Standard Normal Distribution

The standard normal distribution is a probability distribution that has a mean of zero and a standard deviation of one. It is used as a benchmark to compare other probability distributions.

Properties
- The standard normal distribution is a continuous probability distribution.
- The total area under the curve of the standard normal distribution is equal to one.
- The standard normal distribution is symmetric about the mean, which is zero.
- The standard deviation of the standard normal distribution is one.
- The standard normal distribution is also known as the Gaussian distribution or the bell curve.

Formula
The formula to calculate the standard normal distribution is:

z = (x - μ) / σ

Where:
- z is the standard score or standard deviation from the mean
- x is the raw score or value
- μ is the mean
- σ is the standard deviation

Uses
- The standard normal distribution is used in hypothesis testing and statistical inference because it is easier to calculate probabilities using this distribution.
- It is also used in quality control to determine if a process is within specification limits.
- The standard normal distribution is used in finance to calculate the probability of a certain rate of return or loss.

Example
Suppose a company wants to know the probability of a machine producing a certain part within a specified tolerance. The mean and standard deviation of the machine's output are known. The company can use the standard normal distribution to calculate the probability of the part being within the tolerance.

Conclusion
The standard normal distribution is a probability distribution that has important applications in statistics, finance, and quality control. It is used as a benchmark to compare other probability distributions and is easier to calculate probabilities using this distribution.