Spotting the Shape

# Spotting the Shape Every metric you ever analyze has an underlying distribution. Most analysts never bother to look - and then they're confused when their "average" doesn't match what users actually experience. This lesson is about training your eye. ## The four shapes you'll see most **Normal (bell-shaped)** - symmetric, single hump. Most data clustered around the mean, tapering off in both directions. Mean = median = mode. Examples: adult height, IQ scores, measurement errors. **Log-normal** - right-skewed with a long tail of high values. Mean > median. Mean is dragged up by the tail. Examples: income, house prices, time-to-complete a task, customer lifetime value. **Long-tailed / power-law** - extreme right-skew. A tiny fraction of observations account for most of the mass. Examples: city populations, website traffic per page, revenue per customer in B2B. **Bimodal** - two distinct humps. Usually a sign you're mixing two populations. Examples: order values from a freemium product (free users near zero, paid users much higher), session durations (bouncers vs engaged users). ## Spotting shapes & log transforms in Python ```python import numpy as np import pandas as pd import matplotlib.pyplot as plt import seaborn as sns # Generate right-skewed Customer Lifetime Value (LTV) data np.random.seed(42) raw_ltv = np.random.lognormal(mean=4.5, sigma=1.2, size=5000) df = pd.DataFrame({"ltv": raw_ltv}) # Log-transformation to normalize skewed distributions df["log_ltv"] = np.log1p(df["ltv"]) print("--- Raw LTV vs Log-Transformed LTV ---") print(f"Raw LTV: Mean = ${df['ltv'].mean():.2f}, Median = ${df['ltv'].median():.2f}") print(f"Log LTV: Mean = {df['log_ltv'].mean():.2f}, Std = {df['log_ltv'].std():.2f}") # Plotting distribution comparison fig, axes = plt.subplots(1, 2, figsize=(10, 4)) sns.histplot(df["ltv"], kde=True, ax=axes[0], color="crimson") axes[0].set_title("Raw LTV (Right-Skewed / Log-Normal)") axes[0].set_xlabel("LTV ($)") sns.histplot(df["log_ltv"], kde=True, ax=axes[1], color="teal") axes[1].set_title("log1p(LTV) (Transformed to Bell Shape)") axes[1].set_xlabel("Log(LTV + 1)") plt.tight_layout() # plt.show() ``` ## Why the shape changes which stats you trust | Shape | Use | Avoid | |---|---|---| | Normal | Mean + std | Nothing - both work | | Log-normal | Median + percentiles, or log-transform then use mean | Raw mean (it lies) | | Long-tailed | Percentiles only (p50, p90, p99) | Mean, std (both useless) | | Bimodal | Split groups, analyze separately | Any single summary statistic | ## Takeaway Before you summarize any metric: 1. Plot its histogram (`sns.histplot(df['col'], kde=True)`). 2. Identify the shape. 3. Pick the summary statistic that fits the shape. Two lines of Python save you from a meeting where someone says "but the average is fine?!" and you have to explain that the average is meaningless because the distribution is bimodal.