# NAG Library Chapter Introductiong01 – Simple Calculations on Statistical Data

## 1  Scope of the Chapter

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

## 2  Background to the Problems

### 2.1  Summary 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.

### 2.2  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 arguments 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 eries), and Chapter e04 (for nonlinear least squares problems).
Pseudorandom numbers from many statistical distributions can be generated by functions in Chapter g05.

### 2.3  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 ${y}_{i}$ 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.

### 2.4  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 $\mu $ and variance-covariance matrix $\Sigma $ then for an $n$ by $n$ 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.

### 2.5  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 argument $\kappa $ is less than $0.01$ and greater than $10.0$ respectively.

### 2.6  Vectorized 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 nag_prob_gamma_vector (g01sfc), a vectorized version of nag_gamma_dist (g01efc), which calculates the probabilities for a gamma distribution. The gamma distribution has two parameters $\alpha $ and $\beta $ therefore nag_prob_gamma_vector (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:
|                                                                             |                                                                             |                                                                             |                                                                             |                                                                             |
|-----------------------------------------------------------------------------|-----------------------------------------------------------------------------|-----------------------------------------------------------------------------|-----------------------------------------------------------------------------|-----------------------------------------------------------------------------|
| $i$ | Tail                                                                        | $g$ | $\alpha $ | $\beta $ |
| $1$ | ${\mathbf{tail}}\left[1\right]$ | ${\mathbf{g}}\left[1\right]$ | ${\mathbf{a}}\left[1\right]$ | ${\mathbf{b}}\left[1\right]$ |
| $2$ | ${\mathbf{tail}}\left[1\right]$ | ${\mathbf{g}}\left[2\right]$ | ${\mathbf{a}}\left[2\right]$ | ${\mathbf{b}}\left[2\right]$ |
| $3$ | ${\mathbf{tail}}\left[1\right]$ | ${\mathbf{g}}\left[1\right]$ | ${\mathbf{a}}\left[3\right]$ | ${\mathbf{b}}\left[3\right]$ |
| $4$ | ${\mathbf{tail}}\left[1\right]$ | ${\mathbf{g}}\left[2\right]$ | ${\mathbf{a}}\left[1\right]$ | ${\mathbf{b}}\left[4\right]$ |

## 3  Recommendations on Choice and Use of Available Functions

Descriptive statistics / Exploratory analysis:

summaries:

frequency / contingency table,

one variable  nag_frequency_table (g01aec)

mean, variance, skewness, kurtosis (one variable),

from frequency table  nag_summary_stats_freq (g01adc)

from raw data  nag_summary_stats_1var (g01aac)

median, hinges / quartiles, minimum, maximum  nag_5pt_summary_stats (g01alc)

quantiles:

approximate:

large  data stream of fixed size  nag_approx_quantiles_fixed (g01anc)

large data stream of unknown size  nag_approx_quantiles_arbitrary (g01apc)

unordered vector  nag_double_quantiles (g01amc)

Distributions:

Beta:

central:

deviates  nag_deviates_beta (g01fec)

probabilities and probability density function  nag_prob_beta_dist (g01eec)

vectorized deviates  nag_deviates_beta_vector (g01tec)

vectorized probabilities  nag_prob_beta_vector (g01sec)

non-central:

probabilities  nag_prob_non_central_beta_dist (g01gec)

binomial:

distribution function  nag_binomial_dist (g01bjc)

vectorized distribution function  nag_prob_binomial_vector (g01sjc)

Durbin–Watson statistic:

probabilities  nag_prob_durbin_watson (g01epc)

Energy loss distributions:

Landau:

density  nag_prob_density_landau (g01mtc)

derivative of density  nag_prob_der_landau (g01rtc)

distribution  nag_prob_landau (g01etc)

first moment  nag_moment_1_landau (g01ptc)

inverse distribution  nag_deviates_landau (g01ftc)

second moment  nag_moment_2_landau (g01qtc)

Vavilov:

distribution  nag_prob_vavilov (g01euc)

initialization  nag_init_vavilov (g01zuc)

F:

central:

deviates  nag_deviates_f_dist (g01fdc)

probabilities  nag_prob_f_dist (g01edc)

vectorized deviates  nag_deviates_f_vector (g01tdc)

vectorized probabilities  nag_prob_f_vector (g01sdc)

non-central:

probabilities  nag_prob_non_central_f_dist (g01gdc)

gamma:

deviates  nag_deviates_gamma_dist (g01ffc)

probabilities  nag_gamma_dist (g01efc)

probability density function  nag_gamma_pdf (g01kfc)

vectorized deviates  nag_deviates_gamma_vector (g01tfc)

vectorized probabilities  nag_prob_gamma_vector (g01sfc)

vectorized probability density function  nag_gamma_pdf_vector (g01kkc)

Hypergeometeric:

distribution function  nag_hypergeom_dist (g01blc)

vectorized distribution function  nag_prob_hypergeom_vector (g01slc)

Kolomogorov–Smirnov:

probabilities:

one-sample  nag_prob_1_sample_ks (g01eyc)

two-sample  nag_prob_2_sample_ks (g01ezc)

Normal:

bivariate:

probabilities  nag_bivariate_normal_dist (g01hac)

multivariate:

probabilities  nag_multi_normal (g01hbc)

quadratic forms:

cumulants and moments  nag_moments_quad_form (g01nac)

moments of ratios  nag_moments_ratio_quad_forms (g01nbc)

univariate:

deviates  nag_deviates_normal (g01fac)

probabilities  nag_prob_normal (g01eac)

probability density function

scalar  nag_normal_pdf (g01kac)

vectorized  nag_normal_pdf_vector (g01kqc)

reciprocal of Mill's Ratio  nag_mills_ratio (g01mbc)

Shapiro and Wilk's test for Normality  nag_shapiro_wilk_test (g01ddc)

vectorized deviates  nag_deviates_normal_vector (g01tac)

vectorized probabilities  nag_prob_normal_vector (g01sac)

Poisson:

distribution function  nag_poisson_dist (g01bkc)

vectorized distribution function  nag_prob_poisson_vector (g01skc)

Student's t:

central:

bivariate:

probabilities  nag_bivariate_students_t (g01hcc)

univariate:

deviates  nag_deviates_students_t (g01fbc)

probabilities  nag_prob_students_t (g01ebc)

vectorized deviates  nag_deviates_students_t_vector (g01tbc)

vectorized probabilities  nag_prob_students_t_vector (g01sbc)

non-central:

probabilities  nag_prob_non_central_students_t (g01gbc)

Studentized range statistic:

probabilities  nag_prob_studentized_range (g01emc)

von Mises:

probabilities  nag_prob_von_mises (g01erc)

χ 2:

central:

deviates  nag_deviates_chi_sq (g01fcc)

probabilities  nag_prob_chi_sq (g01ecc)

probability of linear combination  nag_prob_lin_chi_sq (g01jdc)

non-central:

probabilities  nag_prob_non_central_chi_sq (g01gcc)

probability of linear combination  nag_prob_lin_non_central_chi_sq (g01jcc)

vectorized deviates  nag_deviates_chi_sq_vector (g01tcc)

vectorized probabilities  nag_prob_chi_sq_vector (g01scc)

Scores:

Normal scores, ranks or exponential (Savage) scores  nag_ranks_and_scores (g01dhc)

Normal scores:

accurate  nag_normal_scores_exact (g01dac)

variance-covariance matrix  nag_normal_scores_var (g01dcc)
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:  nag_ranks_and_scores (g01dhc) computes either ranks, approximations to the Normal scores, Normal, or Savage scores for a given sample. nag_ranks_and_scores (g01dhc) also gives you control over how it handles tied observations. nag_normal_scores_exact (g01dac) computes the Normal scores for a given sample size to a requested accuracy; the scores are returned in ascending order. nag_normal_scores_exact (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.

## 4  Functions Withdrawn or Scheduled for Withdrawal

|                                   |                   |                              |
|-----------------------------------|-------------------|------------------------------|
| WithdrawnFunction                 | Mark ofWithdrawal | Replacement Function(s)      |
| nag_deviates_normal_dist (g01cec) | 24                | nag_deviates_normal (g01fac) |

## 5  References

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
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