Mathematical and statistical applications for HP Prime

Schrausser, D. G. (2025). Mathematical and statistical applications for HP Prime. 1 Mathematical and statistical applications for HP Prime Dietmar G. Schrausser orcid.org/0000-0002-4924-8280 Correspondence: dietmar.schrauss[email protected] Karl-Franzens University, Graz, Austria Abstract Applications for HP Prime CAS, User functions and Applications, an overview of the methods and their origins is given. 1. Introduction Mathematical and statistical applications HP_Prime_MATH 1 for (1) the Computer Algebra System CAS, by means of the Pascal based HP Prime Programming Language (HP PPL), (2) the HP Prime User functions and (3) HP Prime Applications (s. HP Inc., 2017), including methods for (1) correlation, (2) exposure, (3) integration, (4) distribution, (5) probability, (6) combinatorics, (7) resampling and (8) complex plane calculations (Schrausser, 2025a). The description of the underlying algorithms and functions is deliberately omitted, as these are presented and discussed in detail in Schrausser (2025b). Instead, an overview of the implemented methods is given and additionally their historical development is outlined (c.f. Tab. 1). 2. Functions 2.1. Correlation To measure the degree of a linear relation between variables, Karl Pearson (1904) was developing statistical procedures for biometry including the correlation and regression coefficients based on the works of Bravais (1844) and Galton (1877) who introduced the symbol 𝑟, on the then designation of the term reversion. The methodological apparatus of factor analysis as a further and broader concept, based on multiple regression and matrix calculation was first discussed by Charles Edward Spearman (1904), later the initial developed took place by Louis Leon Thurstone (1931, 1934, 1935; s. also Cattell, 1966). The following functions for correlationand regression-techniques are implemented, c.f. also Schrausser (2025b): (1) Pearson product-moment correlation coefficient 𝑟𝑥𝑦, see Pearson (1904, 1905). (2) Spearman’s 𝜌, being equivalent to the product moment correlation when rank values are present (s. Spearman, 1904). 1 https://github.com/Schrausser/HP_Prime_MATH Creative Commons Attribution 4.0 International Schrausser, D. G. (2025). Mathematical and statistical applications for HP Prime. 2 (3) Kendall’s tau 𝜏𝑎, i.e. without adjustment for ties (s. Kendall, 1938). (4) Somers’ 𝐷, for binary data [0,1] (s. Somers, 1962). (5) Point biserial correlation coefficient 𝑟𝑝𝑏 or also point biseral. (6) Biserial correlation coefficient 𝑟𝑏𝑖𝑠, Pearson (1909), s. Tate (1955), also called biseral. (7) Rank biserial correlation coefficient 𝑟𝑏𝑖𝑠𝑅 or rank biseral, corresponding to the effect size for the Mann–Whitney 𝑈-test (Mann and Whitney, 1947). (8) Phi coefficient 𝛷, Yule (1912). (9) Tetrachoric correlation 𝑟𝑡𝑒𝑡, Pearson (1900a), Everitt (1910, 1912), s. e.g. Brown (1977), Digby (1983), also Bonett and Price (2005) or Long et al. (2009), proposed approximate algorithm. (10) Partial correlation 𝑟𝑥𝑦⋅𝑧. (11) Fisher 𝑍-transformation, Fisher (1915). (12) Fisher 𝑍 difference, also Cohen’s 𝑞 (Cohen, 1988, p. 110). (13) Averaged Fisher 𝑍. (14) Coefficient of multiple correlation 𝑅𝑐,12, for 𝑅 𝑐,12 2 see Olkin and Pratt (1958), with the effect size for multiple regression 𝑓2 (Cohen, 1988, p. 410). 2.2. Exposure To determine the appropriate time-aperture-speed combination for given light values on a logarithmic scale (c.f. Allbright, 1991; Marsden and Weinstein, 1985; Howie, 2001 and Sobot, 2021), following functions are included for the calculation of (1) exposure values 𝐸𝑣, where 𝐸𝑣 =log(𝑇𝑣⋅𝐴𝑣2) log(2), (2) aperture 𝐴𝑣 for time 𝑇𝑣 or speed 𝑆 with given 𝐸𝑣, (3) aperture 𝐴𝑣 shift from time 𝑇𝑣 or speed 𝑆 in steps 𝑘 and (4) speed 𝑆 in logarithmic 𝐼𝑆𝑂° or arithmetic 𝐼𝑆𝑂 values. 2.3. Functions of integration for 𝛑 and 𝜞 Gottfried Wilhelm Leibniz (1684, 1686, 1693) along with Sir Isaac Newton (1687, 1713, 1726) are considered the discoverers of differential and integral calculus. According to current consensus, both developed the methods independently of each other, see the so-called Leibniz-Newton calculus controversy (c.f. Cajori, 1919; Cassirer, 1943; Rosenthal, 1951; Schrader, 1962; Kossovsky, 2020). Newton began working on a geometric form of calculus (the method of fluxions and fluents) in 1666, published in 1687 (c.f. Roero, 2005), yet, it was Leibniz who introduced the symbols ∫and ∂. Here, the functions are primarily intended to display and calculate π and 𝛤 within the coordinate system: (1) Circular function for π, where Weierstraß (1894, p. 53) describes π 2=∫1 1−𝑥2 ∞ 0𝑑𝑥, which may be less heuristic (s. Schrausser, 2025b). (2) Spherical functions for π, for source codes to volume integrals of the sphere see Schrausser (2024d). (3) Gamma function 𝛤, meant to extend the factorial to non-integer arguments, was first considered by Daniel Bernoulli and Christian Goldbach (Bernoulli, 1729), later Leonhard Euler (1738) and Johann Carl Friedrich Gauss (s. Remmert, 1998), first tables were given by Jahnke and Emde (1909, 1933, 1938, 1945), Knoll (1939) and Jahnke et al. (1966). Schrausser, D. G. (2025). Mathematical and statistical applications for HP Prime. 3 2.4. Distribution functions The discovery of the normal distribution is attributed to Abraham de Moivre (1738), later Gauss (1809) described the arithmetic mean as an estimator in context with the normal law of errors. Beneath the normal distribution, Gauss (1823) also introduces several important statistical concepts, such as the methods of least squares and of maximum likelihood. The 𝑡-distribution first derived as a posterior distribution by Lüroth (1876), appearing later as Pearson Type IV (Pearson, 1895), however gets its name as Student’s 𝑡-distribution from William Sealy Gosset (1908), who published it using the pseudonym Student, though it was actually through the extensive works of Sir Ronald Aylmer Fisher that the distribution became well known. The 𝜒2-distribution was first described by Friedrich Robert Helmert (1876) and independently rediscovered by Pearson (1900b) in context with the goodness of fit paradigm, where he developed the 𝜒2-test with computed table of values, published by Elderton (1902), s. further Pearson (1914) or Plackett (1983). Fisher (1918, 1921, 1925) introduced the term variance and proposed its formal analysis, as well as the 𝐹-distribution (Fisher, 1924; s. also Snedecor, 1934 and Scheffé, 1959). The methods became widely known from Methods for Research Workers (Fisher, 1925, 1954, 1973, 2017). Following functions for the most relevant methods are available: (1) Standardizing, i.e. 𝑧-values and 𝜁-values. (2) Quantity proportion of 𝑎 at 𝑁 for 𝑛≥𝑝. (3) Weighted arithmetic mean 𝑥󰇘. (4) Geometric mean 𝑥󰇗, for the weighted geometric mean 𝑥󰇗󰇘 s. Siegel (1942). (5) Harmonic mean 𝑥. (6) Coefficient of variation 𝜔. (7) Mean dispersion 𝑑, Schrausser (2022a, p. 33). (8) Standard normal distribution 𝑓(𝑥 = 𝑧), de Moivre (1738), Gauss (1809, 1823). (9) Bivariate normal distribution 𝑓(𝑧1,𝑧2). (10) Student’s 𝑡-distribution 𝑓(𝑥 = 𝑡), Lüroth (1876), Pearson (1895), Gosset (1908). (11) 𝜒2-distribution 𝑓(𝑥 = 𝜒2), Helmert (1876), Pearson (1900b, 1914), Elderton (1902), Plackett (1983). (12) 𝐹-distribution 𝑓(𝑥 = 𝐹), Fisher (1924), Snedecor (1934), Scheffé (1959). (13) Third standardized moment, skewness 𝛼3. (14) Fourth standardized moment, excess kurtosis 𝛼4. (15) Estimated standard error of mean 𝜎𝑥, confidence interval 𝐶𝐼𝑝. Neyman (1937) introduced the confidence interval into statistical hypothesis testing vs. Fisher’s null hypothesis testing, the Neyman–Pearson lemma (Neyman and Pearson, 1933; Lehmann, 1993). (16) Standard error of prediction 𝜎𝑦 𝑥 , confidence interval 𝐶𝐼𝑝. (17) Effect size 𝜖, Cohen’s 𝑑 (Cohen, 1977, 1988, p. 20, p. 49, 1992), Borenstein et al. (1997), Borenstein et al. (2001). (18) Optimal effect size 𝜖𝑝. (19) Optimal alpha level. (20) Variance difference 𝑡-test for paired samples (𝑥1|𝑥2). (21) Paired 2-sample 𝑡-test. (22) Unpaired 2-sample 𝑡-test. (23) One-sample 𝑡-test. (24) 𝜒2-test for independence. Schrausser, D. G. (2025). Mathematical and statistical applications for HP Prime. 4 (25) 2 × 2 𝜒2-test for independence, for Yates’s correction for continuity see Yates (1934). (26) McNemar’s 𝜒2-test for paired 2 × 2 contingency tables with dichotomous trait, McNemar (1947). 2.5. Probability Since until the Renaissance a probable opinion was merely confirmed by an authority and hence there was no further concept of inductive evidence (see Hacking, 1975; Hald, 2003, p. 31), an objective representation of probability as such was first discussed by Antoine Arnauld and Pierre Nicole (1662, 1682, 1693; c.f. also Arnauld et al., 1970; van Evra, 1997; Dessì and Albury, 1997 or Finocchiaro, 1997). The binomial distribution is primarily attributable to de Moivre (1711, 1718, 1738) and Jacob Bernoulli (1713), see also Schneider (2005a, b). Although not included as function, due to its considerability in this context, the configuration frequency analysis, CFA should be mentioned particularly (c.f. Krauth, 1973; Krauth and Lienert, 1993). An account of the systematics and logic of dependent probabilities within the framework of Bayes’ theorem (Bayes and Price, 1763; c.f. Stigler, 2018) can be found in Schrausser (2024c). The arguably most important methods regarding the calculation of probability parameters are implemented as follows: (1) Arcsine transformation, Cohen’s ℎ (Cohen, 1988, p. 181). (2) Additive probability for independent events 𝑢𝑝(∪𝑛𝐴), which corresponding to the geometric distribution 𝑓(𝑋 ≤ 𝑟|𝑝). (3) Geometric distribution 𝑓(𝑋 ≤ 𝑟|𝑝), corresponding to the additive probability 𝑢𝑝(∪𝑛𝐴). (4) Negative binomial distribution 𝑓(𝑋 ≤ 𝑟|𝑟,𝑝), with 𝑘 = 1 it corresponds to the geometric distribution 𝑓(𝑋 ≤ 𝑟|𝑝) and the additive probability 𝑢𝑝(∪𝑛𝐴). (5) Exact binomial test. (6) Exact hypergeometric 2 × 2 test, the so-called Fisher Exact test (Fisher, 1922; Agresti, 1992). 2.6. Combinatorics After Gersonides’ pioneering work from 1321 dealing with arithmetical operations and combinatorics (s. Abraham Bar Hiyya Savasorda, 1450; Rabinovitch, 1970), the methods, being a fundamental part for probability calculations, are mainly based on Blaise Pascal (1665), Bernoulli (1713) and Euler (1753), c.f. Ettingshausen (1826). See further Sylvester (1904, 1908, 1909, 1912) and MacMahon (1915, 1916), giving fundamental contributions to matrix-theory and combinatorics. The following functions to generate permutation and variation matrices are available, primarily to support the resampling procedures described below: (1) Permutation matrix 𝑷𝒏, with 𝑛 elements to 𝑘 = 1 class. (2) Variation matrix 𝒘𝑽𝟐 𝒎 for the dependent 2 sample design, with 𝑛 = 2 elements to class 𝑚. (3) Variation matrix 𝒘𝑽𝒏 𝒎, with 𝑛 elements to class 𝑚. (4) Permutation matrix 𝒘𝑷𝒏 (𝒌𝒎,𝒌𝒏−𝒎), with 𝑛 elements to class 𝑚. Schrausser, D. G. (2025). Mathematical and statistical applications for HP Prime. 5 2.7. Resampling Permutation or randomization tests were first mentioned by Fisher (1935), based on experiments in agriculture (Fisher, 1926; Neyman, 1923). In this context see Pitman (1937a, b, 1938), Fisher (1966, 1971, res.), especially Eugene Sinclair Edgington (1964, 1980, 1987, 2011) or Edgington and Onghena (2007). The bootstrap method was introduced by Bradley Efron (1979, 1981, 1982) as a further development (Quenouille, 1949; Metropolis and Ulam, 1949), for software solutions see e.g. Solomon (1982), Dallal (1986, 1988), Peladeau (1993), Wooff and Peladeau (1994), Mehta et al. (2014), also Schrausser (2024d). Following functions were developed: (1) Permutation test P for 2 paired samples (𝑥1|𝑥2). Random sampling model, systematic permutation, 𝑝-value not randomized, variation matrix 𝒘𝑽𝟐 𝒎 required, s. Scambor (1997), Scambor and Schrausser (2022, p. 7), respectively. (2) Randomized permutation test mP for 2 paired samples (𝑥1|𝑥2). Random sampling model, 𝑝-value not randomized. (3) Permutation test P for 2 independent samples (𝑥|𝑔). Random sampling model, systematic permutation, 𝑝-value not randomized, permutation matrix 𝒘𝑷𝒏 (𝒌𝒎,𝒌𝒏−𝒎) required, see Schrausser (1996, 1998b, 2022b, p. 2). (4) Randomized permutation test mP for 2 independent samples (𝑥|𝑔). Random sampling model, 𝑝-value not randomized. (5) Bootstrap test Bt for 2 independent samples (𝑥|𝑔), c.f. Quenouille (1949), Efron (1979, 1981, 1982). 2.8. Complex plane It was the Italian mathematician Gerolamo Cardano (1545a, b) who first conceived the term imaginary, for the further historical development of imaginary or complex numbers see René Descartes (1664, 2012, res.) and Gauss (1828, 1832), c.f. also Wirtinger (1927). Here finally realized are (1) the geometric representation of complex numbers 𝑧 in the complex plane, the Argand diagram (s. Argand, 1813, 1874, res.) and (2) the graph of the complex function, where 𝑧 = ℜ + ℑ. At this point, one should recall the definitional importance of geometry and trigonometry in context with the calculation of complex numbers itself, where |𝑧| is calculated according to Pythagoras (c.f. Ratdolt, 1482, propositio 46) by |𝑧|= 𝑟 = √𝑥2+ 𝑦2. After the fundamental change in mathematics from geometric to algebraic representation took place in the 16th century (c.f. Heath, 1908a, b, c; Bochner, 1978; Anglin and Lambek, 1995; Malet, 2006 or Alten et al., 2014), the origins of trigonometric series of tangents and sine can be seen following early attempts (s. Jyesthadeva, 1530; Whish, 1834; Gupta, 1974 or Divakaran, 2007) during the European reinvention in the works of Gregory (1671, 1668a, b), Leibniz (1682, 2012), Newton (1669, 1711) and Brook Taylor (1715, 1717) with the definition of the Taylor series of sine, where sin⁡𝑥 = ∑(−1)𝑛 (2⋅𝑛+1)! ∞ 𝑛=0 ⋅ 𝑥2⋅𝑛+1 (c.f. Gregory and Collins, 1939; Boyer, 1968, p. 422 ff.; Feigenbaum, 1985). Finally, Euler (1748a, b) established the analytic treatment of trigonometric functions, defining them in relation with complex exponential functions by e𝒊𝑥 =cos 𝑥 + 𝒊sin⁡𝑥, where Schrausser, D. G. (2025). Mathematical and statistical applications for HP Prime. 6 e = ∑1 𝑛! ∞ 𝑛=0 and thus laid the foundation of modern mathematical analysis (c.f. Finkel, 1897; Walter, 1982; Koyama and Kurokawa, 2005; Calinger, 2016 and Schrausser, 2024b). 3. Conclusion In addition to the source codes of the functions, raw data sets are provided for correlationas well as resampling-methods. CAS programs, HP Prime User functions and functions for HP Prime Applications in comparison to corresponding SCHRAUSSER-MAT functions (Schrausser, 2022a) are displayed in Schrausser (2025b). Furthermore, the application FunktionWin for a precise calculation of probability distributions can additionally be considered (Schrausser, 2023c) as well as the author’s further software applications for mathematical and statistical analyses (Schrausser, 2023a, b, d). On mathematical statistical methods in general see e.g. Cox and Hinkley (1974), Bortz and Weber (2005), Lehmann and Romano (2008) or Bortz and Schuster (2010), Schrausser (2024a) provides a comprehensive overview of the most important distribution functions and corresponding algorithms. For calculus and theory of functions see e.g. Meyberg and Vachenauer (2001a, b) or Remmert and Schumacher (2002), on complex numbers in the complex plane see e.g. Burckel (2021) and Vince (2021), introducing works on resampling methods are given by e.g. Good (2006) or Beasley and Rodgers (2009). For the history of statistical inference in general see e.g. Stigler (1986) and Hald (1990, 1998, 2003, 2007), the historical foundations of mathematics are thematized and discussed in e.g. Suter (1887), Heath (1921a, b), Boyer (1968), Neugebauer (1969), Ewald (1996a, b), Katz (2009) or Merzbach and Boyer (2011), c.f. Tab. 1. Table 1. Timeline (year) of initial work on the methods, corresponding authors with origin and field of expertise. year n Name origin from to field n method work 1280 1290 1 Rabbi Levi ben Gershon France 1288 1344 Theologian 1300 1310 1320 1330 1 Combinatorics 1321 1340 : : 1500 1510 2 Gerolamo Cardano Italy 1501 1576 Polymath 1520 1530 1540 1550 2 "imaginary" 1545 1560 1570 1580 1590 1600 3 René Descartes France 1596 1650 Philosopher 1610 1620 4 Antoine Arnauld France 1618 1698 Theologian 5 Blaise Pascal France 1623 1662 Philosopher 1630 4 Pierre Nicole France 1625 1685 Theologian 1640 6 Sir Isaac Newton England 1643 1727 Polymath 1650 7 Gottfried Wilhelm Leibniz Germany 1646 1716 Polymath 1660 8 Jacob Bernoulli Switzerland 1655 1705 Mathematician 4 Probability 1662 Schrausser, D. G. (2025). Mathematical and statistical applications for HP Prime. 7 1670 9 Abraham de Moivre France 1667 1754 Mathematician 3 Complex numbers 1664 5 Combinatorics 1665 1680 1690 10 Brook Taylor England 1685 1731 Mathematician 6,7 Calculus 1684 1700 11 Daniel Bernoulli Switzerland 1700 1782 Mathematician 12 Rev. Thomas Bayes England 1701 1761 Theologian 13 Leonhard Euler Switzerland 1707 1783 Mathematician 1710 9,8 Binomial distribution 1711 1720 10 Taylor series of sine 1715 1730 11 Gamma 1729 1740 9 Normal distribution 1738 1750 13 Complex exponential functions 1748 1760 12 Bayes' theorem 1763 1770 14 Jean-Robert Argand Switzerland 1768 1822 Polymath 1780 15 Johann Carl Friedrich Gauss Germany 1777 1855 Mathematician 1790 1800 1810 15 Estimator of mean 1809 1820 14 Argand diagram 1813 1830 16 Sir Francis Galton England 1822 1911 Anthropology 1840 1850 17 Friedrich Robert Helmert Germany 1843 1917 Geodesy, mathematics 18 Jacob Lüroth Germany 1844 1910 Mathematics 1860 19 Karl Pearson England 1857 1936 Biology, mathematics 1870 17,18 𝑡-, 𝜒2-distribution 1876 1880 16 Reversion 1877 1890 20 Louis Leon Thurstone USA 1887 1955 Psychophysics 1900 21 Sir Ronald Aylmer Fisher England 1890 1962 Biology, mathematics 1910 19 Correlation 1904 1920 1930 22 Jacob Cohen USA 1923 1998 Psychology, statistics 21 𝐹-distribution 1924 1940 20 Factor analysis 1931 21 Permutation test 1935 23 Bradley Efron USA 1938 Statistics 1950 1960 1970 1980 23 Bootstrap 1979 1990 22 Effect size 1988 2000 References Abraham Bar Hiyya Savasorda, & et al. 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