Normal (Gaussian) Distribution

The Normal Distribution — also called the Gaussian Distribution — is a continuous probability distribution that describes data that cluster around a central mean.

It is the classic “bell-shaped curve” that appears in nature, data, and model errors.

In simple terms:

“Most observations are close to the mean, and the probability of extreme values decreases symmetrically on both sides.”

Probability Density Function (PDF)

Where:

• μ = mean (centre of the distribution)
• σ = standard deviation (spread/width of the curve)
• e = 2.718 (Euler’s number)

The total area under the curve = 1

68–95–99.7 Rule (Empirical Rule)

For a normal distribution:

• ~68% of values lie within 1σ (1 standard deviation) of the mean
• ~95% within 2σ (1 standard deviation) of the mean
• ~99.7% within 3σ (1 standard deviation) of the mean

Examples

• Human height (mean 170cm) - Most people near 170 cm
• IQ scores (mean 100) - 68% of people score between 85–115
• Measurement errors (mean 0) - Random noise around true value

A bell-shaped curve centred at μ=0 with shaded regions for:

• 68% within ±1σ
• 95% within ±2σ
• 99.7% within ±3σ

The total area under the curve = 1
The red dashed line marks the mean (also the median and mode)

In Machine Learning

• Model errors / residuals - Assumed to follow Normal distribution in regression
• Feature normalisation - Many ML algorithms perform better on normally distributed features
• Gaussian Naive Bayes - Uses the Normal PDF to model continuous features
• Statistical tests (Z-test, t-test) - Based on normality assumptions
• Initialisation / noise models - Random weight initialisation, dropout noise, etc.

Python code

Test if all numerical columns in a DataFrame are normally distributed with pg.normality():

import pingouin as pgimport numpy as npimport pandas as pd​# Example datasets# normal distributionnormal_data = np.random.normal(loc=50, scale=5, size=100) # skewed distributionskewed_data = np.random.exponential(scale=5, size=100) ​# Combine into a DataFramedf = pd.DataFrame({'Normal': normal_data, 'Skewed': skewed_data})​# Test for normalityresults = pg.normality(df, alpha=0.05)print(results)

The arguments we parse are: data, alpha=0.05 for the significance level

Output:

               W          pval  normalNormal  0.987236  4.536763e-01    TrueSkewed  0.830377  2.382588e-09   False