The **standard normal distribution** is a special case of the **normal distribution**. It is a probability distribution that has a **mean (μ)** of 0 and a **standard deviation (σ)** of 1. The standard normal distribution is denoted as **N(0, 1)**, and its graph is symmetric, bell-shaped, and centered around 0.

Key characteristics of the standard normal distribution:

1. The **mean (μ)** is 0.
2. The **standard deviation (σ)** is 1.
3. The total area under the curve equals 1, representing the total probability of all outcomes.
4. The values on the x-axis represent **z-scores**, which indicate how many standard deviations a point is from the mean.
5. The **empirical rule** applies: 
   - Approximately **68%** of the data falls within ±1 standard deviation of the mean.
   - Approximately **95%** falls within ±2 standard deviations.
   - About **99.7%** falls within ±3 standard deviations.

### Standard Normal Distribution Formula:
The probability density function (PDF) of a standard normal distribution is given by the formula:
\[
f(x) = \frac{1}{\sqrt{2\pi}} e^{-\frac{x^2}{2}}
\]
Where:
- \( x \) is the value on the horizontal axis (z-score),
- \( e \) is Euler's number (approximately 2.718).

This distribution is commonly used in statistics to compute probabilities and is essential in **hypothesis testing** and the creation of **confidence intervals**.

Do you want more details or have any questions about this?

### Related Questions:
1. What is the relationship between z-scores and the standard normal distribution?
2. How do you calculate probabilities using the standard normal distribution?
3. What is the empirical rule, and how does it apply to normal distributions?
4. How does the standard normal distribution differ from other normal distributions?
5. What is the significance of the area under the curve in a standard normal distribution?

### Tip:
You can convert any normal distribution to a standard normal distribution using the z-score formula:
\[
z = \frac{x - \mu}{\sigma}
\]
where \(x\) is a value from the original distribution, \( \mu \) is the mean, and \( \sigma \) is the standard deviation.

Learn about the standard normal distribution, its characteristics, and how z-scores are used to calculate probabilities. This article covers key concepts in statistics, including the empirical rule and probability distribution functions. Suitable for Grades 11-12 students.

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