Standard Normal Distribution

Standard normal distribution, also known as the z-distribution, is a special type of normal distribution. In this distribution, the mean (average) is 0 and the standard deviation (a measure of spread) is 1. This creates a bell-shaped curve that is symmetrical around the mean.

In this article we have covered, Standard Normal Distribution definitions, examples, and others in detail

Before starting with Standard Normal Distribution let's first learn what is meant by Normal Distribution.

Normal Distribution Definition

Normal Distribution ( or normal random variable) is the most significant continuous probability distribution in the field of statistics. It can also be said as the parent topic to study Standard Normal Distribution. It has a bell-shaped curved graph that roughly characterizes a variety of events seen in science, business, and the natural world.

Now there comes another term which is normal random variable, which is a continuous random variable X with a bell-shaped distribution. The two parameters μ ( symbol for mean for population ) and σ ( symbol for standard deviation for population ) respectively, determine the mathematical formula for the probability distribution of that variable. Therefore, we represent the density values of X by n(x; μ, σ).

Density of normal random variable X, with mean (μ) and variance (σ²), is

n(x; μ, σ) = /√(2πσ), -∞ < x < ∞,

where,

• π = 3.14159...
• e = 2.71828...

Once σ and μ are given, the normal distribution curve can be easily interpreted.

What is Standard Normal Distribution?

Standard Normal Distribution, often referred to as the Z-Distribution, is a specific type of normal distribution that has a mean (μ) of 0 and a standard deviation (σ) of 1.

Let Z = [(x-μ)/σ], and μ = 0, σ = 1 then we can say n(x; μ, σ) = e^(-z2/2) / √(2π), -∞ < x < ∞ and this distribution of a normal random variable with mean 0 (μ = 0) and variance 1 (σ² = 1 ) is what we call as a standard normal distribution. This is the basic difference between Normal Distribution and Standard Normal Distribution.

Standard Normal Distribution Table

Standard Normal Distribution Table which is also called, 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:

The above table can be used to solve problems related to Standard Normal Distribution in statistical mathematics.

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

The image added below represents the area of Standard Normal Distribution

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.

Note: 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).

Standard Normal Distribution Function

Standard Normal Distribution Function is added below:

F(Z) = _-∞∫^Z / √(2π)

where,

• Z = [(x-μ)/σ]
• mean = μ=0
• Standard Deviation = σ=1

Note: "Standard Normal Distribution Table" is used to easily calculate F(Z).

Application of Standard Normal Distribution

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.

• Modeling Phenomena: Analyzing real-world data in natural and social sciences.

• Survey Analysis: Estimating and comparing population proportions.

• Psychometrics: Using standard scores in psychological assessments.

• Operations Research: Optimizing inventory management based on demand modeling.

Characteristics of Standard Normal Distribution

Standard normal distribution is defined by the following characteristics:

• Mean: The mean (average) is 0 that is symbolically represented as μ = 0.

• Standard Deviation: The standard deviation is 1 that is symbolically represented as σ = 1.

• 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 Examples

Let’s solve some problems on Standard Normal Distribution

Example 1: Find the probability density function of the standard normal distribution of the following data. x = 2, μ = 3 and σ = 4.

Solution:

Given,

• Variable (x) = 2
• Mean = 3
• Standard Deviation = 4

Using formula of probability density of standard normal distribution

F(Z) = _-∞∫^Z/ √(2π) and Z = [(x-μ)/σ]

Here Z = -0.25

Simplifying using Standard Normal Distribution Table, we get

F(-0.25) = 1 - F(0.25) = 1 - 0.5987 = 0.4013

Example 2: If the value of the random variable is 1, the mean is 0 and the standard deviation is 1, then find the probability density function of the Gaussian distribution.

Solution:

Given,

• Variable (x) = 1
• Mean = 0
• Standard Deviation = 1

Using formula of probability density of standard normal distribution

F(Z) = _-∞∫^Z/ √(2π) and Z = [(x-μ)/σ]

Here Z = 1

Simplifying using Standard Normal Distribution Table, we get

F(1) = 0.8413