Standard Normal Distribution Calculator

Formula

To calculate the z-score in a standard normal distribution:

\[ z = \frac{X - \mu}{\sigma} \]

Where:

\( z \) is the z-score (standard normal distribution)
\( X \) is the normal random variable
\( \mu \) is the mean of the distribution
\( \sigma \) is the standard deviation of the distribution

Standard Normal Distribution Definition

The standard normal distribution is a normal distribution with a mean of 0 and a standard deviation of 1. It is used to determine the probability of a random variable falling within a certain range under the standard normal curve. The z-score represents the number of standard deviations a data point is from the mean, allowing for the comparison of different datasets on a standardized scale.

Example Calculation 1

Let's assume the following values:

Normal Random Variable (X) = 75
Mean (μ) = 70
Standard Deviation (σ) = 5

Using the formula:

\[ z = \frac{75 - 70}{5} = 1.0000 \]

The standard normal distribution (z-score) is 1.0000.

Example Calculation 2

Let's assume the following values:

Normal Random Variable (X) = 50
Mean (μ) = 45
Standard Deviation (σ) = 10

Using the formula:

\[ z = \frac{50 - 45}{10} = 0.5000 \]

The standard normal distribution (z-score) is 0.5000.