQ plot
2. Average (or sum)
Even if the population distribution is not normal, the central limit theorem states that the average's distribution is approximately normal if the data meets the following criteria:
• Reasonably symmetrical histogram
• The sample size is 30, or bigger

### Normal distribution formula

##### Normal probability density function formula (PDF)
 f(𝑥) = 1 exp(- (𝑥 - μ)2 ) σ√(2π) 2σ2
##### Normal cumulative distribution function formula (CDF)
 1 [1 + erf( x - μ )] 2 σ√2
Z - Standard distribution statistic - normal distribution with μ=0 and σ=1.
 Z = 𝑥 - μ σ