graph rather than a single number. A low Allan variance is a characteristic of a clock with good stability over the measured period. Allan deviation is widely used for plots (conventionally in log–log format) and presentation of numbers. It is preferred, as it gives the relative amplitude stability, allowing ease of comparison with other sources of errors. An Allan deviation of 1.3 at observation time 1 s (i.e. τ = 1 s) should be interpreted as there being an instability in frequency between two observations 1 second apart with a relative root mean square (RMS) value of 1.3. For a 10 MHz clock, this would be equivalent to 13 mHz RMS movement. If the phase stability of an oscillator is needed, then the time deviation variants should be consulted and used. One may convert the Allan variance and other time-domain variances into frequency-domain measures of time (phase) and frequency stability. Definitions M-sample variance The -sample variance is defined (here in a modernized notation form) as where is the clock reading (in seconds) measured at time , or with average fractional frequency time series where is the number of frequency samples used in variance, is the time between each frequency sample, and is the time length of each frequency estimate. An important aspect is that -sample variance model can include dead-time by letting the time be different from that of . An alternative (and equivalent) way to view this formula that makes the connection to the typical sample variance formula more explicit is obtained by multiplying by and dividing the 2 terms inside the curly braces by : Now, the coefficient can be interpreted as Bessel's correction to the biased sample variance which is what appears inside the curly braces in the form . Allan variance The Allan variance is defined as where denotes the expectation operator. This can be conveniently expressed as where is the observation period, is the nth fractional frequency average over the observation time . The samples are taken with no dead-time between them, which is achieved by letting Allan deviation Just as with standard deviation and variance, the Allan deviation is defined as the square root of the Allan variance: Supporting definitions Oscillator model The oscillator being analysed is assumed to follow the basic model of The oscillator is assumed to have a nominal frequency of , given in cycles per second (SI unit: hertz). The