Normal Distribution

A normal distribution is a continuous probability distribution that is symmetric about its mean. In this distribution, values close to the mean occur more frequently than values farther away from the mean.

• The area under the curve of a normal distribution is always equal to 1.
• It is also known as the bell curve or Gaussian distribution.

Example: Imagine a class where students take a math test. Most students score close to the average mark, and only a few score very low or very high. If we want to describe how these marks are spread around the average in a bell-shaped pattern, this is a situation where we use the normal distribution

The formula for the probability density function of the Normal Distribution (Gaussian Distribution) is:

f(x)=\frac{1}{\sigma\sqrt{2\pi}}e^{-\frac{(x-\mu)^2}{2\sigma^2}}

where:

• x = value of the random variable
• μ = mean of the distribution
• σ = standard deviation
• π ≈ 3.14159
• e ≈ 2.71828

Characteristics

• Symmetry: The normal distribution is symmetric around its mean. This means the left side of the distribution mirrors the right side.
• Mean, Median, and Mode: In a normal distribution, the mean, median, and mode are all equal and located at the center of the distribution.
• Bell-shaped Curve: The curve is bell-shaped, indicating that most of the observations cluster around the central peak and the probabilities for values further away from the mean taper off equally in both directions.
• Standard Deviation: The spread of the distribution is determined by the standard deviation. About 68% of the data falls within one standard deviation of the mean, 95% within two standard deviations, and 99.7% within three standard deviations.
• Asymptotic tails: The curve never touches the x-axis but extends infinitely.

Parameters of Normal Distribution

A normal distribution is completely determined by two parameters:

Mean (μ)

The mean represents the center of the distribution and determines the location of the peak.

• Increasing μ shifts the curve to the right.
• Decreasing μ shifts the curve to the left.

Standard Deviation (σ)

The standard deviation determines how spread out the data is around the mean.

• Smaller σ produces a narrower and taller curve.
• Larger σ produces a wider and flatter curve.

Empirical Rule of Standard Deviation

Generally, the normal distribution has a positive standard deviation, and the standard deviation divides the area of the normal curve into smaller parts, and each part defines the percentage of data that falls into a specific region. This is called the Empirical Rule of Standard Deviation in Normal Distribution.

Empirical Rule states that,

• Approximately 68% of data lies within 1 standard deviation of the mean.
• Approximately 95% of data lies within 2 standard deviations of the mean.
• Approximately 99.7% of data lies within 3 standard deviations of the mean.

This rule is known as the Empirical Rule or 68–95–99.7 Rule.

Normal Distribution Table

Normal Distribution Table, which is also called Normal Distribution Z Table, is the table of z-values for normal distribution. This Normal Distribution Z Table is given as follows:

Applications

• Feature Scaling in Machine Learning: Data is standardized using Z-score normalization to improve model performance.
• Bayesian Inference: Gaussian distributions are widely used as prior distributions in Bayesian models.
• Anomaly Detection: Values that lie far from the mean are treated as anomalies or outliers.
• Gaussian Mixture Models: Used in clustering and unsupervised learning.
• Image Processing: Gaussian blur applies a normal-distributed kernel to reduce image noise.
• Generative AI: Diffusion models use Gaussian noise during image generation.

➣Practice: Solved Examples

Related Articles

• Poisson Distribution
• Binomial Distribution
• Probability Distribution