Normal Distribution Probability Calculator

The Normal distribution, also known as Gaussian distribution, is a theoretical continuous frequency distribution represented by a bell-shaped curve symmetrical about the mean as shown in the diagram below. This distribution is inarguably the most important and the most frequently used distribution in both the theory and application of statistics.

The Normal distribution, also known as Gaussian distribution, is a theoretical continuous frequency distribution represented by a bell-shaped curve symmetrical about the mean as shown in the diagram below. This distribution is inarguably the most important and the most frequently used distribution in both the theory and application of statistics.

The Normal distribution, also known as Gaussian distribution, is a theoretical continuous frequency distribution represented by a bell-shaped curve symmetrical about the mean as shown in the diagram below. This distribution is inarguably the most important and the most frequently used distribution in both the theory and application of statistics.

Probability Distribution Function of Normal Distribution Calculator

The probability distribution function (pdf) for the normal distribution is given by the normal equation mentioned below and the total area bounded by the normal curve is equal to 1.

[Normal Equation]
Mathematical and Graphical Representation of Probabilities Calculator

Three types of probabilities are possible here.
Type 1 Probability that X is less than x1, ie, P(X Type 2 Probability that X is greater than x2, ie, P(X > x2)
Type 3 Probability that X is greater than x1 and less than x2 , ie, P(x1 A Simple Three-step Procedure for Evaluation of Probabilities Calculator

Steps 1 and 2 will be common for all the three types mentioned above. Step3, however, will be different.

Step 1: Find the z-scores corresponding to x = x1 and x= x2 using the formula

.

Let z1 and z2 be the z-scores for x1 and x2 respectively.

Step 2: Discard the negative sign, if any, of z1 and z2 for now and find the probabilities Q1 and Q2 corresponding to |z1| and |z2| by looking up in the Standard Normal Distribution table.

Step 3: The required probability Q can be evaluated from Q1 and Q2 by using the appropriate formula for each type.

For Type 1 probabilities, z2 and Q2 are not applicable. The required probability Q will depend on the sign of z1 as mentioned below:
Condition Required Probability Q = P(X z1 z1 > 0 Q = 0.5 + Q1

For Type 2 probabilities, the required probability Q will depend on the sign of z2 as mentioned below:
Condition Required Probability Q = P(x1 z1 z1 0 Q = Q1 + Q2
z1 > 0 and z2 > 0 Q = Q2 – Q1

For Type 3 probabilities, z1 and Q1 are not applicable. The required probability Q will depend on the sign of z2 as mentioned below:
Condition Required Probability Q = P(X > x2)
z1 z1 > 0 Q = 0.5 – Q2