completely allow all types of noise to be distinguished, it provided a means to meaningfully separate many noise-forms for time-series of phase or frequency measurements between two or more oscillators. Allan provided a method to convert between any M-sample variance to any N-sample variance via the common 2-sample variance, thus making all M-sample variances comparable. The conversion mechanism also proved that M-sample variance does not converge for large M, thus making them less useful. IEEE later identified the 2-sample variance as the preferred measure. An early concern was related to time- and frequency-measurement instruments that had a dead time between measurements. Such a series of measurements did not form a continuous observation of the signal and thus introduced a systematic bias into the measurement. Great care was spent in estimating these biases. The introduction of zero-dead-time counters removed the need, but the bias-analysis tools have proved useful. Another early aspect of concern was related to how the bandwidth of the measurement instrument would influence the measurement, such that it needed to be noted. It was later found that by algorithmically changing the observation , only low values would be affected, while higher values would be unaffected. The change of is done by letting it be an integer multiple of the measurement timebase : The physics of crystal oscillators were analyzed by D. B. Leeson, and the result is now referred to as Leeson's equation. The feedback in the oscillator will make the white noise and flicker noise of the feedback amplifier and crystal become the power-law noises of white frequency noise and flicker frequency noise respectively. These noise forms have the effect that the standard variance estimator does not converge when processing time-error samples. This mechanics of the feedback oscillators was unknown when the work on oscillator stability started, but was presented by Leeson at the same time as the set of statistical tools was made available by David W. Allan. For a more thorough presentation on the Leeson effect, see modern phase-noise literature. Interpretation of value Allan variance is defined as one half of the time average of the squares of the differences between successive readings of the frequency deviation sampled over the sampling period. The Allan variance depends on the time period used between samples, therefore, it is a function of the sample period, commonly denoted as τ, likewise the distribution being measured, and is displayed as a