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14.3 Are Two Distributions Different generate, create none none in none projectsdata matrix creating Equation (14.3.2) and th none none e routine chstwo both apply to the case where the total number of data points is the same in the two binned sets.

For unequal numbers of data points, the formula analogous to (14.3.2) is 2 =.

Beaware of Malicious QR Codes S/RRi R/SSi )2 Ri + Si (14.3.3).

where R (14.3.4).

are the respective numbers of data points. corresponding change in chstwo. It is straightforward to make the Kolmogorov-Smirnov Test The Kolmogorov-Smirnov ( none for none or K S) test is applicable to unbinned distributions that are functions of a single independent variable, that is, to data sets where each data point can be associated with a single number (lifetime of each lightbulb when it burns out, or declination of each star). In such cases, the list of data points can be easily converted to an unbiased estimator SN (x) of the cumulative distribution function of the probability distribution from which it was drawn: If the N events are located at values xi , i = 1, . .

. , N , then SN (x) is the function giving the fraction of data points to the left of a given value x. This function is obviously constant between consecutive (i.

e., sorted into ascending order) xi s, and jumps by the same constant 1/N at each xi . (See Figure 14.

3.1.) Different distribution functions, or sets of data, give different cumulative distribution function estimates by the above procedure.

However, all cumulative distribution functions agree at the smallest allowable value of x (where they are zero), and at the largest allowable value of x (where they are unity). (The smallest and largest values might of course be .) So it is the behavior between the largest and smallest values that distinguishes distributions.

One can think of any number of statistics to measure the overall difference between two cumulative distribution functions: the absolute value of the area between them, for example. Or their integrated mean square difference. The KolmogorovSmirnov D is a particularly simple measure: It is de ned as the maximum value of the absolute difference between two cumulative distribution functions.

Thus, for comparing one data set s SN (x) to a known cumulative distribution function P (x), the K S statistic is D=. <x< SN (x) P (x). (14.3.5).

while for comparing two none none different cumulative distribution functions SN1 (x) and SN2 (x), the K S statistic is D=. <x< SN1 (x) SN2 (x). (14.3.6).

14. . Statistical Description of Data cumulative probability distribution SN (x). P(x). Figure 14.3.1.

Kolmogoro v-Smirnov statistic D. A measured distribution of values in x (shown as N dots on the lower abscissa) is to be compared with a theoretical distribution whose cumulative probability distribution is plotted as P (x). A step-function cumulative probability distribution SN (x) is constructed, one that rises an equal amount at each measured point.

D is the greatest distance between the two cumulative distributions.. What makes the K S stati none for none stic useful is that its distribution in the case of the null hypothesis (data sets drawn from the same distribution) can be calculated, at least to useful approximation, thus giving the signi cance of any observed nonzero value of D. A central feature of the K S test is that it is invariant under reparametrization of x; in other words, you can locally slide or stretch the x axis in Figure 14.3.

1, and the maximum distance D remains unchanged. For example, you will get the same signi cance using x as using log x. The function that enters into the calculation of the signi cance can be written as the following sum:.

QKS ( ) = 2. ( 1)j 1 e 2j (14.3.7).

which is a monotonic fun none none ction with the limiting values QKS (0) = 1 QKS ( ) = 0 (14.3.8).

In terms of this functio n, the signi cance level of an observed value of D (as a disproof of the null hypothesis that the distributions are the same) is given approximately [1] by the formula Probability (D > observed ) = QKS Ne + 0.12 + 0.11/ Ne D (14.

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