Java Program for Standard Normal Distribution (SND) - Z-Score and nextGaussian

In Java, the Standard Normal Distribution (SND) is a special case of the normal distribution. It occurs when a normal random variable has a mean of 0 and a standard deviation of 1. The normal random variable of a standard normal distribution is called a standard score or a z-score. A conversion from normally distributed to standard normally distributed values occurs via the formula.

Formula:

Z = (X - u) / s

where,

• Z: Value on the standard normal distribution
• X: Value on the original distribution
• u: Mean of the original distribution
• s: Standard deviation of the original distribution

Why Use the Standard Normal Distribution?

With the help of the Standard Normal Distribution, we can compare different datasets, even if they have different averages. We can do this by converting values into Z-scores. A Z-score tells how far a data point is from the average, using standard deviation as a measure. For example, if a Z-score is +2, then it simply means the value is 2 standard deviations above the average. If it is -1, then it simply means the value is 1 standard deviation below the average. It becomes easier for us to understand where a data point stands compared to the rest of the data.

Note: Z-scores are useful in things like testing, analyzing data, and drawing conclusions from statistics.

Java Program to Calculate Z-Score

Now, we are going to calculate the Z-score using the formula we have discussed above.

Example:

// Java program to demonstrate the naive method
// of finding Z-value
import java.io.*;
import java.util.*;

class Geeks {
    public static void main(String[] args)
    {
        // initialization of variables
        double Z, X, s, u;
        X = 26;
        u = 50;
        s = 10;

        // formula
        Z = (X - u) / s;

        System.out.println("The Z-value obtained is: " + Z);
    }
}

The Z-value obtained is: -2.4

Generating Random Numbers Using the Standard Normal Distribution

In Java, the nextGaussian() method from the Random class is used to generate the random numbers, normally distributed double value with mean 0.0 and standard deviation 1.0.

Declaration of nextGaussian() method:

public double nextGaussian()

• Parameter: This method does not take any parameter.
• Return Type: The method call returns the random, Normally distributed double value with mean 0.0 and standard deviation 1.0.

Now, we are going to discuss how to use nextGaussian() method to generate a random value from the standard normal distribution.

Example:

// Java program to demonstrate the working
// of nextGaussian()
import java.util.*;

public class Geeks {

    public static void main(String[] args)
    {
        // create random object
        Random r = new Random();

        // generating integer
        double n = r.nextGaussian();

        // Printing the random Number
        System.out.println("The next Gaussian value generated is: " + n);
    }
}

The next Gaussian value generated is : -0.9494975084124126

Applications of Standard Normal Distribution

The application of standard normal distribution are listed below:

• Z-Tests in Hypothesis Testing: The Z-test uses the Z-score to determine if a sample mean differs from the population mean by comparing the Z-score.
• Confidence Intervals: It helps to calculate the margin of error for sample data.
• Data Normalization: Used in machine learning to scale data so that all features contribute equally to the model.