Statistical Distributions for Data Analysts

Why Distributions Matter for Data Analysts

A statistical distribution describes the pattern of values in a dataset — what values are possible, how likely each is, and the shape of the spread. Understanding distributions helps analysts choose the right summary statistics, select appropriate statistical tests, identify anomalies, and build better models. Misidentifying a distribution leads to flawed conclusions: averaging skewed data, applying normal-theory tests to count data, or missing the long tail that causes most of the business impact.

Key Distribution Properties

The Normal (Gaussian) Distribution

The normal distribution is the most commonly assumed distribution for continuous measurements. It is symmetric and bell-shaped, defined by its mean (μ) and standard deviation (σ).

Analyst applications: z-score outlier detection, A/B test analysis, control charts, many ML model assumptions. Check normality with histograms, Q-Q plots, or the Shapiro-Wilk test.

Common Continuous Distributions

Common Discrete Distributions

Identifying Distributions in Practice

Before assuming normality, always visualize the data:

Python: Fitting and Comparing Distributions

import numpy as np
import pandas as pd
from scipy import stats
import matplotlib.pyplot as plt

# Basic statistics
data = df['revenue']
print(data.describe())
print('skewness:', data.skew())
print('kurtosis:', data.kurtosis())

# Test for normality
stat, p = stats.shapiro(data.sample(min(len(data), 5000)))
print(f'Shapiro-Wilk p={p:.4f} — {"normal" if p > 0.05 else "not normal"}')

# Fit log-normal (if data is right-skewed and positive)
log_data = np.log(data[data > 0])
mu, sigma = log_data.mean(), log_data.std()

# Generate percentiles
percentiles = np.percentile(data, [10, 25, 50, 75, 90, 95, 99])
print('Percentiles:', percentiles)

Skewed Data: When Not to Use the Mean

Many business metrics — revenue, session duration, load time — follow right-skewed or log-normal distributions. The arithmetic mean is pulled toward outliers and may not represent the typical user experience.

Central Limit Theorem

The Central Limit Theorem (CLT) states that the distribution of sample means approaches a normal distribution as sample size grows, regardless of the underlying distribution. This is why normal-theory tests (t-tests, z-tests) are valid for comparing sample means even when the raw data is skewed — as long as the sample size is large enough (typically n ≥ 30). This underpins virtually all frequentist A/B testing.

Summary

Recognizing which distribution governs your data is a foundational data analysis skill. It determines which summary statistics are meaningful, which tests are appropriate, and how to interpret variability. The normal distribution is a useful default for many measurements, but revenue, durations, rates, and counts each follow their own distributional patterns. Visualizing data before modeling — with histograms, Q-Q plots, and log-scale views — reveals the true shape and prevents the common mistake of applying normal-theory methods to non-normal data.