Standard Normal Distribution (Z-Score)

The Standard Normal Distribution is a special case of the Normal Distribution where μ=0 and σ=1.

That means:

• The mean (centre) = 0
• The standard deviation (spread) = 1

It’s used to measure how far a value is from the mean in standard deviation units — this distance is called the Z-score.

Z-Score Formula

Where:

• X = original data value
• μ = population mean
• σ = population standard deviation

The Z-score tells you how many standard deviations away a value is from the mean.

Interpretation

Why Use Z-Scores?

Z-scores standardise data — making different variables comparable even if they have different units or scales.

For example:
A test score of 75 in Math and 82 in English — which is better?
Z-scores let you compare them on the same relative scale.

Probability Density Function (PDF)

For the standard normal distribution:

• The curve is symmetric around 0
• The total area = 1 (represents 100% probability)

Cumulative Distribution Function (CDF)

Represents the probability that a standard normal variable is less than or equal to a given Z-score.

• Left plot (PDF):
• A bell-shaped curve centered at 0
• Colored areas show probabilities for ±1σ, ±2σ, ±3σ
• Right plot (CDF):
• Smooth S-shaped curve
• Represents cumulative probability up to each Z-score

The area under the curve corresponds to probability
Z-scores make probability lookup easy using Z-tables

In Machine Learning

• Feature standardisation - Convert all features to mean=0, std=1 before training
• Distance-based algorithms - Important for SVMs, KNNs, PCA, clustering
• Outlier detection - Extreme Z-scores
• Probability modelling - Z-scores convert any normal variable to standard form
• Evaluation metrics - Used in Z-tests, confidence intervals, etc.