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Histogram

Histogram

Definition

Core Statement

A Histogram is a graphical representation of the distribution of a continuous numerical variable. It groups data into adjacent intervals called bins and displays the frequency (count) of observations in each bin as a bar.

Purpose

  1. Visualize Distribution Shape: Is it Normal? Skewed? Bimodal?
  2. Detect Outliers: Isolated bars far from the main group.
  3. Assess Spread: How wide is the data range?
  4. Check Center: Where is the peak (mode)?
Histogram vs. Bar Chart

  • Histogram: For continuous data (e.g., Height, Salary). Bars touch (no gaps). X-axis is a number line.
  • Bar Chart: For categorical data (e.g., Country, Car Brand). Bars usually have gaps. X-axis is categories.

Theoretical Background

Binning Choices

The shape of a histogram depends heavily on the number of bins (k) or bin width (h).

  1. Too few bins: Oversmoothing. Hides local details (underfitting).
  2. Too many bins: Jagged/Comb-like. Shows random noise (overfitting).

Rules of Thumb

Worked Example: Salary Distribution

Problem

You have salaries for 1000 employees. Min = 30k, Max = 200k.
You observe a histogram with a huge peak at 40k and a long tail to the right.

Interpretation:

  1. Shape: Right-Skewed (Postive Skew). Most people earn low, a few earn millions.
  2. Center: Median < Mean. The high earners pull the mean up.
  3. Action: A log-transformation (log⁡(Salary)) might make this distribution look more Normal for regression analysis.

Assumptions

Limitations & Pitfalls

Pitfalls

  1. Bin Bias: Changing bin width can completely change the story. A "flat" distribution can look "peaked" just by shifting bin edges. Even governments manipulate this. Always try multiple bin widths.
  2. Sample Size: For small n (<30), histograms are misleading. Use a Dot Plot or Rug Plot instead.
  3. Zero-Count Bias: In some softwares, empty bins are hidden, making the x-axis discontinuous. Ensure the axis represents the full linear range.

Python Implementation

import matplotlib.pyplot as plt
import seaborn as sns
import numpy as np

data = np.random.gamma(2, 2, 1000) # Skewed data

# 1. Matplotlib Basic
plt.hist(data, bins=30, edgecolor='black')
plt.title("Basic Histogram")
plt.show()

# 2. Seaborn (Better defaults)
# kde=True adds a Density curve
sns.histplot(data, bins='auto', kde=True) 
plt.title("Histogram with Density Curve")
plt.show()

# 3. Log-Transformed to fix Skew
sns.histplot(np.log(data), kde=True)
plt.title("Log-Transformed Data")
plt.show()

Interpretation Guide

Shape Meaning
Bell Curve Normal distribution. Mean ≈ Median.
Right Skew (Positive) Long tail to the right. Mean > Median. (e.g., Income).
Left Skew (Negative) Long tail to the left. Mean < Median. (e.g., Age at Death).
Bimodal (Two peaks) Two distinct groups mixed together (e.g., Men's and Women's heights).
Uniform (Flat) No central tendency (e.g., Rolling a die).