10. Class Statistics and the K-S Test
Statistical Description of Data
Cf. NRiC 14. Statistics provides tools for understanding data. In the wrong hands these tools can be dangerous! Heres a typical data analysis cycle: 1. Apply some formula to data to compute a statistic. 2. Find where that value falls in a probability distribution computed on the basis of some null hypothesis. 3. If it falls in an unlikely spot (on distribution tail), conclude null hypothesis is false for your data set.
Statistics
Statistics and probability theory are closely related. Statistics can never prove things, only disprove them by ruling out hypotheses. Distinguish between model-independent statistics (this class, e.g., mean, median, mode) and model-dependent statistics (next class, e.g., least-squares tting). Will make use of special functions (e.g., gamma function) described in NRiC 6.
Moments of a Distribution
The mean, median, and mode of distributions are called measures of central tendency. The most common description of data involves its moments, sums of integer powers of the values. The most familiar moment is the mean: x = <x> = Variance The width of the central value is estimated by its second moment, called the variance,
N 1 Var = (xi x)2 , N 1 i=1
1 N
N
xi .
i=1
or its square root, the standard deviation, = Var. 1
Why N 1? If the mean is known a priori, i.e., if its not measured from the data, then use N, else N 1. If this matters to you, then N is probably too small! A clever way to minimize round-o error when computing the variance is to use the corrected two-pass algorithm. First compute x, then do: Var = 1 1 (xi x)2 N 1 i=1 N
N N i=1 2
(xi x)
.
The second sum would be zero if x were exact, but otherwise it does a good job of correcting RE in Var. Proof: EFTS (hint: set x x + ). Other moments Higher moments, like skewness (3rd moment) and kurtosis (4th moment) are also sometimes used, but can be unreliable. Cf. NRiC 14.1.
Distribution Functions
A distribution function (DF) p(x) gives the probability of nding a value between x 2 and x + dx, e.g., the familiar normal (Gaussian) distribution p(x) dx = 1 ex /2 dx. 2 The expected mean data value is: <x> = For a discrete DF: <x> =
i x p(x) dx . p(x) dx
xi pi . i pi
Similar to weighted means, e.g., center of mass.
Median
The median of a DF is the value xmed for which larger and smaller values of x are equally probable: xmed 1 p(x) dx = = p(x) dx. 2 xmed For discrete values, sort in ascending order (i = 1, 2, ..., N), then: xmed = x(N +1)/2 , if N is odd, 1 (xN/2 + xN/2+1 ), if N is even. 2
2
Mode
The mode of a probability DF p(x) is the value of x where the DF takes on a maximum value. Most useful when there is a single, sharp max, in which case it estimates the central value. Sometimes a distribution will be bimodal, with two relative In maxima. this case the mean and median are not very useful since they give only a compromise value between the two peaks.
Comparing Distributions
Often want to know if two distributions have dierent means or variances (NRiC 14.2): 1. Students t-test for signicantly dierent means. (a) Find number of standard errors /N 1/2 between two means. (b) Compute statistic using nasty formula: probability that the two means are dierent by chance. (c) Small numerical value indicates signicant dierence. 2. F -test for signicantly dierent variances. (a) Compute F = Var1 /Var2 and plug into nasty formula (the distribution of F in the case that the variances are the samethe null hypothesisis related to the incomplete beta function). (b) Small value indicates signicant dierence. Given two sets of data, can generalize to a single question: Are the sets drawn from the same DF? E.g., are stars distributed uniformly in the sky? Do two brands of lightbulbs have the same distribution of burn-out times? Recall can only disprove (to a certain condence level), not prove. May have continuous or binned data. May want to compare one data set with known DF, or two unknown data sets with each other. Popular technique for binned data is the 2 test. For continuous data, use the KS test. Cf. NRiC 14.3. Chi-square (2 ) test Suppose have Ni events in ith bin but expect ni : 2 =
i
(Ni ni )2 . ni
3
Large value of 2 indicates unlikely match (i.e., Ni s probably not drawn from population represented by ni s). Compute probability Q(2 |) from incomplete gamma function, where is the number of degrees of freedom. Typically = NB , where NB is the number of bins, or NB 1, if the ni s are normalized such that i ni = i Ni . Null hypothes...