Exponential moving average calculator

This calculator takes a chronological series of finite numbers and a factor α in the open-closed interval (0, 1], seeds the path with the first observation, then applies EMA_t = α · x_t + (1 − α) · EMA_{t−1} for every later index. The response returns the full EMA array, the final last_ema, the series length n, the factor used, and a compact formula string. Empty series, non-finite entries, and factors outside (0, 1] fail with invalid_input so callers never receive a silent NaN. The arithmetic is fully deterministic, runs free in your browser, and is available over the API at $0.002 per successful request with no language model and no external market-data service on the hot path.

What an exponential moving average is and why it differs from a simple average

At each time step the new estimate is a convex combination of the new observation and the previous estimate: EMA_t equals α times x_t plus (1 − α) times EMA_{t−1}.

How this calculator validates input and builds the EMA path

Call the capability with two required fields: series (a non-empty array of finite numbers in chronological order, oldest first) and factor (the smoothing coefficient α). Validation rejects a missing field, a non-array series, an empty series, lists longer than the published maximum, any entry that is not a finite number, and any factor that is not a finite number strictly greater than zero and at most one. The engine also accepts aliases values, data, x, and prices for the series, and alpha, smoothing, and smoothing_factor for the factor, so scripts that already speak in those terms do not need a rename step. When the factor is out of range the error message says so explicitly under invalid_input, and validation failures do not consume prepaid balance. The response returns n, factor, series, ema, last_ema, and formula so downstream code can chart the path, assert the final level, or recompute intermediate steps without re-sending the request.

Practical use, factor choice, pricing, and delivery

Use the browser tool when you want an immediate EMA while checking a trading notebook, smoothing a noisy lab sensor, or explaining geometric weights in a statistics lesson without installing a package. Choose α from domain convention: short reactive filters use values near 0.3–0.5; slow trend filters use values near 0.05–0.1; the common span mapping α = 2 / (N + 1) turns a 9-period EMA into α = 0.2 and a 19-period EMA into α = 0.1. If you already store span N rather than α, convert once before calling. There is no language model, no external market feed, and no randomness, so the capability is safe to pin in continuous-integration checks and reproducible lesson materials. Each successful request is one billable item at $0.002; failed validation does not consume balance. Accepted series stay bounded for Worker isolates: up to one hundred thousand points cover classroom tables, daily price histories, and typical analytics excerpts while still finishing quickly. Results are fully deterministic: the same series and factor always yield the same ema array and last_ema. Wire the endpoint into charting helpers, quantitative tutorials, quality-control dashboards, coding-interview sandboxes, or preprocessing steps where you already prefer a closed-form EMA recurrence over opaque black-box smoothers that hide their seed rule or silently clamp α.