Statistical Description of DataStatistical Description of DataCf. NRiC, Chapter 14.Statistics provides tools for understanding data.In the wrong hands these tools can be dangerous!Here's a typical data analysis cycle:1. Apply some formula to data to compute a "statistic".2. Find where 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.StatisticsStatistics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 fitting).Will make use of special functions (e.g. gamma function) described in NRiC, Chapter 6.Moments of a DistributionMoments of a DistributionCf. NRiC §14.1.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: =  =∑=VarianceVarianceThe width of the central value is estimated by its second moment, called the variance: or its square root, the standard deviation:Why N -1? If the mean is known a priori, i.e. if it's not measured from the data, then use N, else N -1. If this matters to you, then N is probably too small! = −∑= − =More on MomentsMore on MomentsA clever way to minimize round-off error when computing the variance is to use the corrected two-pass algorithm. First compute <x>, then do:The second sum would be zero if <x> were exact, but otherwise it does a good job of correcting RE in Var.Higher moments, like skewness (3rd moment) and kurtosis (4th moment) are also sometimes used. = −{∑= −−[∑= −]}Distribution FunctionsDistribution FunctionsA distribution function (DF) p(x) gives the probability of finding value between x & x + dx.The expected mean data value is:For a discrete DF:Similar to weighted means, e.g. center of mass.〈  〉 =∫−∞∞  ∫−∞∞ 〈  〉 =∑∑MedianMedianThe median of a DF is the value xmed for which larger & smaller values of x are equally probable:For discrete values, sort in ascending order, then:∫−∞ ==∫∞= /  / /  ModeMode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 maxima. In this case the mean and median are not very useful since they give only a "compromise" value between the two peaks.Comparing DistributionsComparing DistributionsOften want to know if two distributions have different means or variances (NRiC §14.2):1. Student's t-test for significantly different means.a) Find no. of standard errors ~ /N1/2 between two means.b) Compute statistic using nasty formula.c) Small numerical value indicates significant difference.2. F-test for significantly different variances.a) Compute F = Var1/Var2 and plug into nasty formula.b) Small value indicates significant difference.Comparing Distributions, Cont'dComparing Distributions, Cont'dGiven two sets of data, can generalize to a single question: Are the sets drawn from the same DF?Recall can only disprove, 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. NRiC §14.3.Chi-Square (Chi-Square (22) Test) TestSuppose have Ni events in ith bin but expect ni:Large value of 2 indicates unlikely match.Compute probability Q(2|) from incomplete gamma function, where  is # degrees of freedom.For two binned data sets with events Ri and Si:=∑ −=∑−Kolmogorov-Smirnov (KS) TestKolmogorov-Smirnov (KS) TestAppropriate for unbinned distributions.From sorted list of data points, construct estimate SN(x) of the cumulative DF of the probability DF from which it was drawn...SN(x) gives fraction of data points to the left of x.Constant between xi's, jumps 1/N at each xi.Note SN(xmin) = 0, SN(xmax) =1.Behavior between xmin & xmax distinguishes distributions.KS Test, Cont'dKS Test, Cont'dStatistic is maximum value of absolute difference between two cumulative DFs.To compare data set to known cumulative DF:To compare two unknown data sets:Plug D and N (or Ne = N1N2/(N1 + N2)) into nasty formula to get numerical value of significance. = ≤≤∣ −  ∣ = ≤≤∣ −