Standard Normal Distribution

Standard Normal Distribution (Z-distribution) is a normal distribution with a mean (μ) of 0 and a standard deviation (σ) of 1. In other words, a standard normal distribution satisfies the following:

• Mean (μ) = 0
• Standard Deviation (σ) = 1
• Total area under the curve = 1

It is commonly denoted by Z∼N(0,1), where Z represents the standard normal random variable.

The probability density function (PDF) of the standard normal distribution is:

f(z)=\frac{1}{\sqrt{2\pi}}e^{\frac{-z^2}2}

Where:

• e is Euler's number (≈ 2.718)
• π ≈ 3.1416
• z is the standardized variable

This formula describes the shape of the standard normal curve.

Standard normal distribution is defined by the following characteristics:

• Symmetry: It is symmetric around the mean (μ = 0).
• Bell-Shaped Curve: The graph is bell-shaped, that means most values cluster around the mean (μ = 0).
• Total Area Under Curve: The total area under the curve is 1, representing the total probability.
• 68-95-99.7 Rule: Approximately 68% of data falls within 1 standard deviation of the mean, 95% within 2 standard deviations, and 99.7% within 3 standard deviations.
• Asymptotic: The tails of the distribution approach, but never touch, the horizontal axis.
• Unimodal: It has a single peak at the mean (μ = 0).
• Standard Scores (Z-Scores): Any normal distribution can be transformed into the standard normal distribution using z-scores where z = (x - μ)/σ.

Standard Normal Distribution Table

Standard Normal Distribution Z Table is the table of z-value for standard normal distribution where Z = [(x-μ)/σ]. The Standard Normal Distribution Z Table is given as follows:

Area of Standard Normal Distribution

The table given above is used to calculate the "Area of Standard Normal Distribution" curve. It is basically used to find the area from -∞ to Z. So we can say

F(Z) = Area Under Standard Normal Distribution from -∞ to Z

where,

• F stands for cumulative (or collected) area
• F(Z) is calculated using the table

We simply need to find the value corresponding the required Z - value and that is equal to F(Z).

• Case 1: If we need to calculate the area from -∞ to Z, then the area is calculated as F(Z).

For example, for Z < 1, area is -∞ to 1 and this is equal to F(1) found from table which is equal to 0.8413.

• Case 2: If we need to calculate the area from Z to +∞, then the area is calculated as 1 - F(Z).

For example,for Z > 1, area is 1 to +∞ and this is equal to 1-F(1) found from table which is equal to 1-0.8413=0.1587.

• Case 3: If we need to calculate the area from -Z to +Z, then the area is calculated as F(Z) - F(-Z) (where F(-Z) is calculated as 1-F(Z) due to symmetry ) or we can simply calculate 2F(Z) - 1.

For example, for -1<Z<1, area is -1 to +1 and this is equal to F(1) - F(-1) = 2F(1)-1 found from table which is equal to 2(0.8413)-1=0.6826.

Since standard normal distribution curve is symmetric about the mean, the area under the curve to the left of a negative z-value is the same as the area to the right of the corresponding positive z-value. To find probability, look up the area to the right of the corresponding positive z-value and subtract it from 1. Therefore F(-Z) = 1 - F(Z).

Applications

Standard Normal Distribution has a wide range of applications and usage in several fields. Here are some important applications:

• Hypothesis Testing: Performing Z-tests and constructing confidence intervals.
• Probability Calculations: Determining standard normal distribution density probabilities and areas under the curve.
• Data Analysis: Standardizing scores (z-scores) and making suitable analysis.
• Quality Control: Monitoring processes using control charts.
• Risk Management: Calculating financial risks like Value at Risk (VaR).
• Machine Learning: Normalizing features to improve algorithm performance.

➣Practice: Solved Examples