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Probability calculations in JavaScript, with code listing

This JavaScript library is for probability calculations.

It currently contains functions for calculating the Probability Density Function on any normal curve and for calculating areas under the Standard Normal Curve.

There are interactive calculators for doing the calculations on this web page.

Wikipedia has an article about normal distributions.

You can copy the code below, or download (Save) the file (3 KB).

There is a separate page with calculators for basic statistics such as mean and standard deviation.

Normal Probability Density Function and standard score (z-score) calculator

Calculates Z(x), which is the Y-axis value (height) of an X-axis point x on any normal curve. This is known as the Probability Density Function, and in Excel 2003 (NORMDIST Help) as the Probability Mass Function.

It also calculates the z-score (standard score) of x, the number of standard deviations that x is above or below the Mean. In spite of its similar-looking name, the z-score is a completely different calculation from Z(x). It is included in this calculator because the required inputs are exactly the same.

z-score example:

Given a distribution with Mean = 100 and Standard Deviation = 20,
An x value of 80 has a z-score of -1 because it is 1 SD below the mean.
An x value of 110 has a z-score of 0.5 because it is 0.5 SD above the mean.

Instructions:

Enter the known values into the first three boxes below. Allowed characters are .-+0123456789eE.
All others will be converted to spaces.
Click Calculate.

	x	Mean	Standard Deviation

z-score = (x - Mean) / StdDev
Z(x)

Standard Normal Curve areas P(x), Q(x), A(x)

Calculates areas under the Standard Normal Curve for a z-score, X₀.

If a second z-score, X₁, is provided, it also calculates the area between X₀ and X₁.

Z(x) = Probability Density Function, same calculation as above, but only for the Standard Normal Curve.
P(x) = Area under the curve to the left of x.
Q(x) = Area under the curve to the right of x.
A(x) = Area under the curve between -x and +x.
A(x₀->x₁) = Area under the curve between x₀ and x₁.

Standard Normal Curve characteristics: Mean = 0; Standard Deviation = 1; Total Area = 1; P(x) + Q(x) = 1.

Instructions:

Enter the score X₀ and optionally X₁ into the top boxes below. Allowed characters are .-+0123456789eE.
All others will be converted to spaces.
Click Calculate.

	X₀	X₁

Z(x)
P(x)
Q(x)
A(x)
A(x₀->x₁)

Probability JavaScript

 
/*	probability.js		12-25-2008		JavaScript
	Copyright (C)2008 Steven Whitney.
	Initially published by http://25yearsofprogramming.com.

	This program is free software; you can redistribute it and/or
	modify it under the terms of the GNU General Public License (GPL)
	Version 3 as published by the Free Software Foundation.
	This program is distributed in the hope that it will be useful,
	but WITHOUT ANY WARRANTY; without even the implied warranty of
	MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
	GNU General Public License for more details.
	You should have received a copy of the GNU General Public License
	along with this program; if not, write to the Free Software
	Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA  02110-1301, USA.

Functions related to probability calculations.
	
*/
//----------------------------------------------------------------------------------------------
// Calculates a point Z(x), the Probability Density Function, on any normal curve. 
// This is the height of the point ON the normal curve.
// For values on the Standard Normal Curve, call with Mean = 0, StdDev = 1.
function NormalDensityZx(x, Mean, StdDev)
{
	var a = x - Mean;
	return Math.exp(-(a * a) / (2 * StdDev * StdDev)) / (Math.sqrt(2 * Math.PI) * StdDev); 
}
//----------------------------------------------------------------------------------------------
// Calculates Q(x), the right tail area under the Standard Normal Curve. 
function StandardNormalQx(x)
{
	if(x === 0)	// no approximation necessary for 0
		return 0.50;
		
	var t1,t2,t3,t4,t5,qx;
	var negative = false;
	if(x < 0)
	{
		x = -x;
		negative = true;
	}
	t1 = 1 / (1 + (0.2316419 * x)); 
	t2 = t1 * t1; 
	t3 = t2 * t1; 
	t4 = t3 * t1; 
	t5 = t4 * t1;
	qx = NormalDensityZx(x,0,1) * ((0.319381530 * t1) + (-0.356563782 * t2) + 
								(1.781477937 * t3) + (-1.821255978 * t4) + (1.330274429 * t5));
	if(negative == true)
		qx = 1 - qx;
	return qx;
}
//----------------------------------------------------------------------------------------------
// Calculates P(x), the left tail area under the Standard Normal Curve, which is 1 - Q(x). 
function StandardNormalPx(x)
{
	return 1 - StandardNormalQx(x);
}
//----------------------------------------------------------------------------------------------
// Calculates A(x), the area under the Standard Normal Curve between +x and -x. 
function StandardNormalAx(x)
{
	return 1 - (2 * StandardNormalQx(Math.abs(x)));
}
//----------------------------------------------------------------------------------------------