Distributions: Normal, Power-Law, and Bimodal

Plot any column as a histogram and you will see its distribution. Some distributions are bell-shaped, some are long-tailed, some have two humps. The shape is not cosmetic; it determines what statistical tools work.

The three shapes you must know

Normal distributions

Normal distributions show up whenever many small, independent causes add together (Central Limit Theorem). Height is the sum of many genetic and environmental factors, each tiny. Polling errors are the sum of many small deviations. Mean and standard deviation fully describe a normal distribution.

Power-law distributions

Power-laws appear when outcomes multiply rather than add. Rich people can invest and get richer. Popular videos get recommended and get more popular. A tiny fraction of items (the head) dominates everything else (the long tail). In a power-law, mean is essentially meaningless because a handful of extreme values dominate.

Bimodal distributions

A bimodal distribution has two peaks, usually because your data contains two different populations mashed together. Commute times bimodal: car commuters and transit commuters. Restaurant sizes bimodal: independent cafes and chain restaurants. The trick: if you see bimodality, split the data into two populations and analyze each separately.

import matplotlib.pyplot as plt
import numpy as np
import pandas as pd

df = pd.read_csv('some_data.csv')

# Plot histogram to see the shape
for col in df.select_dtypes('number').columns:
    df[col].hist(bins=50)
    plt.title(col)
    plt.show()

# Check for skew
from scipy.stats import skew, kurtosis
print('Skew:', {c: skew(df[c]) for c in df.select_dtypes('number')})
# |skew| > 1 means substantial asymmetry

The big idea: plot first, average later. The shape of your data tells you which tools to trust and which will give you confidently wrong answers.