Here, < sub > n </ sub > denotes the sample mean of the first n samples ( x < sub > 1 </ sub >, ..., x < sub > n </ sub >), s < sup > 2 </ sup >< sub > n </ sub > their sample variance, and σ < sup > 2 </ sup >< sub > n </ sub > their population variance.

Although the sample values x < sub > i </ sub > will be concentrated about the central value x < sub > 0 </ sub >, the sample mean will become increasingly variable as more samples are taken, because of the increased likelihood of encountering sample points with a large absolute value.

In fact, the distribution of the sample mean will be equal to the distribution of the samples themselves ; i. e., the sample mean of a large sample is no better ( or worse ) an estimator of x < sub > 0 </ sub > than any single observation from the sample.

Reconstruction of the original signal is an interpolation process that mathematically defines a continuous-time signal x ( t ) from the discrete samples x ( nT ) and at times in between the sample instants nT.

The Poisson summation formula shows that the samples, x ( nT ), of function x ( t ) are sufficient to create a periodic summation of function X ( f ).

The time error sample series let N denote the number of samples ( x < sub > 0 </ sub > ... x < sub > N-1 </ sub >) in the series.

A common choice of estimate is the one provided by the principle of maximum likelihood, and using this yields the predictive density over a future sample x < sub > n + 1 </ sub >, conditioned on the observed samples x = ( x < sub > 1 </ sub >, ..., x < sub > n </ sub >) given by

The sampling theorem states that, under certain limiting conditions, a function x ( t ) can be recovered exactly from its samples, x

If we assume that P < sub > X </ sub >( x ) is Gaussian with variance σ < sup > 2 </ sup >, and if we assume that successive samples of the signal X are stochastically independent ( or, if you like, the source is memoryless, or the signal is uncorrelated ), we find the following analytical expression for the rate – distortion function:

For instance, we could take 8 samples in the x direction at x = 0. 1, 0. 2, ..., 0. 8 and 0. 9, and 8 samples in the y direction at similar coordinates.

A sufficient condition for recovering s ( t ) ( and therefore S ( ƒ )) from just these samples is that the non-zero portion of s ( t ) be confined to a known interval of duration P, which is the frequency domain dual of the Nyquist – Shannon sampling theorem.

Thus we have the important result that when a discrete data sequence, s, is proportional to samples of an underlying continuous function, s ( t ), one can deduce something about the continuous Fourier transform, S ( ƒ ).

* Welch's t test, a statistical test intended for use with two samples having possibly unequal variances.

Further samples of L. t.

A Gaussian process is a stochastic process X < sub > t </ sub >, t ∈ T, for which any finite linear combination of samples has a joint Gaussian distribution.

Overall, the robustness makes the MWW more widely applicable than the t test, and for large samples from the normal distribution, the efficiency loss compared to the t test is only 5 %, so one can recommend MWW as the default test for comparing interval or ordinal measurements with similar distributions.

where T represents the time interval at which a function s ( t ) is sampled, and 1 / T is the rate of samples / sec.

Often the x sequence represents the values ( aka samples ) of a continuous-time function, x ( t ), at discrete moments in time: t = nT, where T is the sampling interval ( in seconds ), and is the sampling rate ( samples per second ).

To understand this, consider the Poisson summation formula, which indicates that a periodic summation of function X ( f ) can be constructed from the samples of function x ( t ).< ref group =" note "> The same result can be obtained by observing that:

For seven samples ( with six degrees of freedom ) the t value for a 99 % confidence interval is 3. 14.

The data reported here are either from spectra from which the adsorbed water resonance could easily be eliminated or from spectra of samples evacuated and sealed off at 375-degrees-C which contain no adsorbed water.

groups are simply random samples of the same population.

The artists are Chenek " DJ Bean " ( turntables, samples and percussion ), Yaotl ( vocals, indigenous percussion ), Joe " Peps " ( bass, rattles ), Alonzo Beas ( guitars, synth ), Caxo ( drums, indigenous percussion ), and Bulldog ( vocals, flute ).

The peoples of the Tarim Basin in northern China are now known to be Caucasian and there were widespread attempts to keep western archeologists from testing samples such as deliberately replacing the mummies or decapitating them, even defacing ancient cave painting depictions of these people with large amounts of body hair and features such as red colored hair and blue eyes.

The samples they collected there, although their origin is still not certain, are, according to geologist Don Wilhelms, " a reasonable bet to be Descartes.

In the direct elemental analysis of solid samples, the new leaders are laser-induced breakdown and laser ablation mass spectrometry, and the related techniques with transfer of the laser ablation products into inductively coupled plasma.

Liquid or dissolved samples are typically used with flame atomizers.

The requirement for samples of glow discharge atomizers is that they are electrical conductors.

Consequently, atomizers are most commonly used in the analysis of metals and other conducting samples.

Atom probe samples are shaped to implicitly provide a highly curved electric potential to induce the resultant magnification, as opposed to direct use of lensing, such as via magnetic lenses.

The samples used in atom probe are usually a metallic or semi-conducting material, with the needle geometry produced by electropolishing, or focused ion beam methods.

Bioinformatics is very much involved in making sense of protein microarray and HT MS data ; the former approach faces similar problems as with microarrays targeted at mRNA, the latter involves the problem of matching large amounts of mass data against predicted masses from protein sequence databases, and the complicated statistical analysis of samples where multiple, but incomplete peptides from each protein are detected.

However, some virus samples are still available in Russian and American laboratories.

Procedures for testing hypotheses about probabilities ( using finite samples ) are due to Ramsey ( 1931 ) and de Finetti ( 1931, 1937, 1964, 1970 ).

After amplifying and then concentrating any PrP < sup > Sc </ sup >, the samples are labelled with a fluorescent dye using an antibody for specificity and then finally loaded into a micro-capillary tube.

Relatively pure samples of isolated elements are uncommon in nature.

While all of the 98 naturally occurring elements have been identified in mineral samples from the Earth's crust, only a small minority of elements are found as recognizable, relative pure minerals.

Census counts are necessary to adjust samples to be representative of a population by weighting them as is common in opinion polling.

Crossbows are used for shooting sports and bowhunting in modern archery and for blubber biopsy samples in scientific research.

For example, if n samples are taken from a Cauchy distribution, one may calculate the sample mean as:

Similarly, calculating the sample variance will result in values that grow larger as more samples are taken.

Knit and knotted textiles survive from very early periods, but there are no surviving samples of crocheted fabric in any ethnological collection, or archeological source prior to 1800.

Stool and swab samples collected in the acute stage of the disease, before antibiotics have been administered, are the most useful specimens for laboratory diagnosis.

* A " linear " filter is a linear transformation of input samples ; other filters are " non-linear ".