What is a moving average?

A moving average smooths a time series by averaging over a window of recent observations. It’s widely used in finance, operations, and forecasting.

What is the difference between SMA and EMA?

SMA gives equal weight to all values in the window, while EMA gives more weight to recent values using a smoothing factor α = 2/(n+1). EMA reacts faster to changes.

Which period should I use?

Shorter periods (e.g. 5, 10) react faster but are noisier; longer periods (e.g. 20, 50) are smoother but slower. It depends on your data and goal.

Moving Average Calculator (SMA & EMA)

Compute Simple Moving Average (SMA) and Exponential Moving Average (EMA) from your time series. Paste prices, sales, or any numeric series (oldest to newest), choose the period, and the tool will show the latest averages, a full table, and an inline chart.

Data (one value per line, oldest to newest)

SMA period

e.g. 3, 5, 10

Blue: original data; Orange: selected moving average. First 30 points shown.

Latest MA

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Current price/value

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Diff (value - MA)

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Method

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t	Value	MA	Diff

Formulas

Simple moving average (SMA):

\( \text{SMA}_t = \dfrac{X_t + X_{t-1} + \dots + X_{t-n+1}}{n} \)

Exponential moving average (EMA):

\( \alpha = \dfrac{2}{n+1} \quad\Rightarrow\quad \text{EMA}_t = \alpha X_t + (1 - \alpha)\text{EMA}_{t-1} \)

with initialization \( \text{EMA}_1 = X_1 \).

When to use SMA vs EMA

SMA: easy to interpret, good for smoothing.
EMA: reacts faster to recent moves (used a lot in trading, e.g. 12/26 EMA).

Common periods

Finance: 5, 10, 20, 50, 100, 200. Operations/time series: 3, 4, 6, 12 depending on seasonality.

Audit: Complete

Formula (LaTeX) + variables + units

This section shows the formulas used by the calculator engine, plus variable definitions and units.

Formula (extracted LaTeX)

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Formula (extracted text)

Simple moving average (SMA): \( \text{SMA}_t = \dfrac{X_t + X_{t-1} + \dots + X_{t-n+1}}{n} \) Exponential moving average (EMA): \( \alpha = \dfrac{2}{n+1} \quad\Rightarrow\quad \text{EMA}_t = \alpha X_t + (1 - \alpha)\text{EMA}_{t-1} \) with initialization \( \text{EMA}_1 = X_1 \).

Variables and units

No variables provided in audit spec.

Sources (authoritative):

NIST — Weights and measures — nist.gov · Accessed 2026-01-19
https://www.nist.gov/pml/weights-and-measures
FTC — Consumer advice — consumer.ftc.gov · Accessed 2026-01-19
https://consumer.ftc.gov/

Changelog

Version: 0.1.0-draft
Last code update: 2026-01-19

0.1.0-draft · 2026-01-19

Initial audit spec draft generated from HTML extraction (review required).
Verify formulas match the calculator engine and convert any text-only formulas to LaTeX.
Confirm sources are authoritative and relevant to the calculator methodology.

Verified by Ugo Candido on 2026-01-19
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