# Moving average

A moving average is a statistic computed by averaging successive subsets of a data series, producing a new series that tracks the underlying level while suppressing short-term fluctuation. It is widely applied in statistics, signal processing, econometrics, and mathematical finance.<sup>[1](https://ar5iv.labs.arxiv.org/html/1206.3610)</sup> The simplest form, the simple moving average (SMA), weighs all observations in the window equally: the average of the values 3, 4, and 5 is 4.<sup>[2](https://www.itl.nist.gov/div898/handbook/pmc/section4/pmc42.htm)</sup>

| Key facts | |
|---|---|
| Simple moving average | Arithmetic mean of the last k observations plus the current one, spanning k + 1 values; all weighted equally<sup>[3](https://smallake.kr/wp-content/uploads/2016/04/SSRN-id2585056.pdf)</sup><sup> • </sup><sup>[2](https://www.itl.nist.gov/div898/handbook/pmc/section4/pmc42.htm)</sup> |
| Lag | A causal SMA has group delay (M − 1)/2 samples; forecasting notes also describe average data age as (k + 1)/2 periods, using a different window-counting convention<sup>[4](https://numerical.recipes/whp/notes/movingaverages.pdf)</sup><sup> • </sup><sup>[5](https://people.duke.edu/~rnau/411avg.htm)</sup> |
| Exponential variant | First-order recursive filter \( y_t = \alpha x_t + (1-\alpha) y_{t-1} \) with geometrically decaying weights; average data age 1/α periods<sup>[6](https://dsp-workshop.nl/basics/08-smoothing/index.html)</sup><sup> • </sup><sup>[5](https://people.duke.edu/~rnau/411avg.htm)</sup> |
| Signal-processing identity | The SMA is convolution with a rectangular pulse of unit area, a low-pass filter with group delay (N − 1)/2 samples and about 13 dB of sidelobe rejection<sup>[7](https://jensd.dk/doc/filter/dsp_book_ch15.pdf)</sup><sup> • </sup><sup>[6](https://dsp-workshop.nl/basics/08-smoothing/index.html)</sup> |
| Cost | A running-sum update costs one addition and one subtraction per sample regardless of window length; the cumulative average updates in O(1) via \( \mathrm{CMA}_{n+1} = (x_{n+1} + n \cdot \mathrm{CMA}_n)/(n+1) \)<sup>[6](https://dsp-workshop.nl/basics/08-smoothing/index.html)</sup><sup> • </sup><sup>[8](https://mathresearch.utsa.edu/wiki/index.php?title=Weighted_moving_average_graphs)</sup> |
| Finance convention | The 50-day and 200-day moving averages are widely followed; a 50-day SMA lags price by about 24.5 trading days and a 200-day SMA by about 99.5<sup>[9](https://www.investopedia.com/terms/m/movingaverage.asp)</sup><sup> • </sup><sup>[10](https://vankyle00.github.io/trading-models/docs/models/01-sma-crossover-spy.html)</sup> |

## Variants and formulas

**Simple.** The SMA of order m estimates the level at time t by averaging m consecutive observations. In [trend estimation](https://www.edgepedia.org/trend-estimation) the order is usually odd, m = 2k + 1, so the window is symmetric: \( \hat{T}_t = (1/m) \sum_{j=-k}^{k} y_{t+j} \), averaging values within k periods of t.<sup>[11](https://otexts.com/fpp3/moving-averages.html)</sup> In finance the window is trailing, using the current price and k lagged prices, k + 1 observations in all.<sup>[3](https://smallake.kr/wp-content/uploads/2016/04/SSRN-id2585056.pdf)</sup>

**Weighted.** A linearly weighted moving average assigns weights that decrease in arithmetic progression: in LMA(k) the latest observation has weight k + 1, the second latest k, and so on down to 1.<sup>[3](https://smallake.kr/wp-content/uploads/2016/04/SSRN-id2585056.pdf)</sup> Combinations of simple averages also produce weighted averages: a 2×4-MA (a 2-MA of a 4-MA) is equivalent to a weighted 5-MA with weights [1/8, 1/4, 1/4, 1/4, 1/8], which yields a smoother trend estimate.<sup>[11](https://otexts.com/fpp3/moving-averages.html)</sup> Surveys of practice identify the simple, weighted, and exponential averages as the most commonly used types.<sup>[12](https://arxiv.org/html/2410.04149)</sup><sup> • </sup><sup>[3](https://smallake.kr/wp-content/uploads/2016/04/SSRN-id2585056.pdf)</sup>

**Exponential.** The exponential moving average (EMA), also called an exponentially weighted moving average (EWMA), is a first-order infinite impulse response filter: \( y[n] = \alpha x[n] + (1-\alpha) y[n-1] \), whose impulse response decays geometrically as \( h[n] = \alpha (1-\alpha)^n \).<sup>[6](https://dsp-workshop.nl/basics/08-smoothing/index.html)</sup> Expanding the recursion shows the weights directly: \( S_t = \alpha y_{t-1} + \alpha (1-\alpha) y_{t-2} + (1-\alpha)^2 S_{t-2} \), and so on, with geometrically decaying weight on older data.<sup>[13](https://www.itl.nist.gov/div898/handbook/pmc/section4/pmc431.htm)</sup> Hyndman writes the weights as \( a_j = \lambda (1-\lambda)^j \), which approximately sum to one for large t.<sup>[14](https://robjhyndman.com/papers/movingaverage.pdf)</sup> In mathematical terms the SMA uses a uniform kernel \( \rho(t) = \mathbb{1}_{[0,T]}(t)/T \) while the EMA uses an exponential kernel \( \rho(t) = \lambda e^{-\lambda t} \).<sup>[15](https://ar5iv.labs.arxiv.org/html/1710.09476)</sup> Unlike a fixed window, where a spike leaves entirely after the window passes, exponential smoothing counts every past value, so a spike never leaves the series entirely, it only fades.<sup>[16](https://michaelnocito.github.io/analyst-prep-kit/guides/moving-averages/)</sup>

Software parameterizes α in several ways. pandas accepts com, span, halflife, or alpha with \( 0 < \alpha \le 1 \); with adjust=True the weights are \( w_i = (1-\alpha)^i \) normalized, and with adjust=False the recursion is \( y_t = (1-\alpha) y_{t-1} + \alpha x_t \).<sup>[17](https://pandas.pydata.org/pandas-docs/stable/reference/api/pandas.DataFrame.ewm.html)</sup> Trading literature uses the multiplier \( 2/(\text{period} + 1) \), applied recursively as EMA = (Close − previous EMA) × multiplier + previous EMA.<sup>[9](https://www.investopedia.com/terms/m/movingaverage.asp)</sup><sup> • </sup><sup>[18](https://blog.quantinsti.com/moving-average-trading-strategies/)</sup>

**Cumulative.** The cumulative moving average (CMA) averages all data from the start through the current point, updated in constant time by \( \mathrm{CMA}_{n+1} = (x_{n+1} + n \cdot \mathrm{CMA}_n)/(n+1) \), avoiding recomputation over all stored data.<sup>[8](https://mathresearch.utsa.edu/wiki/index.php?title=Weighted_moving_average_graphs)</sup>

**Centered versus trailing.** A centered (symmetric) moving average uses data equally spaced on both sides of the computed point; odd-sized windows place an observation at the center.<sup>[8](https://mathresearch.utsa.edu/wiki/index.php?title=Weighted_moving_average_graphs)</sup> A trailing average labels the window at its right-hand edge and is therefore systematically late by half the window; that lag is the direct cost of using only information that would actually have been available at the time. A centered average removes the lag but uses future data, so it suits historical description such as seasonal decomposition, not live dashboards.<sup>[16](https://michaelnocito.github.io/analyst-prep-kit/guides/moving-averages/)</sup> In pandas, rolling window labels default to the right edge of the window, with a center keyword available.<sup>[19](https://pandas.pydata.org/docs/user_guide/window)</sup>

## How it works mathematically

Within signal processing, the moving average filter operates by convolving the input signal with a rectangular pulse whose area equals one; in a 5-point filter, each of five adjacent samples receives a weight of 1/5.<sup>[7](https://jensd.dk/doc/filter/dsp_book_ch15.pdf)</sup> This makes it optimal for one common task, reducing random noise while retaining a sharp step response, which suits time-domain encoded signals, but it is the worst filter for separating frequency bands.<sup>[7](https://jensd.dk/doc/filter/dsp_book_ch15.pdf)</sup> Its frequency response is sinc-like with the first null at \( f_s / N \), and the −13.3 dB sidelobe peak is intrinsic to the rectangular window: a longer window narrows the main lobe but leaves the sidelobe level unchanged.<sup>[6](https://dsp-workshop.nl/basics/08-smoothing/index.html)</sup> The sin(nω/2) numerator of the response forces ripples rather than a monotonic roll-off, so the SMA leaks some high-frequency energy back into the smoothed series.<sup>[10](https://vankyle00.github.io/trading-models/docs/models/01-sma-crossover-spy.html)</sup>

Every frequency component is delayed by the same (n − 1)/2 samples, so the SMA has constant group delay, but its frequency-dependent attenuation can also distort the signal's shape.<sup>[10](https://vankyle00.github.io/trading-models/docs/models/01-sma-crossover-spy.html)</sup>

**Constant-time updates.** The direct form costs M operations per sample, but the recursive running-sum form \( y[n] = (s[n-1] + x[n] - x[n-M])/M \) is identical in output and costs one addition and one subtraction per sample regardless of window length M; with a FIFO or circular buffer the filter needs only about 3 arithmetic steps.<sup>[6](https://dsp-workshop.nl/basics/08-smoothing/index.html)</sup><sup> • </sup><sup>[8](https://mathresearch.utsa.edu/wiki/index.php?title=Weighted_moving_average_graphs)</sup> The EMA needs only one state variable regardless of its effective window length.<sup>[6](https://dsp-workshop.nl/basics/08-smoothing/index.html)</sup>

## Choosing the window and parameters

The fundamental trade-off across all smoothing methods is smoothness versus lag: more aggressive smoothing removes more noise but delays features and can distort their shape, a consequence of the uncertainty principle relating time and frequency resolution.<sup>[6](https://dsp-workshop.nl/basics/08-smoothing/index.html)</sup> In statistical terms, smoothing with window half-width k trades bias against variance: bias grows with k (approximately proportional to k(k+1) times the second derivative of the underlying function) while variance falls as \( \sigma^2/(2k+1) \).<sup>[14](https://robjhyndman.com/papers/movingaverage.pdf)</sup>

The lag has a precise size. The estimator lag of the SMA is (M − 1)/2, the center of mass of its window of compact support.<sup>[4](https://numerical.recipes/whp/notes/movingaverages.pdf)</sup> Duke's forecasting notes express the same idea as an average data age of (k + 1)/2 periods, so the estimate of the local mean lags the true local mean by about that much.<sup>[5](https://people.duke.edu/~rnau/411avg.htm)</sup> (The two expressions count window members differently and differ by half a period; both appear in the literature.) For exponential smoothing the average age of the data is 1/α periods, so SES lags turning points by about 1/α periods: α = 0.5 gives a 2-period lag, α = 0.2 gives 5, and α = 0.1 gives 10.<sup>[5](https://people.duke.edu/~rnau/411avg.htm)</sup> For weekly business data, α around 0.2 to 0.3 is a reasonable starting point.<sup>[16](https://michaelnocito.github.io/analyst-prep-kit/guides/moving-averages/)</sup>

Two optimality results frame the choice. Among fixed-support M-sample weighted averages, the SMA maximizes the variance reduction factor, achieving \( N_{\mathrm{eff}} = M \).<sup>[4](https://numerical.recipes/whp/notes/movingaverages.pdf)</sup> The EWMA is instead the weighted average that minimizes timestep variance for a specified variance reduction factor, and its estimator lag \( L_1 = (N_{\mathrm{eff}} - 1)/2 \) equals the SMA's at the same \( N_{\mathrm{eff}} \).<sup>[4](https://numerical.recipes/whp/notes/movingaverages.pdf)</sup>

Practical conventions follow the horizon: shorter moving averages of 5 to 20 periods suit short-term trends and trading, 20 to 60 periods suit medium-term trends, and long-term investors often use 100 or more periods.<sup>[20](https://chartschool.stockcharts.com/table-of-contents/technical-indicators-and-overlays/technical-overlays/moving-averages-simple-and-exponential)</sup> Long windows buy stability at the price of inertia: a 252-day rolling estimate may remain dominated by conditions from many months ago.<sup>[21](https://www.cube.exchange/what-is/rolling-window-analysis)</sup> A 2025 Springer chapter critically assesses the standard lag-time measure itself, exposing its limitations in accurately detecting trend-changing times.<sup>[22](https://link.springer.com/chapter/10.1007/978-3-031-90907-8_2)</sup>

## By the numbers

A worked example shows the smoothing–lag trade in concrete terms. A 3-week trailing average of the values 660, 850, and 880 equals 796.67; smoothing reduces the weekly standard deviation from 236.81 (raw) to 100.56 with a 3-week window and 61.95 with a 5-week window, at the cost of 2 and 4 weeks lost at the start of the series.<sup>[16](https://michaelnocito.github.io/analyst-prep-kit/guides/moving-averages/)</sup> In trading terms, the 50-day SMA lags price by approximately 24.5 trading days and the 200-day SMA by approximately 99.5.<sup>[10](https://vankyle00.github.io/trading-models/docs/models/01-sma-crossover-spy.html)</sup> For a 20-day EMA the multiplier is \( 2/(20+1) = 0.0952 \).<sup>[9](https://www.investopedia.com/terms/m/movingaverage.asp)</sup>

## How it compares with other smoothers

**Exponential versus simple.** For a given average age of data, simple exponential smoothing is somewhat superior to a simple moving average because it places relatively more weight on the most recent observation; an SMA with 9 terms and SES with α = 0.2 both have an average data age of 5.<sup>[5](https://people.duke.edu/~rnau/411avg.htm)</sup> EMAs lag less than SMAs of the same length and turn before them.<sup>[20](https://chartschool.stockcharts.com/table-of-contents/technical-indicators-and-overlays/technical-overlays/moving-averages-simple-and-exponential)</sup> The cost is robustness: EMA may be more sensitive to outliers than SMA and WMA.<sup>[23](https://pdfs.semanticscholar.org/7062/610200e7fb0cf75a800f694da02b69d2e962.pdf)</sup>

**Shape-preserving alternatives.** Savitzky–Golay filters fit a local polynomial over a window and preserve peaks, shoulders, and other features better than a moving average of the same length.<sup>[6](https://dsp-workshop.nl/basics/08-smoothing/index.html)</sup> Relatives of the moving average filter, including the Gaussian, Blackman, and multiple-pass moving average, have better frequency-domain performance at the expense of increased computation time.<sup>[7](https://jensd.dk/doc/filter/dsp_book_ch15.pdf)</sup> Tapered weights that decay smoothly toward the past while summing to unity can also mitigate the SMA's lag and discontinuity features.<sup>[4](https://numerical.recipes/whp/notes/movingaverages.pdf)</sup>

**Robustness and state-space methods.** When the goal is trend estimation, the moving average is susceptible to rare events such as rapid shocks or anomalies; the simple moving median is a more robust trend estimate.<sup>[8](https://mathresearch.utsa.edu/wiki/index.php?title=Weighted_moving_average_graphs)</sup> Median smoothing is a nonlinear, statistically stable procedure that is robust to local outliers and, with an optimally chosen window, preserves sharp transitions without distortion.<sup>[24](https://ceur-ws.org/Vol-3970/PAPER18.pdf)</sup><sup> • </sup><sup>[31](https://www.gnu.org/software/gsl/doc/html/filter.html)</sup> The Kalman filter offers a model-based alternative but can require significant computation for large datasets or complex models and relies on assumptions about the statistical properties of the data, such as the distribution of noise.<sup>[23](https://pdfs.semanticscholar.org/7062/610200e7fb0cf75a800f694da02b69d2e962.pdf)</sup>

## Applications in practice

**Finance.** The 50-day and 200-day moving averages are widely followed by investors and traders and are considered important trading signals.<sup>[9](https://www.investopedia.com/terms/m/movingaverage.asp)</sup> Crossovers carry the signal: upward momentum is confirmed with a bullish crossover, when a short-term average crosses above a longer-term one.<sup>[9](https://www.investopedia.com/terms/m/movingaverage.asp)</sup> The death cross occurs when the 50-day SMA moves below the 200-day SMA, a bearish signal, and the golden cross occurs when the 50-day SMA crosses above the 200-day SMA, a bullish signal.<sup>[25](https://www.investopedia.com/ask/answers/071414/whats-difference-between-moving-average-and-weighted-moving-average.asp)</sup> Definitions vary across practitioners: Fidelity describes the golden cross as the 50-day exponential moving average crossing above a 200-day moving average, and cautions that a golden cross or death cross does not suggest that one should mechanically buy or sell.<sup>[26](https://www.fidelity.com/viewpoints/active-investor/moving-averages)</sup> Moving averages also serve as price support and resistance levels, and a common chart setting applies a 50-day EMA and a 200-day EMA together.<sup>[26](https://www.fidelity.com/viewpoints/active-investor/moving-averages)</sup>

Backtest evidence tempers the signal's value. A study of the 50/200-day SMA crossover on the Nifty 50 using 3,077 daily observations from January 2010 to June 2022 documented eleven Golden Cross and eleven Death Cross events; the mechanical strategy underperformed buy-and-hold substantially on raw return (4.01% versus 9.90% CAGR) while achieving materially lower realized volatility (12.54% versus 22.52% annualized), netting a lower [Sharpe ratio](https://www.edgepedia.org/sharpe-ratio) in that sample.<sup>[27](https://zenodo.org/records/21308597)</sup> The crossover signal always has a time lag and therefore cannot be considered a true leading indicator.<sup>[27](https://zenodo.org/records/21308597)</sup>

**Forecasting and signals.** The simple average of all past observations is a useful forecasting estimate only when there are no trends; with trends, estimates that take the trend into account are needed.<sup>[2](https://www.itl.nist.gov/div898/handbook/pmc/section4/pmc42.htm)</sup> In signal processing, moving averages reduce noise in audio, video, and sensor data and act as simple low-pass filters removing high-frequency components.<sup>[12](https://arxiv.org/html/2410.04149)</sup>

## Edge behavior and missing data

Every finite-window smoother has a boundary problem at the start and end of the signal; mirror padding, constant extrapolation, or shrinking the window are the usual fixes, and none is fully satisfactory.<sup>[6](https://dsp-workshop.nl/basics/08-smoothing/index.html)</sup> In forecasting terms, moving averages do not allow estimates of the trend near the ends of the series.<sup>[14](https://robjhyndman.com/papers/movingaverage.pdf)</sup> In pandas, rolling(N).mean() produces NaN for the first N − 1 rows because not enough data exists.<sup>[28](https://www.codegiz.com/blog/pandas-for-finance-episode-11-rolling-windows-moving-avg-bollinger-volatility-drawdown/)</sup> One comparative study addresses edge loss by sequentially applying windows of gradually increasing size, with odd window sizes in practice.<sup>[24](https://ceur-ws.org/Vol-3970/PAPER18.pdf)</sup> For exponentially weighted calculations, pandas exposes min_periods (default 0) and ignore_na (default False), which controls whether missing values are ignored when calculating weights.<sup>[17](https://pandas.pydata.org/pandas-docs/stable/reference/api/pandas.DataFrame.ewm.html)</sup>

## What has changed in practice

Modern tooling has made the mechanics cheaper and more uniform. pandas supports four windowing types: rolling, weighted, expanding, and exponentially weighted; rolling windows support time-based windows, and online operations for exponentially weighted windows were added as of version 1.3.<sup>[19](https://pandas.pydata.org/docs/user_guide/window)</sup> The .rolling() object computes statistics including count, sum, mean, median, variance, standard deviation, correlation, covariance, quantiles, and custom functions through apply or aggregate, alongside .expanding() and .ewm().<sup>[21](https://www.cube.exchange/what-is/rolling-window-analysis)</sup> The three window memories differ in kind: a fixed window has a hard cutoff and a step change at the boundary, an expanding window has growing memory with no discontinuity, and an exponentially weighted window has fading memory with smooth decay.<sup>[21](https://www.cube.exchange/what-is/rolling-window-analysis)</sup>

Analytical engines now compete on speed. DuckDB computes moving averages in SQL with AVG() OVER ... ROWS BETWEEN; a benchmark reports DuckDB 3 to 8 times faster than pandas groupby().rolling() for moving averages, with 60 to 80 percent memory savings from columnar storage.<sup>[29](https://duckdblab.org/en/post/duckdb-financial-signal-engine/)</sup> On the research side, a 2025 Springer chapter reexamines the standard lag-time measure of moving averages, and post-2023 empirical work on the [S&P 500](https://www.edgepedia.org/s-and-p-500) finds trailing averages' aggregate adaptation share falling below 50% in 5 of 18 rolling decade-windows (one negative, 1991–2001), with the modern era (1990–2026) showing mild but statistically significant attraction toward a rate of 53.1%.<sup>[22](https://link.springer.com/chapter/10.1007/978-3-031-90907-8_2)</sup><sup> • </sup><sup>[30](https://papers.ssrn.com/sol3/papers.cfm?abstract_id=7342898)</sup>

## References

1. [On moving averages (arXiv:1206.3610)](https://ar5iv.labs.arxiv.org/html/1206.3610)
2. [NIST/SEMATECH e-Handbook: What are Moving Average or Smoothing Techniques?](https://www.itl.nist.gov/div898/handbook/pmc/section4/pmc42.htm)
3. [Averages (SSRN working paper)](https://smallake.kr/wp-content/uploads/2016/04/SSRN-id2585056.pdf)
4. [Principled Derivation of Some Moving Average Estimators (Numerical Recipes note)](https://numerical.recipes/whp/notes/movingaverages.pdf)
5. [Moving average and exponential smoothing models (Robert Nau, Duke University)](https://people.duke.edu/~rnau/411avg.htm)
6. [Smoothing — DSP Workshop](https://dsp-workshop.nl/basics/08-smoothing/index.html)
7. [The Scientist and Engineer's Guide to Digital Signal Processing, Chapter 15: Moving Average Filters](https://jensd.dk/doc/filter/dsp_book_ch15.pdf)
8. [Weighted moving average graphs (UTSA Mathematics)](https://mathresearch.utsa.edu/wiki/index.php?title=Weighted_moving_average_graphs)
9. [Moving Average (MA): Purpose, Uses, Formula, and Examples (Investopedia)](https://www.investopedia.com/terms/m/movingaverage.asp)
10. [The Moving-Average Crossover as a Lagged Trend Filter](https://vankyle00.github.io/trading-models/docs/models/01-sma-crossover-spy.html)
11. [Forecasting: Principles and Practice (3rd ed.), §3.3 Moving averages](https://otexts.com/fpp3/moving-averages.html)
12. [Mov-Avg: Codeless time series analysis using moving averages (arXiv, 2024)](https://arxiv.org/html/2410.04149)
13. [NIST/SEMATECH e-Handbook: Single Exponential Smoothing](https://www.itl.nist.gov/div898/handbook/pmc/section4/pmc431.htm)
14. [Moving averages (Rob J. Hyndman, technical paper)](https://robjhyndman.com/papers/movingaverage.pdf)
15. [A Mathematical Analysis of Technical Analysis (arXiv)](https://ar5iv.labs.arxiv.org/html/1710.09476)
16. [Moving Averages: Smoothing a Series Without Smoothing Away the Truth — Analyst Prep Kit](https://michaelnocito.github.io/analyst-prep-kit/guides/moving-averages/)
17. [pandas.DataFrame.ewm documentation](https://pandas.pydata.org/pandas-docs/stable/reference/api/pandas.DataFrame.ewm.html)
18. [Moving Average Crossover Strategies (QuantInsti)](https://blog.quantinsti.com/moving-average-trading-strategies/)
19. [pandas Windowing operations user guide](https://pandas.pydata.org/docs/user_guide/window)
20. [Moving Averages — Simple and Exponential (ChartSchool)](https://chartschool.stockcharts.com/table-of-contents/technical-indicators-and-overlays/technical-overlays/moving-averages-simple-and-exponential)
21. [What Is Rolling Window Analysis? (Cube.Exchange)](https://www.cube.exchange/what-is/rolling-window-analysis)
22. [Basics of Moving Averages (Springer chapter, 2025)](https://link.springer.com/chapter/10.1007/978-3-031-90907-8_2)
23. [Comparison of the Effectiveness of Time Series Analysis Methods: SMA, WMA, EMA, EWMA, and Kalman Filter](https://pdfs.semanticscholar.org/7062/610200e7fb0cf75a800f694da02b69d2e962.pdf)
24. [Detection of trends in non-stationary time series: a comparison of moving window smoothing methods (CEUR-WS)](https://ceur-ws.org/Vol-3970/PAPER18.pdf)
25. [Master Moving Averages: Simple, Weighted, and Exponential Techniques (Investopedia)](https://www.investopedia.com/ask/answers/071414/whats-difference-between-moving-average-and-weighted-moving-average.asp)
26. [Stock signals | US stocks indicator & average (Fidelity)](https://www.fidelity.com/viewpoints/active-investor/moving-averages)
27. [Trend-Following Signals in an Emerging Market Index: A Moving-Average Crossover Analysis of the Nifty 50 (2010–2022) (Zenodo)](https://zenodo.org/records/21308597)
28. [Pandas for Finance: Rolling Windows (CodeGiz)](https://www.codegiz.com/blog/pandas-for-finance-episode-11-rolling-windows-moving-avg-bollinger-volatility-drawdown/)
29. [Building a Financial Signal Engine with DuckDB (DuckDB Lab)](https://duckdblab.org/en/post/duckdb-financial-signal-engine/)
30. [Moving Averages Follow Price: A Mathematical Proof and Empirical Validation of the Adaptation Property in Trailing Averages (SSRN)](https://papers.ssrn.com/sol3/papers.cfm?abstract_id=7342898)
31. [Digital Filtering, GSL 2.8 documentation](https://www.gnu.org/software/gsl/doc/html/filter.html)

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*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Statistics and probability › Applied, official, and domain statistics › Applied, official, and domain statistics*

*Initially written Sep 17, 2026 · Reviewed: Oct 15, 2026 · Edited: Oct 15, 2026 · Last review: Oct 15, 2026*

*Copyright 2026 EdgeChat AI, a subsidiary of Biostate AI.*

License: Edgepedia Community License 1.0, https://www.edgepedia.org/edgepedia/license