Histogram Calculator
Print PageA Histogram Calculator (also known as a Histogram Maker, Histogram Generator, Bin Width Calculator, Frequency Density Analyzer, or Sturges’ Rule Chart Utility) converts raw numerical datasets into graphical histograms featuring adjacent rectangular bars, computing optimal bin counts (K = 1 + 3.322 · log10n), class bin widths (w = [Max - Min] ÷ K), class boundaries, absolute frequencies (fi), relative frequencies (rfi), and frequency densities (FDi = fi ÷ wi for unequal bin intervals).
Histograms are essential statistical graphs for displaying continuous continuous probability distributions. Unlike discrete bar charts with gaps, histogram bars touch continuously, making distribution shape features like bimodal peaks, skewness, spread, and outlier clusters instantly visual.
Our free online Histogram Calculator provides instant calculations across all binning algorithms:
- Sturges’ Rule Bin Count (K):
K = ⌈ 1 + 3.322 · log10(n) ⌉(Best for normal distributions,n < 200). - Freedman-Diaconis Bin Width (h):
h = ( 2 · IQR ) ÷ n1/3(Best for skewed data with outliers). - Square Root Rule Bin Count (K):
K = ⌈ √n ⌉. - Equal Bin Width Formula (w):
w = ( Maximum - Minimum ) ÷ K. - Frequency Density Height (FDi):
FDi = Absolute Frequency (fi) ÷ Bin Width (wi). - Total Bar Area Equivalence:
Total Area = ∑ ( Bar Widthi · Heighti ) = Total Sample Size (n).
Master Histogram Reference Table (n = 100 E-Commerce Customer Orders)
The table below displays class boundaries, midpoints (xm), absolute frequencies (f), relative frequencies (rf), and frequency density heights for an e-commerce order dataset (Range = $10.00 to $210.00, Sturges’ K = 8 Bins, Width w = $25.00):
| Bin Interval ($ Order Value) | Bin Midpoint (xm) | Absolute Frequency (f) | Relative Frequency (rf) | Frequency Density (Height = f/25) | Distribution Trajectory Visual |
|---|---|---|---|---|---|
| $10.00 to $34.99 | $22.50 | 18 Orders | 0.180 (18%) | 0.720 | High Starting Volume |
| $35.00 to $59.99 (Modal Bin) | $47.50 | 32 Orders (Peak Bar) | 0.320 (32%) | 1.280 (Peak Density) | Primary Customer Sweet Spot |
| $60.00 to $84.99 | $72.50 | 22 Orders | 0.220 (22%) | 0.880 | Steady Decline |
| $85.00 to $109.99 | $97.50 | 12 Orders | 0.120 (12%) | 0.480 | Moderate Taper |
| $110.00 to $134.99 | $122.50 | 8 Orders | 0.080 (8%) | 0.320 | Tapering Tail |
| $135.00 to $159.99 | $147.50 | 4 Orders | 0.040 (4%) | 0.160 | Low Volume Tail |
| $160.00 to $184.99 | $172.50 | 3 Orders | 0.030 (3%) | 0.120 | Sparse Upper Tail |
| $185.00 to $209.99 | $197.50 | 1 Order | 0.010 (1%) | 0.040 | Right-Skewed High Outlier |
Step-by-Step Histogram Binning & Shape Analysis
To analyze the 100-order e-commerce transaction dataset:
Step 1 (Calculate Sturges' Bin Count K): K = ⌈ 1 + 3.322 · log10(100) ⌉ = ⌈ 1 + 3.322(2.0) ⌉ = ⌈ 7.644 ⌉ = 8 bins
Step 2 (Calculate Bin Width w): w = ($210.00 - $10.00) ÷ 8 = $200.00 ÷ 8 = $25.00 per bin
Step 3 (Identify Peak Modal Bar): The tallest bar occurs at [$35.00 - $59.99] containing 32% of all orders
Step 4 (Evaluate Distribution Skewness): The long right-hand tail extending to $210 indicates a classic right-skewed (positively skewed) distribution
Thus, the histogram reveals that 72% of all customer orders fall below $85.00, with a minority of high-value purchasers creating a long right-hand tail.
Statistical Charts Comparison: Histogram vs. Bar Chart vs. Frequency Polygon
Below is a comparative reference chart detailing when to use a Histogram versus related charts:
| Statistical Chart Type | Variable Type Visualized | Bar Spacing Rule | Primary Visual Function |
|---|---|---|---|
| Histogram | Continuous Numerical Data | NO GAPS between adjacent bars | Visualizing distribution shape, spread, & modal peaks. |
| Bar Chart | Discrete Categorical Data | CLEAR GAPS between categorical bars | Comparing distinct categories (e.g. car brands, states). |
| Frequency Polygon | Continuous Numerical Data | Line graph connecting class midpoints | Overlaying 2+ distributions on a single axis. |
History & Mathematics: 1891 Karl Pearson
1891 Karl Pearson & Coining the Term “Histogram”
In 1891, English mathematician and biometrician Karl Pearson coined the term “histogram” (derived from the Greek histos meaning “mast or vertical pole” and gramma meaning “drawing or record”) to describe vertical bar graphics representing continuous statistical frequency distributions.
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Frequently Asked Questions (FAQ)
What is a Histogram?
A histogram is a graphical representation of continuous numerical data that groups numbers into user-defined bin ranges using touching vertical bars.
How do you determine the optimal number of bins for a Histogram?
Use Sturges’ Rule (K = 1 + 3.322 · log10n) for normal datasets or the Freedman-Diaconis rule (h = 2 · IQR ÷ n1/3) for skewed datasets with outliers.
Why do Histogram bars touch without gaps?
Histogram bars touch because the horizontal axis represents a continuous numerical scale, unlike bar charts which represent discrete categories separated by gaps.