# NAG CL InterfaceG01 (Stat)Simple Calculations on Statistical Data

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## 1Scope of the Chapter

This chapter covers the following topics:
• summary statistics
• statistical distribution functions and their inverses;
• testing for Normality and other distributions.

## 2Background to the Problems

### 2.1Summary Statistics

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 Chapter G07. The second group of summary statistics are based on the order statistics, where the $i$th order statistic in a sample is the $i$th smallest observation in that sample. Examples of such statistics are minimum, maximum, median, hinges and quantiles.

### 2.2Statistical 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 $\left[0,1\right]$ 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 ${\chi }^{2}$, $F$ (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 $\alpha$ 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 Chapter G08, 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 $\alpha$, the value $x$ is to be found, such that the probability that a value not exceeding $x$ is observed is equal to $\alpha$. A confidence interval of size $1-2\alpha$, 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 functions in this chapter, and Chapter G02 (for regression), Chapter G04 (for analysis of designed experiments), Chapter G13 (for time series), and Chapter E04 (for nonlinear least squares problems).
Pseudorandom numbers from many statistical distributions can be generated by functions in Chapter G05.

### 2.3Testing 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 ${y}_{i}$ is the $i$th smallest observation from a sample of size $n$ (i.e., the $i$th order statistic) then in a $Q-Q$ plot for a distribution with cumulative distribution function $F$, the value ${y}_{i}$ is plotted against ${x}_{i}$, where $F\left({x}_{i}\right)=\left(i-\alpha \right)/\left(n-2\alpha +1\right)$, a common value of $\alpha$ being $\frac{1}{2}$. For the Normal distribution, the $Q-Q$ plot is known as a Normal probability plot.
The values ${x}_{i}$ 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 ${\chi }^{2}$ goodness-of-fit test and the Kolmogorov–Smirnov test; both can be found in Chapter G08.

Many test statistics for Normally distributed data lead to quadratic forms in Normal variables. If $X$ is an $n$-dimensional Normal variable with mean $\mu$ and variance-covariance matrix $\Sigma$ then for an $n×n$ matrix $A$ the quadratic form is
 $Q=XTAX.$
The distribution of $Q$ depends on the relationship between $A$ and $\Sigma$: if $A\Sigma$ is idempotent then the distribution of $Q$ will be central or noncentral ${\chi }^{2}$ depending on whether $\mu$ 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.

### 2.5Energy 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:
1. (i)Gaussian (Normal) distribution;
2. (ii)the Landau distribution;
3. (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 $\kappa$ is less than $0.01$ and greater than $10.0$ respectively.

### 2.6Vectorized Functions

A number of vectorized functions are included in this chapter. Unlike their scalar counterparts, which take a single set of parameters and perform a single function evaluation, these functions take vectors of parameters and perform multiple function evaluations in a single call. The input arrays to these vectorized functions 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 g01sfc, a vectorized version of g01efc, which calculates the probabilities for a gamma distribution. The gamma distribution has two parameters $\alpha$ and $\beta$, therefore, g01sfc 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 $\alpha$ (a) and one for $\beta$ (b). The lengths of these arrays are ltail, lg, la and lb respectively.
For sake of argument, lets assume that ${\mathbf{ltail}}=1$, ${\mathbf{lg}}=2$, ${\mathbf{la}}=3$ and ${\mathbf{lb}}=4$, then $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{ltail}},{\mathbf{lg}},{\mathbf{la}},{\mathbf{lb}}\right)=4$ values will be returned. These four probabilities would be calculated using the following parameters:
$\mathbit{i}$ tail $\mathbit{g}$ $\mathbit{\alpha }$ $\mathbit{\beta }$
$1$ ${\mathbf{tail}}\left[0\right]$ ${\mathbf{g}}\left[0\right]$ ${\mathbf{a}}\left[0\right]$ ${\mathbf{b}}\left[0\right]$
$2$ ${\mathbf{tail}}\left[0\right]$ ${\mathbf{g}}\left[1\right]$ ${\mathbf{a}}\left[1\right]$ ${\mathbf{b}}\left[1\right]$
$3$ ${\mathbf{tail}}\left[0\right]$ ${\mathbf{g}}\left[0\right]$ ${\mathbf{a}}\left[2\right]$ ${\mathbf{b}}\left[2\right]$
$4$ ${\mathbf{tail}}\left[0\right]$ ${\mathbf{g}}\left[1\right]$ ${\mathbf{a}}\left[0\right]$ ${\mathbf{b}}\left[3\right]$

## 3Recommendations on Choice and Use of Available Functions

 Descriptive statistics / Exploratory analysis,
 summaries,
 frequency / contingency table,
 one variable g01aec
 mean, variance, skewness, kurtosis (one variable),
 combine summaries g01auc
 from raw data g01atc
 median, hinges / quartiles, minimum, maximum g01alc
 quantiles,
 approximate,
 large  data stream of fixed size g01anc
 large data stream of unknown size g01apc
 unordered vector
 unweighted g01amc
 weighted, Example 2 in m01dsc m01dsc
 rolling window,
 mean, standard deviation (one variable) g01wac
 Distributions,
 Beta,
 central,
 deviates,
 scalar g01fec
 vectorized g01tec
 probabilities and probability density function,
 scalar g01eec
 vectorized g01sec
 non-central,
 probabilities g01gec
 binomial,
 distribution function,
 scalar g01bjc
 vectorized g01sjc
 Dickey–Fuller unit root test,
 probabilities, g01ewc
 Durbin–Watson statistic,
 probabilities g01epc
 energy loss distributions,
 Landau,
 density g01mtc
 derivative of density g01rtc
 distribution g01etc
 first moment g01ptc
 inverse distribution g01ftc
 second moment g01qtc
 Vavilov,
 density g01muc
 distribution g01euc
 initialization g01zuc
 $F$:
 central,
 deviates,
 scalar g01fdc
 vectorized g01tdc
 probabilities,
 scalar g01edc
 vectorized g01sdc
 non-central,
 probabilities g01gdc
 gamma,
 deviates,
 scalar g01ffc
 vectorized g01tfc
 probabilities,
 scalar g01efc
 vectorized g01sfc
 probability density function,
 scalar g01kfc
 vectorized g01kkc
 Hypergeometric,
 distribution function,
 scalar g01blc
 vectorized g01slc
 Kolomogorov–Smirnov,
 probabilities,
 one-sample g01eyc
 two-sample g01ezc
 Normal,
 bivariate,
 probabilities g01hac
 multivariate,
 probabilities g01hbc
 probability density function,
 vectorized g01lbc
 cumulants and moments g01nac
 moments of ratios g01nbc
 univariate,
 deviates,
 scalar g01fac
 vectorized g01tac
 probabilities,
 scalar g01eac
 vectorized g01sac
 probability density function,
 scalar g01kac
 vectorized g01kqc
 reciprocal of Mill's Ratio g01mbc
 Shapiro and Wilk's test for Normality g01ddc
 Poisson,
 distribution function,
 scalar g01bkc
 vectorized g01skc
 Student's $t$:
 central,
 bivariate,
 probabilities g01hcc
 multivariate,
 probabilities g01hdc
 univariate,
 deviates,
 scalar g01fbc
 vectorized g01tbc
 probabilities,
 scalar g01ebc
 vectorized g01sbc
 non-central,
 probabilities g01gbc
 Studentized range statistic,
 deviates g01fmc
 probabilities g01emc
 von Mises,
 probabilities g01erc
 ${\chi }^{2}$:
 central,
 deviates g01fcc
 probabilities g01ecc
 probability of linear combination g01jdc
 non-central,
 probabilities g01gcc
 probability of linear combination g01jcc
 vectorized deviates g01tcc
 vectorized probabilities g01scc
 Scores,
 Normal scores,
 accurate g01dac
 variance-covariance matrix g01dcc
 Normal scores, ranks or exponential (Savage) scores g01dhc
Note:  the Student's $t$, ${\chi }^{2}$, and $F$ functions 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 ${\chi }^{2}$-distribution can be transformed to a gamma distribution, so a higher accuracy can be obtained by calls to the gamma or beta functions.
Note:  g01dhc computes either ranks, approximations to the Normal scores, Normal, or Savage scores for a given sample. g01dhc also gives you control over how it handles tied observations. g01dac computes the Normal scores for a given sample size to a requested accuracy; the scores are returned in ascending order. g01dac 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.

### 3.1Working with Streamed or Extremely Large Datasets

The majority of the functions in this chapter are ‘in-core’, that is all the data required must be held in memory prior to calling the function. 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 functions in this chapter applicable to datasets of this form:
g01atc computes the mean, variance and the coefficients of skewness and kurtosis for a single variable.
g01auc, takes the results from two calls to g01atc and combines them, returning the mean, variance and the coefficients of skewness and kurtosis for the combined dataset. This function allows the easy utilization of more than one processor to spread the computational burden inherent in summarising a very large dataset.
g01anc and g01apc compute the approximate quantiles for a dataset of known and unknown size respectively.
g01wac computes the mean and standard deviation in a rolling window.
In addition, see g02buc and g02bzc for functions to summarise two or more variables.

None.

## 5 Withdrawn or Deprecated Functions

The following lists all those functions that have been withdrawn since Mark 23 of the Library or are in the Library, but deprecated.
Function Status Replacement Function(s)
g01aac Withdrawn at Mark 26 g01atc
g01cec Withdrawn at Mark 24 g01fac

## 6References

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