20.3. Standard Normal Distribution

The Standard Normal Distribution is a special case of the normal distribution with:

• Mean 𝜇 = 0
• Standard deviation 𝜎 = 1

The Standard Normal Distribution is a specialized version of the normal distribution. In this case, the mean (𝜇) is set to 0, and the standard deviation (𝜎) is set to 1. This standardization allows for easier comparisons and calculations involving normally distributed data, regardless of the original mean and standard deviation.

Think of the Standard Normal Distribution like converting temperatures from Celsius to Fahrenheit. Just as Celsius and Fahrenheit measure temperature differently but refer to the same temperature values, the Standard Normal Distribution allows us to convert different normal distributions into a uniform scale where we can analyze and compare them effectively.

To convert a normal distribution to a standard normal distribution, we use the Z-score formula:

$Z = \frac{X - \mu}{\sigma}$

Where:

• 𝑋 = any value from the original normal distribution
• 𝑍 = corresponding value in the standard normal distribution

The Z-score is a method that helps us standardize a value from any normal distribution. By subtracting the mean (𝜇) from the value (𝑋) and dividing by the standard deviation (𝜎), we get the Z-score (𝑍). This Z-score tells us how many standard deviations the value is from the mean. A Z-score of 0 indicates that the value is exactly at the mean, while a positive Z-score means it is above the mean and a negative Z-score means it is below.

Imagine you are measuring the height of students in a class. If you know that the average height is 160 cm, the Z-score can tell you how tall each student is in comparison to this average. If a student is 170 cm tall, the Z-score will show that this student is taller than average. It’s like using a common ruler to see how each height compares to the average height of the class.

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