This chapter covers three topics:
  • plots, descriptive statistics, and exploratory data analysis;
  • statistical distribution functions and their inverses;
  • testing for Normality and other distributions.


public static class G01
Visual Basic
Public NotInheritable Class G01
Visual C++
public ref class G01 abstract sealed
type G01 =  class end

Background to the Problems

Plots, Descriptive Statistics and Exploratory Data Analysis

Plots and simple descriptive statistics are generally used for one of two purposes:
  • the presentation of data;
  • exploratory data analysis.
Exploratory data analysis (EDA) is used to pick out the important features of the data in order to guide the choice of appropriate models. EDA makes use of simple displays and summary statistics. These may suggest models or transformations of the data which can then be confirmed by further plots. The process is interactive between you, the data, and the program producing the EDA displays.
The summary statistics consist of two groups. The first group are those based on moments; for example mean, standard deviation, coefficient of skewness, and coefficient of kurtosis (sometimes called the ‘excess of kurtosis’, which has the value 0 for the Normal distribution). These statistics may be sensitive to extreme observations and some robust versions are available in G07 class. The second group of summary statistics are based on the order statistics, where the ith order statistic in a sample is the ith smallest observation in that sample. Examples of such statistics are minimum, maximum, median, hinges and quantiles.
In addition to summarising the data by using suitable statistics the data can be displayed using tables and diagrams. Such data displays include frequency tables, stem and leaf displays, box and whisker plots, histograms and scatter plots.

Statistical Distribution Functions and Their Inverses

Statistical distributions are commonly used in three problems:
  • evaluation of probabilities and expected frequencies for a distribution model;
  • testing of hypotheses about the variables being observed;
  • evaluation of confidence limits for parameters of fitted model, for example the mean of a Normal distribution.
Random variables can be either discrete (i.e., they can take only a limited number of values) or continuous (i.e., can take any value in a given range). However, for a large sample from a discrete distribution an approximation by a continuous distribution, usually the Normal distribution, can be used. Distributions commonly used as a model for discrete random variables are the binomial, hypergeometric, and Poisson distributions. The binomial distribution arises when there is a fixed probability of a selected outcome as in sampling with replacement, the hypergeometric distribution is used in sampling from a finite population without replacement, and the Poisson distribution is often used to model counts.
Distributions commonly used as a model for continuous random variables are the Normal, gamma, and beta distributions. The Normal is a symmetric distribution whereas the gamma is skewed and only appropriate for non-negative values. The beta is for variables in the range 0,1 and may take many different shapes. For circular data, the ‘equivalent’ to the Normal distribution is the von Mises distribution. The assumption of the Normal distribution leads to procedures for testing and interval estimation based on the χ2, F (variance ratio), and Student's t-distributions.
In the hypothesis testing situation, a statistic X with known distribution under the null hypothesis is evaluated, and the probability α of observing such a value or one more ‘extreme’ value is found. This probability (the significance) is usually then compared with a preassigned value (the significance level of the test), to decide whether the null hypothesis can be rejected in favour of an alternate hypothesis on the basis of the sample values. Many tests make use of those distributions derived from the Normal distribution as listed above, but for some tests specific distributions such as the Studentized range distribution and the distribution of the Durbin–Watson test have been derived. Nonparametric tests as given in (G08 not in this release), such as the Kolmogorov–Smirnov test, often use statistics with distributions specific to the test. The probability that the null hypothesis will be rejected when the simple alternate hypothesis is true (the power of the test) can be found from the noncentral distribution.
The confidence interval problem requires the inverse calculation. In other words, given a probability α, the value x is to be found, such that the probability that a value not exceeding x is observed is equal to α. A confidence interval of size 1-2α, for the quantity of interest, can then be computed as a function of x and the sample values.
The required statistics for either testing hypotheses or constructing confidence intervals can be computed with the aid of methods in this chapter, and G02 class (for regression), (G04 not in this release) (for analysis of designed experiments), G13 class (for time eries), and E04 class (for nonlinear least squares problems).
Pseudorandom numbers from many statistical distributions can be generated by methods in G05 class.

Testing for Normality and Other Distributions

Methods of checking that observations (or residuals from a model) come from a specified distribution, for example, the Normal distribution, are often based on order statistics. Graphical methods include the use of probability plots. These can be either P-P plots (probability–probability plots), in which the empirical probabilities are plotted against the theoretical probabilities for the distribution, or Q-Q plots (quantile–quantile plots), in which the sample points are plotted against the theoretical quantiles. Q-Q plots are more common, partly because they are invariant to differences in scale and location. In either case if the observations come from the specified distribution then the plotted points should roughly lie on a straight line.
If yi is the ith smallest observation from a sample of size n (i.e., the ith order statistic) then in a Q-Q plot for a distribution with cumulative distribution function F, the value yi is plotted against xi, where Fxi=i-α/n-2α+1, a common value of α being 12. For the Normal distribution, the Q-Q plot is known as a Normal probability plot.
The values xi used in Q-Q plots can be regarded as approximations to the expected values of the order statistics. For a sample from a Normal distribution the expected values of the order statistics are known as Normal scores and for an exponential distribution they are known as Savage scores.
An alternative approach to probability plots are the more formal tests. A test for Normality is the Shapiro and Wilk's W Test, which uses Normal scores. Other tests are the χ2 goodness-of-fit test and the Kolmogorov–Smirnov test; both can be found in (G08 not in this release).

Distribution of Quadratic Forms

Many test statistics for Normally distributed data lead to quadratic forms in Normal variables. If X is a n-dimensional Normal variable with mean μ and variance-covariance matrix Σ then for an n by n matrix A the quadratic form is
The distribution of Q depends on the relationship between A and Σ: if AΣ is idempotent then the distribution of Q will be central or noncentral χ2 depending on whether μ is zero.
The distribution of other statistics may be derived as the distribution of linear combinations of quadratic forms, for example the Durbin–Watson test statistic, or as ratios of quadratic forms. In some cases rather than the distribution of these functions of quadratic forms the values of the moments may be all that is required.

Energy Loss Distributions

An application of distributions in the field of high-energy physics where there is a requirement to model fluctuations in energy loss experienced by a particle passing through a layer of material. Three models are commonly used:
(i) Gaussian (Normal) distribution;
(ii) the Landau distribution;
(iii) the Vavilov distribution.
Both the Landau and the Vavilov density functions can be defined in terms of a complex integral. The Vavilov distribution is the more general energy loss distribution with the Landau and Gaussian being suitable when the Vavilov parameter κ is less than 0.01 and greater than 10.0 respectively.

Vectorized Methods

A number of vectorized methods are included in this chapter. Unlike their scalar counterparts, which take a single set of parameters and perform a single function evaluation, these methods take vectors of parameters and perform multiple function evaluations in a single call. The input arrays to these vectorized methods are designed to allow maximum flexibility in the supply of the parameters by reusing, in a cyclic manner, elements of any arrays that are shorter than the number of functions to be evaluated, where the total number of functions evaluated is the size of the largest array.
To illustrate this we will consider (G01SFF not in this release), a vectorized version of g01ef, which calculates the probabilities for a gamma distribution. The gamma distribution has two parameters α and β therefore (G01SFF not in this release) has four input arrays, one indicating the tail required (tail), one giving the value of the gamma variate, g, whose probability is required (g), one for α (a) and one for β (b). The lengths of these arrays are ltaillgla and lb respectively.
For sake of argument, lets assume that ltail=1, lg=2, la=3 and lb=4, then maxltail,lg,la,lb=4 values will be returned. These four probabilities would be calculated using the following parameters:

Recommendations on Choice and Use of Available Methods

Descriptive statistics / Exploratory analysis, 
        frequency / contingency table, 
            one variable g01ae
            two variables, with  χ 2 and Fisher's exact test g01af
        mean, variance, skewness, kurtosis (one variable), 
            from frequency table g01ad
            from raw data g01aa
        median, hinges / quartiles, minimum, maximum g01al
            unordered vector g01am
                scalar g01fe
            probabilities and probability density function, 
                scalar g01ee
            probabilities g01ge
        distribution function, 
            scalar g01bj
    Durbin–Watson statistic, 
        probabilities g01ep
    energy loss distributions, 
            density g01mt
            derivative of density g01rt
            distribution g01et
            first moment g01pt
            inverse distribution g01ft
            second moment g01qt
            density g01mu
            distribution g01eu
            initialization g01zu
                scalar g01fd
                scalar g01ed
            probabilities g01gd
            scalar g01ff
            scalar g01ef
        distribution function, 
            scalar g01bl
            one-sample g01ey
            two-sample g01ez
            probabilities g01ha
            probabilities g01hb
            quadratic forms, 
                cumulants and moments g01na
                moments of ratios g01nb
                scalar g01fa
                scalar g01ea
            reciprocal of Mill's Ratio g01mb
            Shapiro and Wilk's test for Normality g01dd
        distribution function, 
            scalar g01bk
    Student's t: 
                    scalar g01fb
                    scalar g01eb
            probabilities g01gb
    Studentized range statistic, 
        deviates g01fm
        probabilities g01em
    von Mises, 
        probabilities g01er
     χ 2: 
            deviates g01fc
            probabilities g01ec
            probability of linear combination g01jd
            probabilities g01gc
            probability of linear combination g01jc
    Normal scores, 
        accurate g01da
        approximate g01db
        variance-covariance matrix g01dc
    Normal scores, ranks or exponential (Savage) scores g01dh
Note:  the Student's t, χ2, and F methods do not aim to achieve a high degree of accuracy, only about four or five significant figures, but this should be quite sufficient for hypothesis testing. However, both the Student's t and the F-distributions can be transformed to a beta distribution and the χ2-distribution can be transformed to a gamma distribution, so a higher accuracy can be obtained by calls to the gamma or beta methods.
Note:  g01dh computes either ranks, approximations to the Normal scores, Normal, or Savage scores for a given sample. g01dh also gives you control over how it handles tied observations. g01da computes the Normal scores for a given sample size to a requested accuracy; the scores are returned in ascending order. g01da can be used if either high accuracy is required or if Normal scores are required for many samples of the same size, in which case you will have to sort the data or scores.

Working with Streamed or Extremely Large Datasets

The majority of the methods in this chapter are ‘in-core’, that is all the data required must be held in memory prior to calling the method. In some situations this might not be possible, for example, when working with extremely large datasets or where all of the data is not available at once (i.e., the data is being streamed).
There are five methods in this chapter applicable to datasets of this form:
(G01ATF not in this release) computes the mean, variance and the coefficients of skewness and kurtosis for a single variable.
(G01AUF not in this release), takes the results from two calls to (G01ATF not in this release) and combines them, returning the mean, variance and the coefficients of skewness and kurtosis for the combined dataset. This method allows the easy utilization of more than one processor to spread the computational burden inherent in summarising a very large dataset.
(G01ANF not in this release) (G01APF not in this release) compute the approximate quantiles for a dataset of known and unknown size respectively.
(G01WAF not in this release) computes the mean and standard deviation in a rolling window.
In addition, see g02bu (G02BZF not in this release) for methods to summarise two or more variables.


Hastings N A J and Peacock J B (1975) Statistical Distributions Butterworth
Kendall M G and Stuart A (1969) The Advanced Theory of Statistics (Volume 1) (3rd Edition) Griffin
Tukey J W (1977) Exploratory Data Analysis Addison–Wesley

Inheritance Hierarchy


See Also