Allan Variance Calculation

Precision Stability Analysis

Compute Allan variance and Allan deviation from a sequence of fractional frequency deviation samples. Paste your data, choose averaging parameters, and visualize how stability changes as averaging time increases.

Calculator Inputs

Formula used for non-overlapping Allan variance of averaged frequency data: σ²y(τ) = 1 / [2(K – 1)] × Σ (ȳk+1 – ȳk)², where each ȳk is the mean of m adjacent samples and τ = mτ₀.

Results

Expert Guide to Allan Variance Calculation

Allan variance calculation is one of the most important techniques in frequency metrology, inertial sensor analysis, timing systems, and oscillator characterization. Engineers use it because many real-world noise processes are not well described by ordinary variance. Standard variance tends to diverge or become misleading when a data source includes drift, random walk, flicker noise, or other non-stationary components. Allan variance, by contrast, was designed specifically to evaluate how stability changes with averaging time. That makes it ideal for clocks, gyroscopes, accelerometers, crystal oscillators, rubidium references, GPS disciplined oscillators, and laboratory frequency standards.

In practical terms, Allan variance tells you how much one averaged measurement block differs from the next averaged measurement block when the averaging time is τ. If the difference between adjacent block averages is small, the system is stable at that averaging interval. If the difference is large, stability is poor. The square root of Allan variance is called Allan deviation, and engineers often prefer plotting Allan deviation on a log-log chart because the resulting slope patterns reveal the dominant noise type.

This calculator is built for users who already have a sampled sequence of fractional frequency deviation values. It computes non-overlapping Allan variance across multiple averaging factors and displays both the numeric summary and a chart of Allan deviation versus averaging time. That chart is often the fastest way to identify whether short-term instability is dominated by white noise, whether the midrange region is limited by flicker processes, or whether long-term drift and environmental effects begin to dominate.

What Allan Variance Measures

Suppose you measure the fractional frequency deviation of an oscillator once every second. A simple average of the full record gives one estimate of offset, but it does not tell you how stability behaves at 1 second, 10 seconds, 100 seconds, or 1000 seconds. Allan variance fills that gap by grouping the data into windows of length m samples, averaging each window, and then comparing one average with the next. The averaging time is:

τ = m × τ₀

where τ₀ is the sample interval. For a sequence of fractional frequency values y[k], the non-overlapping Allan variance is commonly written as:

σ²y(τ) = 1 / [2(K – 1)] × Σ (ȳk+1 – ȳk)²

Here, ȳk is the average of each block and K is the number of full blocks available for a given m. The Allan deviation is simply:

σy(τ) = √σ²y(τ)

Because Allan deviation is calculated over multiple averaging times, it reveals how averaging improves or worsens performance. White frequency noise typically averages down with increasing τ, while random walk or drift may eventually cause the curve to flatten or rise.

Why not just use standard deviation?

• Standard deviation assumes a stationary process and can fail on data with drift or random walk.
• Many clock and sensor error sources are scale-dependent, meaning the noise looks different at different averaging times.
• Allan deviation provides direct insight into short-term, mid-term, and long-term stability.
• Noise type identification often depends on the slope of the Allan deviation curve rather than a single summary number.

Step-by-Step Allan Variance Calculation Workflow

1. Collect a uniformly sampled time series. The data must be measured at a known sample interval. Uneven sampling requires preprocessing or different estimators.
2. Convert to the correct metric. In many oscillator applications, the series should be fractional frequency deviation, not raw frequency in hertz. If your instrument provides normalized offset directly, you can use those values.
3. Select an averaging factor m. For m = 1, the averaging time equals the sample period. Larger m values represent longer averaging windows.
4. Form block averages. Average each consecutive group of m samples to create a lower-rate sequence.
5. Difference adjacent block averages. Compute the squared difference between each neighboring pair.
6. Average the squared differences and divide by 2. This gives Allan variance at that τ.
7. Repeat for multiple m values. A full Allan deviation curve is much more informative than a single point.
8. Interpret the slopes. The shape of the curve points toward white frequency noise, flicker noise, random walk, rate random walk, bias instability, or environmental drift depending on the application.

Minimum data requirements

As a rule, you need enough samples to support at least two adjacent blocks for a given averaging factor. In practice, more is better. If your record is too short, long-τ estimates become noisy because very few averaged blocks are available. That is why serious metrology work often uses long observation windows and reports confidence intervals or error bars alongside the Allan deviation plot.

How to Read an Allan Deviation Plot

An Allan deviation chart is usually displayed on logarithmic axes. The x-axis is averaging time τ, and the y-axis is Allan deviation σy(τ). Different noise types produce different slopes. Engineers do not merely look for the lowest point; they also examine how the line moves across decades of averaging time.

• Downward slope: averaging is reducing random noise and improving stability.
• Flat region: a flicker or bias-limited mechanism may be dominant.
• Upward slope: longer averaging is no longer helping because drift, random walk, environmental effects, or control-loop behavior are taking over.
• Lowest point: often the most useful averaging time if you want the best practical stability from the system.

Noise Process	Typical Allan Deviation Trend	Interpretation in Practice
White frequency noise	Falls roughly with τ^-1/2	Common in short-term oscillator measurements and quantization-limited counters
Flicker frequency noise	Approximately flat versus τ	Often seen in real oscillators over intermediate averaging times
Random walk frequency noise	Rises roughly with τ^+1/2	Long-term environmental or control-system instability can appear this way
Bias instability in IMUs	Broad flat minimum region	Useful for estimating gyro or accelerometer bias stability

Representative Stability Statistics from Published Institutional Sources

The table below summarizes representative performance figures commonly cited in institutional and technical literature. Exact values depend on model, environmental control, and measurement method, but these ranges are useful for context when interpreting your own Allan variance output.

Reference Type	Representative Short-Term Stability	Longer-Term or Accuracy Statistic	Context
Laboratory cesium fountain standard	On the order of 1 × 10^-13 at 1 s	Systematic uncertainty near 1 × 10^-16	NIST fountain standards have reported total uncertainties in the low 10^-16 range, illustrating state-of-the-art long-term accuracy.
Hydrogen maser	Commonly around 1 × 10^-13 to a few × 10^-13 at 1 s	Excellent mid-term stability over 10^2 to 10^4 s	Widely used in timing labs because of very strong short- and mid-term stability.
Premium OCXO	Often around 1 × 10^-12 to 1 × 10^-11 at 1 s	Can improve significantly with averaging before aging dominates	High-grade quartz oscillators are strong practical references but do not reach atomic standards.
GPS disciplined oscillator	Usually worse than a free-running OCXO at very short τ	Can outperform quartz alone at long τ because GPS constrains drift	Useful example of why Allan deviation must be inspected over multiple averaging times.

A second important comparison is found in inertial sensors. Navigation-grade gyroscopes can show angle random walk values in the order of 0.001°/√hr to 0.01°/√hr, while many MEMS gyros are closer to 0.05°/√hr or much larger depending on grade. Allan variance is routinely used to estimate angle random walk, bias instability, and rate random walk from long static datasets.

Sensor or Oscillator Class	Representative Published Range	What Allan Analysis Helps Extract
Consumer MEMS gyro	Bias instability often several °/hr to tens of °/hr	Bias floor, angle random walk, repeatability limits
Industrial MEMS gyro	Bias instability often below 1 °/hr to a few °/hr	Mid-range optimum averaging time for navigation filtering
Navigation-grade gyro	Bias instability may reach 0.01 °/hr class or better	Long-term drift terms and calibration strategy
Precision frequency reference	Ranges from 10^-11 class quartz to 10^-16 class atomic accuracy	Short-term noise, flicker floor, long-term drift crossover

Common Mistakes in Allan Variance Calculation

• Using raw phase data with a frequency-data formula. Phase and frequency measurements require related but different forms of the estimator.
• Ignoring sample interval. If τ₀ is wrong, every Allan variance point is assigned to the wrong averaging time.
• Mixing units. Fractional frequency deviation, hertz, radians, and angular rate must be handled consistently.
• Overinterpreting long-τ points. The tail of the plot may be based on very few averages and therefore have high uncertainty.
• Not removing obvious gross errors. Missing values, slips, and instrument resets can distort the curve dramatically.
• Assuming one number tells the whole story. Allan deviation is valuable precisely because it is a function of averaging time, not a single scalar statistic.

When to use overlapping Allan variance

This calculator uses a non-overlapping method for clarity and speed. In high-precision work, overlapping Allan variance is often preferred because it uses more of the data and reduces estimator uncertainty, especially at larger τ. If you need publication-grade uncertainty performance, overlapping forms, confidence bounds, and drift modeling are recommended.

Best Practices for Better Results

1. Record enough data to cover the longest averaging time you care about by a wide margin.
2. Stabilize the environment. Temperature, vibration, supply variation, and airflow can dominate long-term behavior.
3. Use clean timestamps and uniform sampling intervals.
4. Keep the measurement chain better than the device under test. Counters, ADCs, references, and software clocks can otherwise limit the result.
5. Plot on log-log scales and look for slope transitions rather than only the minimum point.
6. Compare the Allan curve before and after filtering or disciplining. The crossover between short-term and long-term improvement is often where design tradeoffs become obvious.

For deeper standards and institutional references, consult authoritative resources such as the National Institute of Standards and Technology time and frequency materials, the NIST handbook on frequency stability analysis, and educational material from universities such as UC Santa Barbara instructional notes on Allan variance. These sources are especially useful if you need formal definitions, confidence intervals, modified Allan deviation, or phase-noise conversion methods.

Final Takeaway

Allan variance calculation is a foundational tool for anyone evaluating stability across time scales. It reveals whether averaging helps, where the optimal averaging interval lies, and what physical noise process is likely dominating the measurement. Whether you are tuning a frequency reference, characterizing a gyroscope, validating a timing subsystem, or comparing control-loop designs, Allan deviation provides actionable insight that ordinary variance simply cannot. Use the calculator above to build intuition quickly, then move to overlapping methods and uncertainty-aware workflows when your application demands metrology-grade rigor.