modified total variance and the Theo variance. These distinguish themselves in better use of statistics for improved confidence bounds or ability to handle linear frequency drift. Confidence intervals and equivalent degrees of freedom Statistical estimators will calculate an estimated value on the sample series used. The estimates may deviate from the true value and the range of values which for some probability will contain the true value is referred to as the confidence interval. The confidence interval depends on the number of observations in the sample series, the dominant noise type, and the estimator being used. The width is also dependent on the statistical certainty for which the confidence interval values forms a bounded range, thus the statistical certainty that the true value is within that range of values. For variable-τ estimators, the τ0 multiple n is also a variable. Confidence interval The confidence interval can be established using chi-squared distribution by using the distribution of the sample variance: where s2 is the sample variance of our estimate, σ2 is the true variance value, df is the degrees of freedom for the estimator, and χ2 is the degrees of freedom for a certain probability. For a 90% probability, covering the range from the 5% to the 95% range on the probability curve, the upper and lower limits can be found using the inequality which after rearrangement for the true variance becomes Effective degrees of freedom The degrees of freedom represents the number of free variables capable of contributing to the estimate. Depending on the estimator and noise type, the effective degrees of freedom varies. Estimator formulas depending on N and n has been found empirically: {| class="wikitable" |+ Allan variance degrees of freedom |- !Noise type !degrees of freedom |- |white phase modulation (WPM) | |- |flicker phase modulation (FPM) | |- |white frequency modulation (WFM) | |- |flicker frequency modulation (FFM) | |- |random-walk frequency modulation (RWFM) | |} Power-law noise The Allan variance will treat various power-law noise types differently, conveniently allowing them to be identified and their strength estimated. As a convention, the measurement system width (high corner frequency) is denoted fH. As found in and in modern forms. The Allan variance is unable to distinguish between WPM and FPM, but is able to resolve the other power-law noise types. In order to distinguish WPM and FPM, the modified Allan variance needs to be employed. The above