Normal Distribution CDF Calculator

This CDF calculator computes the cumulative probability P(X≤x) for a given normal distribution and also offers features including inverse CDF calculations, and solving for the mean (μ) and standard deviation (σ). It can also create a plot of P(X≤x) as a function of any of the paramaters.

See the Statistical Distributions Overview page for more statistical distribution calculators.

Example 1

An investment has returns that are normally distributed with an average return of 5% and a standard deviation of 2%. An investor wants to understand the probability of the return being 4% or less.

Value	Keystrokes	Display	Description
4	x	4.00	Stores the x value.
5	μ	5.00	Stores the μ (mean) value.
2	σ	2.00	Stores the σ (standard deviation) value.
	P(X≤x)	0.309	Calculates the cumulative distribution function.

The History Detail

Normal Distribution CDF
           4.00 : The x value.
           5.00 : The mean of the distribution.
           2.00 : The standard deviation of the distribution.
          0.309 : The normal cumulative distribution function P(X≤x).
Result:	 0.309

Example 2, Inverse

To assess risk, the investor seeks the worst expected return in the bottom 5% of cases.

These keystrokes assume the values from example 1.

Value	Keystrokes	Display	Description
.05	P(X≤x)	0.05	Stores the cdf value.
	x	1.71	Solves for the x value.

The History Detail

Normal Distribution CDF
Solving for x, x value, with initial value (guess) of 4.00
           1.71 : The x value.
           5.00 : The mean of the distribution.
           2.00 : The standard deviation of the distribution.
         0.0500 : The normal cumulative distribution function P(X≤x).
Result:	 1.71

Example 3, Solving for Standard Deviation with Known Mean.

Besides CDF and inverse this calculator can also solve for mean or standard deviation.

With the goal of ensuring that the portfolio's annual return only falls below 4% 20% of the time, and knowing the average return is aimed to be 6%, determine the maximum allowable volatility (standard deviation) of the portfolio.

Value	Keystrokes	Display	Description
.2	P(X≤x)	0.200	Stores the cdf value.
4	x	4.00	Stores the x value.
6	μ	6.00	Stores the μ (mean) value.
	σ	2.38	Solves for the σ (standard deviation) value.

The History Detail

Normal Distribution CDF
Solving for σ, standard deviation of the distribution, with initial value (guess) of 2.00
           4.00 : The x value.
           6.00 : The mean of the distribution.
           2.38 : The standard deviation of the distribution.
          0.200 : The normal cumulative distribution function P(X≤x).
Result:	 2.38

Reference:

Normal distribution - Wikipedia