Normal Distribution Calculator
Normal Distribution Calculator helps to compute the cumulative probability of a value being lower or higher than a given data point. Normal distribution is also known as Gaussian distribution. It is the most significant continuous probability distribution.
What is the Normal Distribution Calculator?
Normal Distribution Calculator is an online tool that determines the probability of a value being higher or lower than a given data point x. A probability bell curve is used to depict a normal distribution. To use the normal distribution calculator, enter the values in the given input boxes.
Normal Distribution Calculator
NOTE: Please enter the values up to three digits only.
How to Use Normal Distribution Calculator?
Please follow the steps below to find the probability of a value being higher or lower than a given data point using the normal distribution calculator.
- Step 1: Go to Cuemath's online normal distribution calculator.
- Step 2: Enter the mean, standard deviation, and data point in the input boxes.
- Step 3: Click on the "Calculate" button to find the probability.
- Step 4: Click on the "Reset" button to clear the fields and enter new values.
How Does Normal Distribution Calculator Work?
There are two parameters that are needed when we calculate the normal distribution. These are the mean and the standard deviation. The Normal distribution is symmetric about the mean. The steps given below are used to calculate the area under the bell curve to establish the probability of a value being higher or lower than the random variable x.
- Step 1: We first calculate the Z score. This is given by Z = (x - u)/sd. Here, u is the mean and sd is the standard deviation of the given data.
- Step 2: Now we use the normal distribution table to determine the value of φ (Z) .This will represent P(X < x). This is because the normal distribution table represents the area under the bell curve to the left of the Z score.
- Step 3: To determine P (X > x) we subtract the value of P(X < x) from 1.
Probability of a value being lower than x is given by
P (X < x) = φ (Z) = φ ((x - u)/sd)
Probability of a value being higher than x is given by
P (X > x) = 1 - P(X < x)
P (X > x) = 1 - φ (Z) = 1 - φ ((x - u)/sd)
Solved Examples on Normal Distribution Calculator
Example 1:
Find P(X < 5) and P(X > 5) when we are given the mean = 4 and the standard deviation = 2 of the normal distribution. Verify it using the normal distribution calculator.
Solution:
x = 5, u = 4, sd = 2
For P (X < 5),
P(X < x) = φ ((x - u)/sd)
Using the normal distribution table
P (X < 5) = φ ((5 - 4)/2) = 0.06915
For P (X > 5),
P (X > x) = 1 - P(X < x)
P (X > x) = 1 - φ ((x - u)/sd)
P (X > 5) = 1 - φ ((5 - 4)/2)
P (X > 5) = 1 - φ (0.5)
Using the normal distribution table
P (X > 5) = 1 - 0.6915 = 0.3085.
Example 2:
Find P(X > 12) when we are given the mean = 8.2 and the standard deviation = 3.1 of the normal distribution. Verify it using the normal distribution calculator.
Solution:
x = 12, u = 8.2, sd = 3.1
For P (X < 12),
P(X < x) = φ ((x - u)/sd)
Using the normal distribution table
P (X < 12) = φ ((12 - 8.2)/3.1) = 0.8907
For P(X > 12),
P (X > x) = 1 - P(X < x)
P (X > x) = 1 - φ ((x - u)/sd)
P (X > 12) = 1 - φ ((12 - 8.2)/3.1)
P (X > 12) = 1 - φ (1.23)
Using the normal distribution table
P (X > 12) = 1 - 0.8907 = 0.1093
Similarly find P(X > x) and P(X < x)for the following values:
-
x = 26, mean = 12, standard deviation = 18
- x = 30.2, mean = 27.8, standard deviation = 5.7