It means that a particular approximation is rigorously defined on first principles ( quantum theory ) and then solved within an error margin that is qualitatively known beforehand.

Another way of proving the chain rule is to measure the error in the linear approximation determined by the derivative.

Here the left-hand side represents the true difference between the value of g at a and at, whereas the right-hand side represents the approximation determined by the derivative plus an error term.

That expansion, in turn, serves as the starting point for one of the derivations of precise error estimates for Stirling's approximation of the factorial function.

Such algorithms trade the approximation error for increased speed or other properties.

As a quick approximation of the error function, the first 2 terms of the Taylor series can be used:

This approximation works out to within a 1 % error margin for up to 14 outs.

) Instead, the theorem states that x / ln ( x ) approximates π ( x ) in the sense that the relative error of this approximation approaches 0 as x approaches infinity.

The prime number theorem is equivalent to the statement that the nth prime number p < sub > n </ sub > is approximately equal to n ln ( n ), again with the relative error of this approximation approaching 0 as n approaches infinity.

Following Kolmogorov's work in the 1950s, advanced statistics uses approximation theory and functional analysis to quantify the error of approximation.

In this approach, the metric geometry of probability distributions is studied ; this approach quantifies approximation error with, for example, the Kullback – Leibler distance, Bregman divergence, and the Hellinger distance.

Taylor's theorem gives quantitative estimates on the error in this approximation.

The error that corresponds to the sinc-function approximation is referred to as interpolation error.

The error in this approximation can be derived from Taylor's theorem.

The discrepancy between the exact mathematical value and the computed value is called the approximation error.

For satellites having a mass less than M μ < sub > err </ sub >/ μ ≈ 10 < sup > 15 </ sup > kg, that is, over a billion times that of the ISS, the error due to the approximation is smaller than the error on the universal geocentric gravitational constant ( and thus negligible ).

When the illuminant is not a point source, the inverse square rule is often still a useful approximation ; when the size of the light source is less than one-fifth of the distance to the subject, the calculation error is less than 1 %.

Alongside his development of the Taylor series of the trigonometric functions, he also estimated the magnitude of the error terms created by truncating these series and gave a rational approximation of an infinite series.

Indeed, there are several versions of it applicable in different situations, and some of them contain explicit estimates on the approximation error of the function by its Taylor-polynomial.

While this reasoning error often can produce a relatively good approximation of the correct hypothesis probability value, it can lead to a completely wrong result and wrong conclusion in case the base rate is very low and the reliability of the test is not perfect.

A mean of zero is needed for finding a basis that minimizes the mean square error of the approximation of the data.

Loss of the positive-definite condition through round-off error is avoided if rather than updating an approximation to the inverse of the Hessian, one updates the Cholesky decomposition of an approximation of the Hessian matrix itself.

There is no room for error or waste.

To my knowledge, Lincoln remains the only Head of State and Commander-in-Chief who, while fighting a fearful war whose issue was in doubt, proved man enough to say this publicly -- to give his foe the benefit of the fact that in all human truth there is some error, and in all our error, some truth.

So great a man could not but understand, too, that the thing that moves men to sacrifice their lives is not the error of their thought, which their opponents see and attack, but the truth which the latter do not see -- any more than they see the error which mars the truth they themselves defend.

Their great error is to mingle the responses typical of each of the three types of change.

Without a precise knowledge of Germanic philology, however, it is debatable whether their use was not more often a source of confusion and error than anything else.

The observational data determining the concentration distribution have a range of error which is magnified in the extension into the micrometeorite region.

Another problem in the area of federal-state relationships is this: what constitutes reversible error in a state decision??

If the polarity is correct, the platform will turn until the heading error angle is zero.

If the drift error is systematic, it can be canceled with a bias circuit which can be arranged and adjusted to supply the required compensating current.

The point is at a latitude **yl, and the platform is at an error in heading east.

Such talk is seriously in error.

He has a pleasant sense of humor and is modest enough to admit mistakes and even `` a cardinal error ''.

He contends this idea doesn't conflict with experiments on which the principle of conservation of matter and energy is based because some slight error must be assumed in such experiments.

It allows compact encoding, but is less reliable for data transmission ; an error in transmitting the shift code typically makes a long part of the transmission unreadable.

( It follows that the mean is also the best single predictor in the sense of having the lowest root mean squared error.

The Type II error rate is a function of several things including

the result of the effect and is partially random error.

Random error is typically assumed to be normally distributed with zero mean and a constant variance.

Random error is also called experimental error.

The number of degrees of freedom DF can be partitioned in a similar way: one of these components ( that for error ) specifies a chi-squared distribution which describes the associated sum of squares, while the same is true for " treatments " if there is no treatment effect.