Kernel Density Estimation (KDE) Silverman’s Rule-of-Thumb & Scott’s Bandwidth Calculator

Compute Silverman’s robust KDE bandwidth h and Scott’s normal-reference bandwidth.

Silverman Optimal KDE Bandwidth (h)
4.007 bandwidth
Scott’s Rule Bandwidth (1.06·s·n^−1/5) 4.989 bandwidth
Robust Spread min(s, IQR/1.34) 13.433 spread
Peak Gaussian Kernel Density at u=0 0.1 density
Inputs
⚡ Scenarios:
SD s
Sample standard deviation s.
IQR
Sample Interquartile Range IQR = Q₃ − Q₁.
n
Total number of observations n.
Results & Graph
Silverman Optimal KDE Bandwidth (h)
4.007 bandwidth
Scott’s Rule Bandwidth (1.06·s·n^−1/5)
4.989 bandwidth
Robust Spread min(s, IQR/1.34)
13.433 spread
Peak Gaussian Kernel Density at u=0
0.1 density

Silverman Optimal KDE Bandwidth (h)

Graph 1 • Silverman & Scott KDE Bandwidth • Domain Architecture & Live Telemetry Maps Sample Std Dev S, Interquartile Range Iqr, Sample Size N through Silverman & Scott KDE Bandwidth Formula.
Silverman Optimal KDE Bandwidt: 4.007 bandwidth Silverman & Scott KDE Bandwidth • Invariants & Metrics Silverman Optimal KDE Bandwi 4.007 bandwidth Scott’s Rule Bandwidth (1.06 4.989 bandwidth Robust Spread min(s, IQR/1.3 13.433 spread Peak Gaussian Kernel Density 0.1 density
Graph 2 • Silverman Optimal KDE Bandwidth (h) Across Sample Std Dev S Sweep (40% → 160%) Evaluates h_Silverman = 0.9 × min(s, IQR / 1.34) × n^(−1/5); h_Scott = 1.06 × s across 7 operating points (current input highlighted).
0 bandwi 2.20 bandwi 4.41 bandwi 1.69 5.68SD s 2.54 8.52SD s 3.39 11.36SD s 4.01 14.20SD s 4.01 17.04SD s 4.01 19.88SD s 4.01 22.72SD s
Graph 3 • Joint Sensitivity: Sample Std Dev S × Interquartile Range Iqr Compares Silverman Optimal KDE Bandwidth (h) when Interquartile Range Iqr is shifted +25% (solid) vs. −25% (dashed).
0 bandwi 2.75 bandwi 5.51 bandwi --- Interquartile Range Iqr @ 75% 2.96 -30% 3.60 -15% 4.24 +0% 4.87 +15% 5.01 +30%
Summary: For n=250, s=14.2, and IQR=18, Silverman’s Rule-of-Thumb bandwidth is h = 4.007 and Scott’s normal reference bandwidth is 4.9889.
Summary: For n=250, s=14.2, and IQR=18, Silverman’s Rule-of-Thumb bandwidth is h = 4.007 and Scott’s normal reference bandwidth is 4.9889.
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Formula & Mathematical Methodology: Silverman & Scott KDE Bandwidth Formula

hSilverman = 0.9 × min(s, IQR / 1.34) × n−1/5
hScott = 1.06 × s × n−1/5
h_Silverman = 0.9 × min(s, IQR / 1.34) × n^(−1/5); h_Scott = 1.06 × s × n^(−1/5)
  • sampleStdDevS: Sample Std Dev S (SD s)
  • interquartileRangeIqr: Interquartile Range Iqr (IQR)
  • sampleSizeN: Sample Size N (n)

Rounding: Internal floating-point calculations run in 64-bit IEEE-754 double precision; display values round to 2–4 decimal places.

Worked Example: Worked Mathematical Example: Silverman & Scott KDE Bandwidth

Compute Silverman’s robust KDE bandwidth h and Scott’s normal-reference bandwidth.

  1. Specify the function parameters, matrix entries, or confusion matrix counts.
  2. Apply the closed-form or iterative formula: h_Silverman = 0.9 × min(s, IQR / 1.34) × n^(−1/5); h_Scott = 1.06 × s × n^(−1/5).
  3. Examine the primary result, error bounds, and 3 interactive SVG plots.
Result: Computes Silverman & Scott KDE Bandwidth via h_Silverman = 0.9 × min(s, IQR / 1.34) × n^(−1/5); h_Scott = 1.06 × s × n^(−1/5).

Frequently Asked Questions

How does the Kernel Density Estimation (KDE) Silverman’s Rule-of-Thumb & Scott’s Bandwidth Calculator compute the solution?

It evaluates h_Silverman = 0.9 × min(s, IQR / 1.34) × n^(−1/5); h_Scott = 1.06 × s × n^(−1/5) directly in your browser and updates three custom SVG visualizations in real time.

Can I solve backwards for an input parameter using Goal Seek?

Yes. Click "🎯 Goal Seek" in the results toolbar to solve backwards for any numeric input field.

Does this calculator run locally offline?

Yes. All calculations run 100% locally in your browser and work offline via the CalculatorGraph PWA.

Mathematics & Statistics · Non-Parametric & Hypothesis Testing

Kernel Density Estimation (KDE) Silverman’s Rule-of-Thumb & Scott’s Bandwidth Calculator: inputs, method, and how to read the result

Compute Silverman’s robust KDE bandwidth h and Scott’s normal-reference bandwidth.

Choose inputs for the case you are checking

The defaults are illustrative. Replace them with values for your case and keep the units shown beside each field.

  • Sample Std Dev S: Sample standard deviation s.
  • Interquartile Range Iqr: Sample Interquartile Range IQR = Q₃ − Q₁.
  • Sample Size N: Total number of observations n.

What the calculation includes

With the default values shown, Silverman Optimal KDE Bandwidth (h) is 4.007 bandwidth. The calculation follows Silverman & Scott KDE Bandwidth Formula; the formula and variable definitions above show how the inputs combine.

One-input check: With the other defaults held constant, changing Sample Std Dev S from 14.2 SD s to 12.78 SD s changes Silverman Optimal KDE Bandwidth (h) from 4.007 bandwidth to 3.812 bandwidth. This is a comparison inside the calculator's stated model, not a prediction.

Where to use caution

  • Uses exact analytical or standard numerical analysis formulations from NIST DLMF.
  • Double-precision floating-point arithmetic has ~15–17 significant decimal digits of precision.

Scope: Global Mathematical & Statistical Standard Check the assumptions and limitations against your situation before relying on the estimate.

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