Key Takeaways: Probability Distributions and the Normal Curve

One-Sentence Summary

Probability distributions are mathematical models that describe how randomness behaves — from the binomial distribution for counting successes to the normal distribution that appears everywhere in nature — and the z-score transformation lets you convert any normal variable into a universal scale for finding exact probabilities, as long as you remember that the model is useful, not true.

Core Concepts at a Glance

Distribution Comparison

The Binomial Distribution

Formula

$$\boxed{P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}}$$

Conditions (BINS)

Quick Formulas

$$E(X) = np \qquad \sigma = \sqrt{np(1-p)}$$

Python

from scipy import stats

# P(X = k)
stats.binom.pmf(k, n, p)

# P(X <= k)
stats.binom.cdf(k, n, p)

# Mean and standard deviation
stats.binom.mean(n, p)
stats.binom.std(n, p)

The Normal Distribution

Properties

1. Bell-shaped and symmetric about $\mu$
2. Mean = median = mode
3. Defined entirely by $\mu$ and $\sigma$
4. 68-95-99.7 rule holds exactly
5. Tails extend to infinity (never touch x-axis)
6. Total area under curve = 1

The Z-Score Transformation

$$\boxed{z = \frac{x - \mu}{\sigma}} \qquad \text{(transforms any } N(\mu, \sigma^2) \text{ to } N(0, 1)\text{)}$$

$$\boxed{x = \mu + z \cdot \sigma} \qquad \text{(transforms back to original scale)}$$

Z-Table Excerpt (Standard Normal Cumulative Probabilities)

Common Z-Scores and Probabilities

Python Quick Reference

from scipy import stats

# --- Normal distribution ---
# P(X <= x)
stats.norm.cdf(x, loc=mu, scale=sigma)

# P(X > x)
1 - stats.norm.cdf(x, loc=mu, scale=sigma)

# P(a < X < b)
stats.norm.cdf(b, loc=mu, scale=sigma) - stats.norm.cdf(a, loc=mu, scale=sigma)

# Find x from probability (inverse/quantile)
stats.norm.ppf(probability, loc=mu, scale=sigma)

# --- Assessing normality ---
from scipy import stats
import matplotlib.pyplot as plt

# QQ-plot
stats.probplot(data, dist="norm", plot=plt)

# Shapiro-Wilk test
stat, p_value = stats.shapiro(data)

Assessing Normality: Decision Guide

Is the data approximately normal?
│
├── Step 1: HISTOGRAM
│   └── Does it look roughly bell-shaped and symmetric?
│       ├── NO → Probably not normal. Check why.
│       └── YES / MAYBE → Continue to Step 2.
│
├── Step 2: QQ-PLOT
│   └── Do points follow the diagonal line?
│       ├── Curves up at right → heavy right tail (right-skewed)
│       ├── Curves down at left → heavy left tail (left-skewed)
│       ├── S-shape at both ends → heavy tails (leptokurtic)
│       ├── Flat S-shape → light tails (platykurtic)
│       └── Close to line → approximately normal
│
└── Step 3: SHAPIRO-WILK TEST
    └── What is the p-value?
        ├── p < 0.05 → Evidence against normality
        │   └── BUT: For large n, trivial departures are detected
        ├── p > 0.05 → No evidence against normality
        │   └── BUT: Doesn't prove normality
        └── ALWAYS interpret alongside the QQ-plot

Continuity Correction

When to use: Only when approximating a discrete distribution with a continuous one. Not needed for genuinely continuous data.

Common Misconceptions

The One Thing to Remember

If you forget everything else from this chapter, remember this:

The normal distribution is a model — the most useful one in statistics, but a model nonetheless. It works beautifully for heights, blood pressure, test scores, and measurement errors because those measurements result from many small, independent, additive effects. It fails for income, wealth, stock crashes, and social media virality because those phenomena involve multiplicative effects, heavy tails, and structural boundaries. Your job as a statistician isn't to assume normality — it's to check it (using QQ-plots and the Shapiro-Wilk test) and to know when it matters (small samples, individual predictions) and when it doesn't (large samples, sample means). George Box was right: all models are wrong, but some are useful. The normal distribution is the most useful wrong model you'll ever learn.

Key Terms