Statistics
Intermediate 14 min read Statistics

Common Distributions

Recognizing the shapes data takes

Real-world data forms recognizable shapes: normal, binomial, Poisson, exponential, log-normal. Identifying the shape tells you which tests and models are valid and where the process behaves.

normal distributionbinomialPoissonlog-normalexponentialcentral limit theorem
Prerequisites: Descriptive Statistics Probability Basics
Before we start

What you will learn

  • Describe the normal, binomial, Poisson and log-normal families
  • Match a business process to its likely distribution
  • Use the central limit theorem correctly
Introduction

What is this?

A distribution is the probability shape of a variable. The normal is the symmetric bell for sums and averages (via the Central Limit Theorem). The binomial counts successes in a fixed number of trials; Poisson counts rare events in time/space; exponential models waiting times; log-normal shows up when a variable multiplies (prices, incomes, app-session durations). Choosing the right family is the first step of nearly every test and model.

Motivation

Why do you need it?

  • The test you run depends on the shape: t-tests assume roughly normal means, Poisson handles counts.
  • Outliers and tails differ — exponential/log-normal variables have heavy right tails the bell cannot hold.
  • The CLT is why sample means of skewed data behave normally at sufficient n.
  • Forecasting averages without knowing the shape makes errors compound wrongly.
Think about it

Mental model

Imagine

Character types in a play

Think of

Distribution families

You wouldn't cast a rom-com lead to play King Lear. Recognize the data's 'character type' first; then casting (the test/model) is appropriate.

Mechanics

How it works

  • Normal(μ, σ): symmetric bell; mean=median=mode; the refuge of sums/averages at moderate n.
  • Binomial(n, p): number of successes in n independent trials, each with probability p.
  • Poisson(λ): counts of rare independent events per interval; variance = mean.
  • Exponential(rate): time between Poisson events — memoryless, right-skewed.
  • Log-normal: a variable whose log is normal; comes from multiplying effects.

Matching a process to its distribution family.

What are we measuring?
Symmetric, additive?
Fixed trials?
Rates in time?
Multiplicative?
Hands-on

Examples

Fit and compare distributions in Python

Fit and compare distributions in Python
import numpy as np, pandas as pd, matplotlib.pyplot as plt
 
# 10k daily support tickets (Poisson-ish counts)
np.random.seed(1)
tickets = np.random.poisson(8, 10000)
print("mean %.2f var %.2f" % (tickets.mean(), tickets.var()))  # close = Poisson
 
# Session lengths (log-normal-like, right-skewed)
sessions = np.random.lognormal(mean=3, sigma=0.8, size=10000)
print("median %.1f  mean %.1f  (skew pulls mean > median)" % (np.median(sessions), sessions.mean()))

Variance≈mean signals Poisson counts; log-normal shows the classic right-skew where the median, not the mean, describes typical sessions.

Real world

In production

  • Web traffic modeled as Poisson counts for capacity and anomaly detection.
  • Order values fit log-normal; median, not mean, drives 'typical basket' reporting.
  • Customer lifetime lengths approximated by exponential for churn forecasting.
Guidelines

Best practices

  • Plot first, test second — the histogram tells you which family to try.
  • Remember the CLT: the average of many independent draws becomes normal even if the raw data isn't.
  • For counts, default to Poisson/negative binomial thinking before 'normal'.
  • Heavy right tail? Consider log transform before using normal-family tools.
Watch out

Common mistakes

  • Forcing the normal distribution on counts or durations (they are skewed).
  • Ignoring the CLT and complaining sample means aren't bell-shaped.
  • Using mean for a log-normal's 'typical' value when the median is correct.
  • Assuming independence between events in binomial/Poisson models.
When things break

Debugging tips

  • Variance >> mean in count data — overdispersion; try negative binomial instead.
  • Normal Q-Q plot bending at both ends — heavy tails; re-think the family.
  • Mean and median far apart is skew, not a bug — pick the right summary.
  • Per-event counts ballooning? Check the measurement window isn't doubling.
Interviews

Interview checkpoint

Practice

Hands-on lab

Identify Distributions by Their Shape Numbers
30 min · intermediate
In progress
Objectives
  • Simulate Poisson, binomial, exponential and log-normal samples
  • Identify each family from shape diagnostics
  • Validate the CLT on a skewed sample

Identify Distributions by Their Shape Numbers

30 min · intermediate

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sandbox · Linux sandbox
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Test yourself

Quick quiz

0/2
1

Number of defect-free successes out of 200 inspected units follows which shape?

2

The mean of a heavily right-skewed sample is always close to the median.

Go deeper

Official documentation

scipy.stats distribution reference
docs