Exponentially Weighted Moving Average (EWMA) Smoothing, Span & Half-Life Calculator

Calculate updated EWMA S_t = α·Y_t + (1−α)·S_{t−1}, equivalent span N, and exponential half-life.

Updated EWMA Smoothed Value (S_t)
106.8 units
Equivalent N-Period Moving Avg Span 9 periods
Exponential Memory Half-Life 3.106 steps
Center of Mass ((1 − α) / α) 4 steps
Inputs
⚡ Scenarios:
α
Smoothing decay weight α (0.01 to 0.99).
Y_t
Latest observed data point Y_t.
S_{t−1}
Previous EWMA smoothed value S_{t−1}.
Results & Graph
Updated EWMA Smoothed Value (S_t)
106.8 units
Equivalent N-Period Moving Avg Span
9 periods
Exponential Memory Half-Life
3.106 steps
Center of Mass ((1 − α) / α)
4 steps

Updated EWMA Smoothed Value (S_t)

Graph 1 • EWMA Smoothing & Half-Life • Domain Architecture & Live Telemetry Maps Smoothing Factor Alpha, Latest Observation Yt, Prior Ewma Level St1 through EWMA Smoothing & Half-Life Formula.
Updated EWMA Smoothed Value (S: 106.8 units EWMA Smoothing & Half-Life • Invariants & Metrics Updated EWMA Smoothed Value 106.8 units Equivalent N-Period Moving A 9 periods Exponential Memory Half-Life 3.106 steps Center of Mass ((1 − α) / α) 4 steps
Graph 2 • Updated EWMA Smoothed Value (S_t) Across Smoothing Factor Alpha Sweep (40% → 160%) Evaluates S_t = α·Y_t + (1 − α)·S_{t−1}; Span_N = (2 / α) − 1; HalfLife = −ln( across 7 operating points (current input highlighted).
0 units 59.66 units 119.3 units 105.1 0.0800α 105.7 0.120α 106.2 0.160α 106.8 0.200α 107.4 0.240α 107.9 0.280α 108.5 0.320α
Graph 3 • Joint Sensitivity: Smoothing Factor Alpha × Latest Observation Yt Compares Updated EWMA Smoothed Value (S_t) when Latest Observation Yt is shifted +25% (solid) vs. −25% (dashed).
0 units 63.42 units 126.8 units --- Latest Observation Yt @ 75% 110.1 -30% 111.4 -15% 112.7 +0% 114.0 +15% 115.3 +30%
Summary: With α = 0.2 (equivalent span N = 9 periods, memory half-life = 3.11 steps), S_t updates from 104 toward 118 at 106.8.
Summary: With α = 0.2 (equivalent span N = 9 periods, memory half-life = 3.11 steps), S_t updates from 104 toward 118 at 106.8.
⚡ Send result to: Scientific →Statistics & SD →Probability & P-Value →
Interactive chart & math engine powered by CalculatorGraph

Formula & Mathematical Methodology: EWMA Smoothing & Half-Life Formula

St = α·Yt + (1 − α)·St−1
SpanN = (2 / α) − 1
HalfLife = −ln(2) / ln(1 − α)
S_t = α·Y_t + (1 − α)·S_{t−1}; Span_N = (2 / α) − 1; HalfLife = −ln(2) / ln(1 − α)
  • smoothingFactorAlpha: Smoothing Factor Alpha (α)
  • latestObservationYt: Latest Observation Yt (Y_t)
  • priorEwmaLevelSt1: Prior Ewma Level St1 (S_{t−1})

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: EWMA Smoothing & Half-Life

Calculate updated EWMA S_t = α·Y_t + (1−α)·S_{t−1}, equivalent span N, and exponential half-life.

  1. Specify the function parameters, matrix entries, or confusion matrix counts.
  2. Apply the closed-form or iterative formula: S_t = α·Y_t + (1 − α)·S_{t−1}; Span_N = (2 / α) − 1; HalfLife = −ln(2) / ln(1 − α).
  3. Examine the primary result, error bounds, and 3 interactive SVG plots.
Result: Computes EWMA Smoothing & Half-Life via S_t = α·Y_t + (1 − α)·S_{t−1}; Span_N = (2 / α) − 1; HalfLife = −ln(2) / ln(1 − α).

Frequently Asked Questions

How does the Exponentially Weighted Moving Average (EWMA) Smoothing, Span & Half-Life Calculator compute the solution?

It evaluates S_t = α·Y_t + (1 − α)·S_{t−1}; Span_N = (2 / α) − 1; HalfLife = −ln(2) / ln(1 − α) 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 · Econometrics & Regression

Exponentially Weighted Moving Average (EWMA) Smoothing, Span & Half-Life Calculator: inputs, method, and how to read the result

Calculate updated EWMA S_t = α·Y_t + (1−α)·S_{t−1}, equivalent span N, and exponential half-life.

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.

  • Smoothing Factor Alpha: Smoothing decay weight α (0.01 to 0.99).
  • Latest Observation Yt: Latest observed data point Y_t.
  • Prior Ewma Level St1: Previous EWMA smoothed value S_{t−1}.

What the calculation includes

With the default values shown, Updated EWMA Smoothed Value (S_t) is 106.8 units. The calculation follows EWMA Smoothing & Half-Life Formula; the formula and variable definitions above show how the inputs combine.

One-input check: With the other defaults held constant, changing Smoothing Factor Alpha from 0.2 α to 0.22 α changes Updated EWMA Smoothed Value (S_t) from 106.8 units to 107.08 units. 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.

More in Mathematics & Statistics: Scientific Calculator, Statistics & Standard Deviation Calculator, Permutations, Combinations, Probability, Sample Size & P-Value Calculator.