Normal Distribution CDF for 2 Values Calculator

This CDF calculator computes the cumulative probability P(x1 <= X <= x2) 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 as a function of any of the paramaters.

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

Example 1

A financial analyst is evaluating the risk of a new loan product with a default rate normally distributed, based on past data showing a 5% average rate and 2% standard deviation. They aim to calculate the chance that the default rate stays within the acceptable 3% to 7% range, to assess the financial stability and risk of introducing this product, ensuring it aligns with the bank's risk tolerance.

Value	Keystrokes	Display	Description
3	x1	5.00	Stores the x1 value.
7	x2	7.00	Stores the x2 value.
5	μ	5.00	Stores the μ (mean) value.
2	σ	2.00	Stores the sd (standard deviation) value.
	P(x₁₂)	0.683	Calculates the cdf between x1 and x2.

The History Detail

Normal Distribution CDF for 2 Values
           3.00 : The x1 value.
           7.00 : The x2 value.
           5.00 : The mean of the distribution.
           2.00 : The standard deviation of the distribution.
          0.683 : The normal cumulative distribution function, two values P(x1≤X≤x2).
Result:	 0.683

Example 2

Building on the prior analysis, the analyst plans to modify the loan's risk profile so 80% of loans have a 3% to 7% default rate. With a 5% mean default rate, the focus is on finding a standard deviation adjustment to meet this target, ensuring 80% of loans stay within the acceptable risk range.

These keystrokes assume the values from example 1.

Value	Keystrokes	Display	Description
.8	P(x₁₂)	0.800	Stores the cdf value.
	σ	1.56	Calculates the σ value.

The History Detail

Normal Distribution CDF for 2 Values
Solving for σ, standard deviation of the distribution, with initial value (guess) of 2.00
           3.00 : The x1 value.
           7.00 : The x2 value.
           5.00 : The mean of the distribution.
           1.56 : The standard deviation of the distribution.
          0.800 : The normal cumulative distribution function, two values P(x1≤X≤x2).
Result:	 1.56

Reference:

Normal distribution - Wikipedia