... estimate.^2.1

Results are sometimes classified according to their use: indication based only on primary measures such as sample means or correlation coefficients; determination based on primary and secondary statistics, so that some estimate of uncertainty is obtained; and inference, in which a specific mathematical model is used to assess uncertainty quantitatively. Often, a considerable amount of information about the underlying distribution must be known (or assumed) before statistical inference is possible. Experimental results are usually appropriately quoted as determinations.

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... noise^2.2

I.e., the error that results when a continuously varying measurement is measured by a digital instrument that must round the measurement to the nearest digital value.

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... events^2.3

When, for example, the average measurement might be the possibly fractional value x but the actual value is an integer number of events

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... variations.^2.4

This is not part of the generally accepted standards, and is a difficult goal to implement because estimates of bias depend on judgments that are difficult to quantify.

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... quadrature,^2.5

i.e., $s^2=\sum_is_i^2$.

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... freedom^2.6

In the case of an average of n values, the number of degrees of freedom is n-1.

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... uncertainties.^2.7

Brackets denote multidimensional quantities and bold-face symbols denote matrices.

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....^2.8

These values do not necessarily minimize the sum of the squared deviations from $\{\overline{Y}\}$, the derived quantities, unless the relationship to the measured quantities is linear.

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... deviation:^3.1

The chisquare distribution is discussed in Chapter 7.

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... problem.^4.1

Linear fits to the logarithm of F are sometimes used, but usually this approach distorts the effects of the measurement errors on the fit.

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... uncertainty.^4.2

Cf. Orear, J., 1958: Notes on statistics for physicists, UCRL-8417, University of California Radiation Laboratory, Berkeley, CA.

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... variables^5.1

A set of measurements $\{y_1, y_2, y_3, \cdots\}$ of a scalar quantity are denoted here by $\{y_i\}$ or $\{y\}$. These measurements may be made a different times or otherwise at different values of the independent variables $\{{\bf x}\}$, where x_i is the ith occurrence of the possibly multidimensional vector of the independent variables.

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... approach.^5.2

An example is the NCAR LOCLIB FORTRAN routine NS01A, which has performed well for the author.

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... is^7.1

Cf., e.g., Brownlee 1965 for a derivation

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... tables^7.2

e.g., Abramowitz, M. and I. A. Stegun, Handbook of Mathematical Functions, 1970, Dover Publications, New York, p. 987

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...and^8.1

This estimator, called the ``unbiased'' estimator for the autocovariance, differs slightly from that given earlier in (8.23). See, e.g., Jenkins and Watts (19XX) for further discussion of the difference between these two estimators.

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... dimensionless.^11.1

For an exception, see Example 11.2.

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... used.^11.2

This example may appear straightforward and standard, but it has been the author's experience that even at the graduate level a surprising number of students benefit from this systematic approach.

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... intervals.^11.3

A convenient way of representing this relationship is with the cumulative distribution function F(x^*), giving the probability for all values of x smaller than x^*. If the corresponding cumulative distribution function in terms of y is G(y), then G(y(x^*)) = F(x^*). But dF(x)/dx=f(x) and dG(y)/dy=g(y) so, for corresponding intervals such that dF(x)=dG(y) it must be true that f(x)dx=g(y)dy.

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....^11.4

To show this, replace x and y by $\langle x\rangle+x^\prime$ and $\langle y\rangle+y^\prime$, then take the product and average.

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... analyst.^11.5

This is the reason that much of this section is written in more personalized form, using ``we" and ``our", than other sections of this book.

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... extrapolation.^11.6

Note that this solution does not permit the best estimates of the autocovariance to differ from the observations, as would be obtained by maximizing (). Instead, the observations are treated as constraints and the maximum-entropy solution found subject to those constraints. This solution thus is appropriate in the case of data without noise.

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