HP_Prime_MATH: Manual

Schrausser, D. G. (2025). HP_Prime_MATH: Manual. https://www.academia.edu/130037807 1 HP_Prime_MATH: Manual Dietmar G. Schrausser Karl-Franzens University, Graz, Austria Introduction Mathematical and statistical applications for HP Prime (s. HP Inc., 2017; Schrausser, 2025). Algorithms are presented in context with the corresponding scope of application (s. Functions). CAS programs (1), HP Prime User functions (2) and functions for HP Prime Applications (3) are listed in alphabetical order (s. Source Codes), for a comparison to corresponding SCHRAUSSER-MAT functions (Schrausser, 2022a) see Table 2. In addition to the source codes of the functions, raw data sets are provided for correlationas well as resampling-methods (s. Data). 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. Introducing works on resampling methods are given by e.g. Good (2006) or Beasley and Rodgers (2009), 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). For the history of statistical inference in general see e.g. Stigler (1986) and Hald (1990, 1998, 2003, 2007), 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). Functions 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. Table 1. Appropriate correlation coefficients; product-moment or Pearson correlation 𝑟𝑥𝑦, Spearman’s rank correlation coefficient rho 𝜌, biserial (or biseral) coefficients 𝑟𝑏𝑖𝑠, 𝑟𝑝𝑏𝑖𝑠, 𝑟𝑏𝑖𝑠𝑅 and phi coefficient 𝛷 at the corresponding scale levels, interval i, ordinal o, and nominal n. i o n i 𝑟𝑥𝑦 o 𝜌 ¹ n 𝑟𝑝𝑏𝑖𝑠, 𝑟𝑏𝑖𝑠 𝑟𝑏𝑖𝑠𝑅 𝛷 ² ¹) also Kendall’s tau 𝜏 (1938) or Somers’ 𝐷 (1962). ²) also tetrachoric correlation 𝑟𝑡𝑒𝑡. Creative Commons Attribution 4.0 International Schrausser, D. G. (2025). HP_Prime_MATH: Manual. https://www.academia.edu/130037807 2 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). [KOR|IC_M] [rxy|RED|tr|TRW|pRW|pRWx] [E01] Pearson product-moment correlation coefficient 𝑟𝑥𝑦 Bravais (1844), Galton (1877), Pearson 1904, 1905). 𝑟𝑥𝑦=𝜎𝑥𝑦 2 𝜎𝑥⋅𝜎𝑦, 𝜎𝑥𝑦 2=∑ (𝑥𝑖−𝑥)⋅(𝑦𝑖−𝑦) 𝑛𝑖=1 𝑛 with 𝑡(𝑑𝑓)=𝑟⋅√𝑛−2 √1−𝑟2 where 𝑟2 = coefficient of determination, redundancy 𝑑𝑒𝑡 𝜎𝑥𝑦 2 = covariance of 𝑥 and 𝑦 𝑑𝑓 = 𝑛−2 [RHO] Spearman’s 𝜌 Equivalent to the product moment correlation when rank values are present (s. Spearman, 1904). 𝑟𝑠=𝜌=1−6⋅∑𝑑𝑖2𝑛𝑖=1 𝑛⋅(𝑛2−2) with 𝑡(𝑑𝑓)=𝜌⋅√𝑛−2 √1−𝜌2;𝑛≥30 where 𝑑𝑖 = rank difference of 𝑥𝑖 and 𝑦𝑖 𝑑𝑓 = 𝑛−2 Schrausser, D. G. (2025). HP_Prime_MATH: Manual. https://www.academia.edu/130037807 3 [TAU] Kendall’s tau 𝜏𝑎 Without adjustment for ties (s. Kendall, 1938). 𝜏𝑎=1− 2⋅𝑛𝑑 0.5⋅𝑛⋅(𝑛−1), with 𝑧=3⋅𝜏𝑎⋅√𝑛⋅(𝑛−1) √2⋅(2⋅𝑛+5);𝑛>10 alternatively 𝑧= 𝑛𝑐−𝑛𝑑 √1 18⋅𝑛⋅(𝑛−1)⋅(2⋅𝑛+5) where 𝑛 = total number of pairs 𝑛𝑑 = number of discordant pairs 𝑛𝑐 = number of concordant pairs, with 𝑛𝑐=(𝑛2)−𝑛𝑑 [DELTA2] Somers’ 𝐷 For binary data [0,1] (s. Somers, 1962). 𝐷𝑌𝑋=𝑛1,1 𝑛−𝑛1,0 𝑛 where 𝑛 = total number of pairs 𝑛1,1 = number of pairs with 𝑌=1,𝑋=1 𝑛1,0 = number of pairs with 𝑌=1,𝑋=0 [rpbis] Point biserial correlation coefficient 𝑟𝑝𝑏 Also point biseral. 𝑟𝑝𝑏=𝑥1−𝑥0 𝜎𝑥⋅√𝑛1⋅𝑛2 𝑛2 with 𝑡(𝑑𝑓)=𝑟𝑝𝑏⋅√𝑛−2 √1−𝑟𝑝𝑏 2 Schrausser, D. G. (2025). HP_Prime_MATH: Manual. https://www.academia.edu/130037807 4 where 𝑑𝑓 = 𝑛−2 [rbis|srbis|zrbis|prbis] Biserial correlation coefficient 𝑟𝑏𝑖𝑠 Pearson (1909), see e.g. Tate (1955), also called biseral. 𝑟𝑏𝑖𝑠=𝑥1−𝑥0 𝜎𝑥⋅𝑛1⋅𝑛2 𝜗⋅𝑛2 with 𝑧=𝑟𝑏𝑖𝑠 𝜎𝑟𝑏𝑖𝑠, 𝜎𝑟𝑏𝑖𝑠=√𝑛1⋅𝑛2 𝜗⋅𝑛⋅√𝑛 where 𝜗= 1 √2⋅π⋅𝑒−𝐹(𝑝=𝑛0 𝑛)2 2 [rbisR|U_1|U_2|zrbisR|prbisR] Rank biserial correlation coefficient 𝑟𝑏𝑖𝑠𝑅 Also rank biseral correlation, corresponds to the effect size for the Mann–Whitney 𝑈 test (Mann and Whitney, 1947). 𝑟𝑏𝑖𝑠𝑅=2𝑛⋅(𝑖1−𝑖2) with 𝑧= 𝑈−𝑛1⋅𝑛2 2 √𝑛1⋅𝑛2⋅(𝑛+1) 12 where 𝑈=𝑛1⋅𝑛2+𝑛12+𝑛1 2−∑𝑥𝑖 𝑛1 𝑖=1 [PHC] [PHI|xPHI|pPHI] Phi coefficient 𝛷 Yule (1912). Schrausser, D. G. (2025). HP_Prime_MATH: Manual. https://www.academia.edu/130037807 5 𝛷= 𝑎⋅𝑑−𝑏⋅𝑐 √(𝑎+𝑐)⋅(𝑏+𝑑)⋅(𝑎+𝑏)⋅(𝑐+𝑑) with 𝜒(𝑑𝑓) 2=𝑛⋅𝛷2 where 𝑑𝑓=1 [PHC] [rtet|srtet|prtet] 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. 𝑟𝑡𝑒𝑡=cosπ 1+√𝑏⋅𝑐 𝑎⋅𝑑 with 𝑧=𝑟𝑡𝑒𝑡 𝜎𝑟𝑡𝑒𝑡, 𝜎𝑟𝑡𝑒𝑡=√𝑎+𝑏 𝑛⋅𝑎+𝑐 𝑛⋅𝑐+𝑑 𝑛⋅𝑏+𝑑 𝑛 𝑛⋅1 𝜗𝑥⋅𝜗𝑦 where 𝜗𝑥=1 √2⋅π⋅e−𝐹(𝑝=𝑐+𝑑 𝑛)2 2 𝜗𝑦=1 √2⋅π⋅e−𝐹(𝑝=𝑏+𝑑 𝑛)2 2 [PKR] [rxy_z|zrxy_z|prxy_z|ry_xz] Partial correlation 𝑟𝑥𝑦⋅𝑧 𝑟𝑥𝑦⋅𝑧=𝑟𝑥𝑦−𝑟𝑥𝑧⋅𝑟𝑦𝑧 √1−𝑟𝑥𝑧 2⋅√1−𝑟𝑦𝑧 2 with 𝑧=𝑍𝑟𝑥𝑦⋅𝑧⋅√𝑛−2 and semi partial correlation 𝑟𝑦(𝑥⋅𝑧)=𝑟𝑥𝑦−𝑟𝑥𝑧⋅𝑟𝑦𝑧 √1−𝑟𝑥𝑧 2, Schrausser, D. G. (2025). HP_Prime_MATH: Manual. https://www.academia.edu/130037807 6 [ZCor] [Zr|rZ] Fisher 𝑍-transformation Fisher (1915). 𝑍=12⋅ln1+𝑟 1−𝑟 with 𝑧= 𝑍 √1 𝑛−3 and 𝑟𝑍=𝑒2⋅𝑍−1 𝑒2⋅𝑍+1 [Zrr|prr] Fisher 𝑍 difference, Cohen’s 𝑞 Cohen (1988, p. 110). 𝜃=𝑑𝑍=𝑍𝑟1−𝑍𝑟2 with 𝑧= 𝑑𝑍 √1 𝑛1−3+1 𝑛2−3 [mZ|mr] Averaged Fisher 𝑍 𝑍=∑ (𝑛𝑖−3) 𝑘𝑖=1 ⋅𝑍𝑖 ∑ (𝑛𝑖−3) 𝑘𝑖=1 [MCORR2] [MCORR|SCR|Cf2|FMCORR|pMCORR] Coefficient of multiple correlation 𝑅𝑐,12, Cohen’s 𝑓2 For 𝑅𝑐,12 2 see Olkin and Pratt (1958). 𝑅𝑐,12=√𝑟1𝑐 2+𝑟2𝑐 2−2⋅𝑟12⋅𝑟1𝑐⋅𝑟2𝑐 1−𝑟12 2, Schrausser, D. G. (2025). HP_Prime_MATH: Manual. https://www.academia.edu/130037807 7 𝑅𝑐,12 2=1− 𝑛−3 𝑛−𝑘−2⋅[(1−𝑅𝑐,12 2)+ 2 𝑛−𝑘⋅(1−𝑅𝑐,12 2)2];𝑘=2 with 𝑓2=𝑅𝑐,12 2 1−𝑅𝑐,12 2, 𝐹(3,𝑑𝑓2)=𝑅𝑐,12 2⋅(𝑛−4) (1−𝑅𝑐,12 2)⋅3 where 𝑓2 = effect size for multiple regression (Cohen, 1988, p. 410) 𝑑𝑓2=𝑛−4 Exposure functions To calculate the appropriate time-aperture-speed combination for given light values on a logarithmic scale, see e.g. Allbright (1991), Marsden and Weinstein (1985), Howie (2001) and Sobot (2021). [Ev|TEv|AEv] [E02|E03] Exposure value 𝐸𝑣 𝐸𝑣=log2𝐴𝑣2 𝑇𝑣−1=log(𝑇𝑣⋅𝐴𝑣2) log(2) hence 𝑇𝑣=2𝐸𝑣 𝐴𝑣2, 𝐴𝑣=√2𝐸𝑣⋅𝑇𝑣 𝑇𝑣 with 𝑇𝑣 = time value with 𝑇𝑣=𝑠−1 𝐴𝑣= aperture value 𝑓 [AvTv] Aperture 𝐴𝑣 for time 𝑇𝑣 with given 𝐸𝑣 𝐴𝑣𝑇𝑣=𝐴𝑣𝑇𝑣0⋅𝑎𝑇𝑣 with 𝑎𝑇𝑣=212⋅log2𝑇𝑣0 𝑇𝑣 Schrausser, D. G. (2025). HP_Prime_MATH: Manual. https://www.academia.edu/130037807 8 =e12⋅log(𝑇𝑣0) 𝑇𝑣 where 𝑇𝑣 = time value with 𝑇𝑣=𝑠−1 𝑇𝑣0 = initial time value with 𝑇𝑣0=𝑠−1 𝐴𝑣= aperture value 𝑓 [AvS] [E03] Aperture 𝐴𝑣 for speed 𝑆 with given 𝐸𝑣 𝐴𝑣𝑆=𝐴𝑣𝑆0⋅𝑎𝑆 with 𝑎𝑆=212⋅log2𝑆 𝑆0 =e12⋅log𝑆 𝑆0 where 𝑆 = arithmetic speed 𝐼𝑆𝑂 𝑆0 = initial arithmetic speed 𝐼𝑆𝑂 𝐴𝑣= aperture value 𝑓 [AvTvk] Aperture 𝐴𝑣 shift from time 𝑇𝑣 in steps 𝑘 𝑇𝑣𝑛−𝑘=𝑇𝑣𝑛⋅2𝑘,𝑇𝑣𝑛+𝑘=𝑇𝑣𝑛 2𝑘, with 𝐴𝑣=𝐴𝑣0⋅√2𝑘 where 𝐴𝑣0 = initial aperture value [AvSk] Aperture 𝐴𝑣 shift from speed 𝑆 in steps 𝑘 𝑆𝑛+𝑘=𝑆𝑛⋅√2𝑘,𝑆𝑛−𝑘=𝑆𝑛 √2𝑘, with 𝐴𝑣=𝐴𝑣0⋅√2𝑘 where Schrausser, D. G. (2025). HP_Prime_MATH: Manual. https://www.academia.edu/130037807 9 𝐴𝑣0 = initial aperture value [ISOL|ISOA] Speed 𝑆 in logarithmic 𝐼𝑆𝑂° or arithmetic 𝐼𝑆𝑂 conversion 𝑆°=10⋅log10(𝑆)+1=10⋅log(𝑆) log(10)+1, 𝑆=10𝑆°−1 10 Functions of integration, 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. [F01|F05] Circular function, π Weierstraß (1894, p. 53) describes π2=∫1 1−𝑥2 ∞ 0𝑑𝑥, which may be less heuristic. 𝑓(𝑥)=√1−(𝑥−𝑏 𝑎)2⋅𝑎+𝑐 with 𝐹(𝑥)=π2=∫𝑓 1 −1 (𝑥)𝑑𝑥;𝑎=1,𝑏=𝑐=0 [F01Z] Spherical functions, π For Source codes to volume integrals of the sphere see Schrausser (2024). 𝑓1(𝑥,𝑦)=√(1−𝑥2)+(1−𝑦2) with Schrausser, D. G. (2025). HP_Prime_MATH: Manual. https://www.academia.edu/130037807 16 𝑝 = probability 𝑛 = number of cases [CIXY] [CIr] [E01] Standard error of prediction 𝜎𝑦𝑥 , confidence interval 𝐶𝐼𝑝 𝜎𝑦𝑥=𝜎𝑦⋅√1−𝑟2 with 𝐶𝐼𝑝=𝑦𝑥±𝑧(1−1−𝑝 2)⋅𝜎𝑦𝑥 𝑝 = probability 𝑟 = correlation 𝑦 = predicted value 𝑦 [EPSILON] [EFG|EFR] [E01] Effect size 𝜖, Cohen’s 𝑑 Cohen (1977, 1988, p. 20, p. 49, 1992), Borenstein et al. (1997), Borenstein et al. (2001). 𝜖=𝑑=𝜇1−𝜇0 𝜎 , 𝑑𝑣=𝑑 √1−𝑟 with 𝑥𝑐𝑟𝑖𝑡 𝛽=𝜇1±𝑡(𝑝𝑐𝑟𝑖𝑡,𝑑𝑓)⋅𝜎𝑥, 𝑡(𝑑𝑓) 𝛼=𝑥01−𝜇0 𝜎𝑥, 𝑡(𝑑𝑓) 𝛽=𝑥01−𝜇1 𝜎𝑥 where 𝑑𝑣 = 𝑑 for paired samples 𝑟 = correlation Power = 𝑝1−𝛽=1−𝑝𝛽 [EPSILON2] Optimal effect size 𝜖𝑝 𝜖𝑝=√(2⋅𝑡(𝑝𝑐𝑟𝑖𝑡,𝑑𝑓))2 𝑛, Schrausser, D. G. (2025). HP_Prime_MATH: Manual. https://www.academia.edu/130037807 17 [EPSILON2] Optimal alpha level 𝑡(𝑜𝑝𝑡,𝑑𝑓) 𝛼=√𝜖2⋅𝑛 2 [TKV] [tTKV|pTKV] Variance difference 𝑡-test For paired samples (𝑥1|𝑥2). 𝜃=𝑑𝜎2=𝜎12−𝜎22 with 𝑡(𝑑𝑓)=𝑑𝜎2⋅√𝑛−2 2⋅√𝜎12⋅𝜎22⋅(1−𝑟2) where 𝑑𝑓 = 𝑛−2 [TV_] [tTV|pTV] Paired 2-sample 𝑡-test 𝜃=𝑥𝑑=∑𝑥(𝑖,1)−𝑥(𝑖,2) 𝑛𝑖=1 𝑛 with 𝑡(𝑑𝑓)=𝑥𝑑 𝜎𝑥𝑑, 𝜎𝑥𝑑=√∑(𝑥(𝑖,1)−𝑥(𝑖,2))2 𝑛𝑖=1 −(∑𝑥(𝑖,1)−𝑥(𝑖,2) 𝑛𝑖=1 )2 𝑛 𝑛−1 ⋅1 √𝑛 where 𝑥𝑑 = mean of the differences of 𝑥1 and 𝑥2 values 𝑑𝑓 = 𝑛−1 [TU_] [tTU_|pTU_|tTUx|pTUx] Unpaired 2-sample 𝑡-test 𝜃=𝑑𝑥=𝑥1−𝑥2 Schrausser, D. G. (2025). HP_Prime_MATH: Manual. https://www.academia.edu/130037807 18 with 𝑡(𝑑𝑓)=𝑑𝑥 𝜎𝑑𝑥, 𝜎𝑑𝑥=√∑(𝑥(𝑖,1)−𝑥1)2 𝑛1 𝑖=1 +∑(𝑥(𝑖,2)−𝑥2)2 𝑛2 𝑖=1 𝑛−2 ⋅√1 𝑛1+1 𝑛2 where 𝑑𝑥 = difference of the means 𝑥1 and 𝑥2 𝑑𝑓 = 𝑛1+𝑛2=𝑛−2 [TT_] [tTT_|pTT_] One-sample 𝑡-test 𝜃=𝑑𝑥𝑦=𝑥−𝑦 with 𝑡(𝑑𝑓)=𝑑𝑥𝑦 √𝜎2 𝑛−1 where 𝑑𝑥𝑦 = difference between sample mean 𝑥 and test value 𝑦 𝑑𝑓 = 𝑛−1 [ABT1] [x2F|p2F|zBN|pzBN] 𝜒2-test for independence 𝜒2=∑(𝑓𝑒𝑖−𝑓𝑏𝑖)2 𝑓𝑏𝑖 𝑛 𝑖=1 with 𝑧=𝑏−𝑏+𝑐 2 √𝑏+𝑐 4 [VFCH] [x4F|p4F|x4FY|p4FY|z4F|pz4F] 2 × 2 𝜒2-test for independence For Yates’s correction for continuity see Yates (1934). Schrausser, D. G. (2025). HP_Prime_MATH: Manual. https://www.academia.edu/130037807 19 𝜒2=𝑁⋅(𝑎⋅𝑑−𝑏⋅𝑐)2 (𝑎+𝑏)⋅(𝑐+𝑑)⋅(𝑎+𝑐)⋅(𝑏+𝑑), 𝜒𝑌𝑎𝑡𝑒𝑠 2=𝑁⋅(|𝑎⋅𝑑−𝑏⋅𝑐|⋅𝑁2)2 (𝑎+𝑏)⋅(𝑐+𝑑)⋅(𝑎+𝑐)⋅(𝑏+𝑑);4<𝑓𝑒<7 with 𝑧= 𝑑−𝑁⋅𝑃𝑑 √𝑁⋅𝑃𝑑⋅(1−𝑃𝑑)−𝑁⋅(𝑁−1)⋅𝑃𝑑⋅(𝑃𝑑−𝑃𝑑) where 𝑓𝑒 = expected frequency 𝑃𝑑=(𝑑+𝑏)⋅(𝑐+𝑑) 𝑁2 𝑃𝑑=(𝑑+𝑏−1)⋅(𝑐+𝑑−1) (𝑁−1)2 𝑑𝑓=1 [VFCH] [xMN|pMN|xMNY|pMNY] McNemar’s 𝜒2-test for paired 2 × 2 contingency tables with dichotomous trait McNemar (1947). 𝜒2=(𝑏−𝑐)2 𝑏+𝑐 , 𝜒2=(|𝑏−𝑐|−12)2 𝑏+𝑐 ;20<(𝑏+𝑐)<30 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). Schrausser, D. G. (2025). HP_Prime_MATH: Manual. https://www.academia.edu/130037807 20 The arguably most important methods regarding the calculation of probability parameters are implemented. [Ch|ChA|ChB] Arcsine transformation, Cohen’s ℎ Cohen (1988, p. 181). 𝜃=𝑝1−𝑝2, ℎ=2⋅sin−1√𝑝1−2⋅sin−1√𝑝2 with 𝑝1=sin(2⋅sin−1√𝑝2+ℎ 2)2, 𝑝2=−sin(−2⋅sin−1√𝑝1+ℎ 2)2 where probabilities = 𝑝1, 𝑝2 [ABT1] [ADDP] [E01] Additive probability for independent events 𝑢𝑝(∪𝑛𝐴) Corresponds to the geometric distribution 𝑓(𝑋≤𝑟|𝑝). 𝑢𝑝(∪𝑛𝐴)=1−(1−𝑝𝐴)𝑛 where 𝑛 = number of events 𝐴 𝑝𝐴 = probability of event 𝐴 [GMVTLG] Geometric distribution 𝑓(𝑋≤𝑟|𝑝) Corresponds to the additive probability 𝑢𝑝(∪𝑛𝐴). 𝑓(𝑋=𝑟|𝑝)=𝑃𝑛=𝑝⋅𝑞𝑟 with 𝑓(𝑋≤𝑟|𝑝)=𝑝𝑛=∑𝑝⋅𝑞𝑖 𝑟 𝑖=0 where Schrausser, D. G. (2025). HP_Prime_MATH: Manual. https://www.academia.edu/130037807 21 𝑝 = probability of event 𝑟+1=𝑛 = number of events [NBNMVTLG] [NBINOM] [E01] 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). 𝑓(𝑋=𝑟|𝑟,𝑝)=𝑃𝑛=(𝑘+𝑟−1)! 𝑟!⋅(𝑘−1)!⋅𝑝𝑘⋅𝑞𝑟 with 𝑓(𝑋≤𝑟|𝑟,𝑝)=𝑝𝑛=∑(𝑘+𝑖−1)! 𝑖!⋅(𝑘−1)!⋅𝑝𝑘⋅𝑞𝑖 𝑟 𝑖=0 where 𝑟+𝑘=𝑛 = number of events 𝑘 = number of successes [ABT1] [BINOM|zBN|pzBN] [E01] Exact binomial test 𝑓(𝑋=𝑏|𝑏,𝑐)=𝑃0=(𝑏+𝑐)! 𝑏!⋅𝑐! ⋅2−𝑏⋅2−𝑐 with 𝑓(𝑋≤𝑏|𝑏,𝑐)=𝑝=𝑝𝑒𝑥𝑎𝑐𝑡1=∑ (𝑏+𝑐)! 𝑖!⋅(𝑏+𝑐−𝑖)! 𝑏 𝑖=0 ⋅2−𝑖⋅2−(𝑏+𝑐−𝑖);𝑝≤12, 𝑝𝑒𝑥𝑎𝑐𝑡1=(1−𝑝)+𝑃0;𝑝>12 also 𝑧=𝑏−𝑏+𝑐 2 √𝑏+𝑐 4 [FX_] [z4F|pz4F] Exact hypergeometric 2 × 2 test Fisher Exact test (Fisher, 1922; Agresti, 1992). 𝑓(𝑋=𝑎|𝑎,𝑏,𝑐,𝑑)=𝑃0=(𝑎+𝑏)!⋅(𝑐+𝑑)!⋅(𝑎+𝑐)!⋅(𝑏+𝑑)! 𝑁!⋅𝑎!⋅𝑏!⋅𝑐!⋅𝑑! Schrausser, D. G. (2025). HP_Prime_MATH: Manual. https://www.academia.edu/130037807 22 with 𝑓(𝑋≤𝑎|𝑎,𝑏,𝑐,𝑑)=𝑝𝑒𝑥𝑎𝑐𝑡1=∑𝑃 𝑎 𝑖=1 𝑖;𝑝≤12, 𝑓(𝑋≥𝑎|𝑎,𝑏,𝑐,𝑑)=𝑝𝑒𝑥𝑎𝑐𝑡1=∑𝑃 𝑛 𝑖=𝑎 𝑖;𝑝>12 where 𝑃𝑖=(𝑎+𝑏)!⋅(𝑐+𝑑)!⋅(𝑎+𝑐)!⋅(𝑏+𝑑)! 𝑁!⋅𝑖!⋅(𝑎+𝑏−𝑖)!⋅(𝑎+𝑐−𝑖)!⋅(2⋅𝑐+𝑑−𝑎−𝑖)! also 𝑧= 𝑑−𝑁⋅𝑃𝑑 √𝑁⋅𝑃𝑑⋅(1−𝑃𝑑)−𝑁⋅(𝑁−1)⋅𝑃𝑑⋅(𝑃𝑑−𝑃𝑑) where 𝑃𝑑=(𝑑+𝑏)⋅(𝑐+𝑑) 𝑁2 𝑃𝑑=(𝑑+𝑏−1)⋅(𝑐+𝑑−1) (𝑁−1)2 𝑑𝑓=1 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 functions generate permutation and variation matrices primarily to support the resampling procedures described below (s. Resampling). [PRM2] Permutation matrix 𝑷𝒏 𝑛 elements to 𝑘=1 class. 𝐏𝐧=[𝑝1(𝑥1)⋯ 𝑝1(𝑥𝑛) ⋮ ⋱ ⋮ 𝑝𝑃(𝑥1)⋯ 𝑝𝑃(𝑥𝑛)] where 𝑃𝑛=𝑛! Schrausser, D. G. (2025). HP_Prime_MATH: Manual. https://www.academia.edu/130037807 23 [PRM5] Variation matrix 𝒘𝑽𝟐𝒎 For the dependent 2 sample design, 𝑛=2 elements to class 𝑚. 𝒘𝑽𝟐𝒎=[𝑣1(𝑥1)⋯ 𝑣1(𝑥𝑚) ⋮ ⋱ ⋮ 𝑣𝑤𝑉(𝑥1)⋯ 𝑣𝑤𝑉(𝑥𝑚)] where 𝑤𝑉2𝑚=2𝑚 [PRM4] Variation matrix 𝒘𝑽𝒏𝒎 𝑛 elements to class 𝑚. 𝒘𝑽𝒏𝒎=[𝑣1(𝑥1)⋯ 𝑣1(𝑥𝑚) ⋮ ⋱ ⋮ 𝑣𝑤𝑉(𝑥1)⋯ 𝑣𝑤𝑉(𝑥𝑚)] where 𝑤𝑉𝑛𝑚=𝑛𝑚;𝑛>𝑚 [PRM3] [nk] Permutation matrix 𝒘𝑷𝒏(𝒌𝒎,𝒌𝒏−𝒎) 𝑛 elements to class 𝑚. 𝒘𝑷𝒏(𝒌𝒎,𝒌𝒎−𝒏)=[𝑝1(𝑥11)⋯ 𝑝1(𝑥𝑘1) 𝑝1(𝑥12)⋯ 𝑝1(𝑥𝑘2) ⋮ ⋱ ⋮ ⋮ ⋱ ⋮ 𝑝𝑤𝑃(𝑥11)⋯ 𝑝𝑤𝑃(𝑥𝑘1) 𝑝𝑤𝑃(𝑥12)⋯ 𝑝𝑤𝑃(𝑥𝑘2)] where 𝑤𝑃𝑛(𝑘𝑚,𝑘𝑛−𝑚)=𝑛! ∏𝑘𝑖 2𝑖=1 !;𝑛≥𝑚 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 Schrausser, D. G. (2025). HP_Prime_MATH: Manual. https://www.academia.edu/130037807 24 e.g. Solomon (1982), Dallal (1986, 1988), Peladeau (1993), Wooff and Peladeau (1994), Mehta et al. (2014), also Schrausser (2024d). [PV_] Permutation test P for 2 paired samples (𝑥1|𝑥2) Random sampling model, systematic permutation, 𝑝-value not randomized, variation matrix 𝒘𝑽𝟐𝒎 required, s. Scambor (1997), Scambor and Schrausser (2022, p. 7), respectively. 𝛩11=∑𝑥1𝑖 𝑛 𝑖=1 ,𝛩21=∑𝑥2𝑖 𝑛 𝑖=1 , 𝛩2=(∑𝑥1𝑖 𝑛 𝑖=1 )2+(∑𝑥2𝑖 𝑛 𝑖=1 )2 with 𝑝𝑒𝑥𝑎𝑐𝑡=∑1 2𝑛 𝑖=1 2𝑛;𝜃𝑖≥𝛩 where 𝛩11,𝛩21 = one-tailed test values 𝛩2 = two-tailed test value [mPV_] Randomized permutation test mP for 2 paired samples (𝑥1|𝑥2) Random sampling model, 𝑝-value not randomized. 𝛩11=∑𝑥1𝑖 𝑛 𝑖=1 ,𝛩21=∑𝑥2𝑖 𝑛 𝑖=1 , 𝛩2=(∑𝑥1𝑖 𝑛 𝑖=1 )2+(∑𝑥2𝑖 𝑛 𝑖=1 )2 with 𝑝=∑1 𝑀 𝑖=1 𝑀;𝜃𝑖≥𝛩 where 𝛩11,𝛩21 = one-tailed test values 𝛩2 = two-tailed test value 𝑀 = simulation cycles over variations 𝑤𝑉2𝑚=2𝑛 Schrausser, D. G. (2025). HP_Prime_MATH: Manual. https://www.academia.edu/130037807 25 [PU_] 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). 𝛩11=∑𝑥𝑔1𝑖 𝑛1 𝑖=1 ,𝛩21=∑𝑥𝑔2𝑖 𝑛2 𝑖=1 , 𝛩2=|𝑥𝑔1−𝑥𝑔2| with 𝑝𝑒𝑥𝑎𝑐𝑡=∑1 𝑛! 𝑛1!⋅𝑛2! 𝑖=1 𝑛! 𝑛1!⋅𝑛2!;𝜃𝑖≥𝛩 where 𝛩11,𝛩21 = one-tailed test values 𝛩2 = two-tailed test value 𝑛=𝑛1+𝑛2 [mPU_] Randomized permutation test mP for 2 independent samples (𝑥|𝑔) Random sampling model, 𝑝-value not randomized. 𝛩11=∑𝑥𝑔1𝑖 𝑛1 𝑖=1 ,𝛩21=∑𝑥𝑔2𝑖 𝑛2 𝑖=1 , 𝛩2=|𝑥𝑔1−𝑥𝑔2| with 𝑝=∑1 𝑀 𝑖=1 𝑀;𝜃𝑖≥𝛩 where 𝛩11,𝛩21 = one-tailed test values 𝛩2 = two-tailed test value 𝑛=𝑛1+𝑛2 𝑀 = simulation cycles over permutations 𝑤𝑃𝑛(𝑘𝑚,𝑘𝑛−𝑚)=𝑛! 𝑛1!⋅𝑛2! [BtU_] Bootstrap test Bt for 2 independent samples (𝑥|𝑔) Quenouille (1949), Efron (1979, 1981, 1982). Schrausser, D. G. (2025). HP_Prime_MATH: Manual. https://www.academia.edu/130037807 32 L3(3) NTHROOT (product((L1(x)(1))^L1(x)(2),x,1,L3(1)))▸L3(5) L3(3)/(Σ(L1(x)(2)/L1(x)(1),x,1,L3(1)))▸L3(6) //n,AM,sumni,GAM,GGM,GHM L3 END; #end // B BNMVTLG.pas //BNMVTLG(p[e],a=k,n)/D.G.SCHRAUSSER/2025 //e.g.BNMVTLG(0.5,5,10) #cas BNMVTLG(P,K,N):= BEGIN B=0; FOR I FROM 0 TO K DO BINOMIAL(N,P,I)▸L4(I) B=B+L4(I) END; D5=L4;L4={} FOR I FROM 0 TO N DO BINOMIAL(N,P,I)▸L5(I); END; D6=L5;L5={}; STARTAPP("Statistiken_1_Var"); STARTVIEW(1); "D5"▸H1(1);5▸H1(3); "D6"▸H2(1);5▸H2(3); //p RETURN(B); END; #end // BtU_.pas //BtU_(simulation cycles B)/D.G.SCHRAUSSER/2025 //Bootstrap method, Bt //2 independent samples (x|g) //e.g.BtU_(1000) #cas BtU_(B):= BEGIN //L1L2 provided {}▸L3;{}▸L4 {}▸L5;{}▸L6 0▸M11 0▸M12 0▸M2 SIZE(L1)▸N1 SIZE(L2)▸N2 N=N1+N2 ABS(mean(L1)-mean(L2))▸Q02 ΣLIST(L1)▸Q011 ΣLIST(L2)▸Q012 // CONCAT(L1,L2)▸L9 MSGBOX("BtU") FOR J FROM 1 TO B DO // FOR A FROM 1 TO N DO L9(RANDINT(N))▸L0(A) END; FOR A FROM 1 TO N1 DO Schrausser, D. G. (2025). HP_Prime_MATH: Manual. https://www.academia.edu/130037807 33 L0(A)▸L3(A) END; FOR A FROM 1 TO N2 DO L0(N1+A)▸L4(A) END; ABS(mean(L3)-mean(L4))▸QJ2 ΣLIST(L3)▸QJ11 ΣLIST(L4)▸QJ12 IF QJ11≥Q011 THEN M11=M11+1 END; IF QJ12≥Q012 THEN M12=M12+1 END; IF QJ2≥Q02 THEN M2=M2+1 END; QJ11▸L5(J) QJ2▸L6(J) END; // SORT(L5)▸L5 SORT(L6)▸L6 {}▸L9 {}▸L0 N1,N2,[Q011,Q012,Q02],M11/B,M12/B,M2/B END; #end // C ch2VTLG.pas //ch2VTLG(chi-squared,df)/D.G.SCHRAUSSER/2025 //e.g.ch2VTLG(2.65,1)[AdvancedGraphing] #cas ch2VTLG(C2569,A7485):= BEGIN G=Gamma(A7485/2) //P=∫((1/(2^(A7485/2)*G))*X^((A7485/2)-1)*e^(-X/2),X,0,C) P=CHISQUARE_CDF(A7485,C2569) A7485▸A C2569▸C "Y=(1/(2^(A/2)*G))*X^((A/2)-1)*e^(-X/2)"▸V1 "Y<(1/(2^(A/2)*G))*X^((A/2)-1)*e^(-X/2) AND Y>0 AND X<C AND X>0"▸V2 STARTAPP("Erweiterte_Grafiken"); STARTVIEW(1); [1-P] END; #end // CIXY.pas //CIXY(x,y'CI)/D.G.SCHRAUSSER/2022 //Standard error of prediction sy'x, CI //e.g.CIXY(3,0.99[ZWERT,Statistics_2_Var,Spreadsheet,AdvancedGraphing] #cas CIXY(X,C):= BEGIN //C1C2 provided STARTAPP("Statistiken_2_Var"); STARTVIEW(−6) A=Corr B=sY D=MeanY PredY(X)▸L3(2); √(1-A^2)*B*NORMALD_ICDF(1-((1-C)/2))▸L3(4) L3(4)+L3(2)▸L3(3) L3(2)-L3(4)▸L3(1) C▸L3(5) ZWERT(L3(1),D,B)▸L4(1) ZWERT(L3(2),D,B)▸L4(2) ZWERT(L3(3),D,B)▸L4(3) L4(1)▸U Schrausser, D. G. (2025). HP_Prime_MATH: Manual. https://www.academia.edu/130037807 34 L4(3)▸O ZWERT(X,MeanX,sX)▸Q STARTAPP("Arbeitsblatt"); "ŷ-"▸A1;L3(1)▸B1;L4(1)▸C1 "ŷ"▸A2;L3(2)▸B2;L4(2)▸C2 "ŷ+"▸A3;L3(3)▸B3;L4(3)▸C3 "±"▸A4;L3(4)▸B4;L3(4)/B▸C4 "CI"▸A5;L3(5)▸B5; STARTAPP("Erweiterte_Grafiken"); STARTVIEW(1) "Y=A*X"▸V3 "Y>0 AND (Y<A^(-1)*X AND Y>A*X) OR Y<0 AND (Y>A^(-1)*X AND Y<A*X)"▸V4 "Y=√(1-X^2)"▸V5 "Y=-1*√(1-X^2)"▸V6 "X<A AND X>0 AND Y<A AND Y>0"▸V7 CAS((X,Y)->((Y<O) AND (Y>U)) AND ((X==Q))▸V0) //y'-,y',y'+,CI,sy'x,CIp RETURN(L3); END; #end // CPLHX.pas //CPLHX(complex number,a+bi)/D.G.SCHRAUSSER/2022 //e.g.CPLHX(2+i/2),[AdvancedGraphing] #cas CPLHX(C):= BEGIN C▸Z1 RE(Z1)▸R IM(Z1)▸I Z1▸L1(1) ABS(Z1)▸L1(2) ARG(Z1)▸L1(3) "Y=R*X"▸V1 "Y=I"▸V2 "Y=√((R*X)^2+I^2)"▸V3 "Y=(I/ABS(I))*(π/2)-ATAN((R/I)*X)"▸V4 STARTAPP("Erweiterte_Grafiken") STARTVIEW(1) RETURN(L1); END; #end // CPLX.pas //CPLX(complex number,a+bi)/D.G.SCHRAUSSER/2022 //e.g.CPLX(2+i/2),[AdvancedGraphing] #cas CPLX(C):= BEGIN C▸Z1 RE(Z1)▸R IM(Z1)▸I ABS(Z1)▸L1(1) R▸X▸J I▸K "Y=√(1-X^2)"▸V5 "Y=-1*√(1-X^2)"▸V6 "Y=I"▸V7 "X=R"▸V8 "X<R AND X>0 AND Y>0 AND Y≤0.01"▸V0 "Y=(I/R)*X AND Y>0 AND X<R"▸V9 IF I<0 THEN "Y=(I/R)*X AND Y<0 AND X<R"▸V9 Schrausser, D. G. (2025). HP_Prime_MATH: Manual. https://www.academia.edu/130037807 35 END; IF R<0 THEN "Y=(I/R)*X AND Y>0 AND X>R"▸V9 END; IF I<0 AND R<0 THEN "Y=(I/R)*X AND Y<0 AND X>R"▸V9 END; ARG(Z1)▸L1(2) CONVERT(L1(2)_rad,1_deg)▸L1(3) RETURN(L1); END; #end // CPLX.pas //CPLX2(complex number,a+bi)/D.G.SCHRAUSSER/2022 //e.g.CPLX2(2+i/2),[CPLX,Spreadsheet]// #cas CPLX2(C):= BEGIN CPLX(C) STARTAPP("Arbeitsblatt"); "z"▸A1;Z1▸B1 "|z|"▸A2;L1(1)▸B2 "∡π"▸A3;L1(2)▸B3 "∡°"▸A4;L1(3)▸B4 END; #end // D DELTA2.pas //DELTA2()/D.G.SCHRAUSSER/2025 //Somers' D for binary values [0,1] #cas DELTA2():= BEGIN SIZE(L1)▸N {}▸L3 0▸X01 0▸X02 FOR I FROM 1 TO N DO IF L1(I)=1 AND L2(I)=1 THEN X01=X01+1 END; IF L1(I)=1 AND L2(I)=0 THEN X02=X02+1 END; END; X01/N▸L3(1) X02/N▸L3(2) L3(1)-L3(2)▸L3(3) //pA,pB,D approx(L3) END; #end // Schrausser, D. G. (2025). HP_Prime_MATH: Manual. https://www.academia.edu/130037807 36 E EPSILON.pas //EPSILON(x1,m1,m2,s12,d)/D.G.SCHRAUSSER/2022 //e.g.EPSILON(106,100,110,15,25) #cas EPSILON(X,M,N,S,D):= BEGIN G=Gamma((D+1)/2)/Gamma(D/2) E=(N-M)/S P=1-∫(G*(D*π)^(-1/2)*(1+(X^2/D))^(-(D+1)/2),X,−∞,E) T=(((N+M)/2)-M)/S H=1-∫(G*(D*π)^(-1/2)*(1+(X^2/D))^(-(D+1)/2),X,−∞,T) Q=(X-M)/S R=1-∫(G*(D*π)^(-1/2)*(1+(X^2/D))^(-(D+1)/2),X,−∞,Q) U=∫(G*(D*π)^(-1/2)*(1+(X^2/D))^(-(D+1)/2),X,−∞,(X-N)/S) B=1-U "X>T AND X≤T"▸V1 "X>Q AND X≤Q"▸V2 // D▸K "X>0 AND X<E AND Y<0 AND Y>-0.01"▸V3 "Y<(G*(K*π)^(-1/2)*(1+((E-X)^2/K))^(-(K+1)/2)) AND Y>0 AND X>Q"▸V6 "Y<(G*(K*π)^(-1/2)*(1+((E-X)^2/K))^(-(K+1)/2)) AND Y>0 AND X<Q"▸V7 "Y=(G*(K*π)^(-1/2)*(1+((E-X)^2/K))^(-(K+1)/2))"▸V8 "Y=(G*(K*π)^(-1/2)*(1+((X)^2/K))^(-(K+1)/2))"▸V0 "Y<(G*(K*π)^(-1/2)*(1+((X)^2/K))^(-(K+1)/2)) AND Y>0 AND X>Q"▸V9 E▸L2(1);P▸L3(1);N▸L1(1) T▸L2(2);H▸L3(2);T*S+M▸L1(2) Q▸L2(3);R▸L3(3);Q*S+M▸L1(3) U▸L4(3);B▸L5(3) STARTAPP("Arbeitsblatt"); "ε"▸A1;L2(3)▸B1;L2(2)▸C1;L2(1)▸D1; "x"▸A2;L1(3)▸B2;L1(2)▸C2;L1(1)▸D2; "α"▸A3;L3(3)▸B3;L3(2)▸C3;L3(1)▸D3; "β"▸A4;L4(3)▸B4; "1-β"▸A5;L5(3)▸B5; STARTAPP("Erweiterte_Grafiken") STARTVIEW(1) RETURN(L2(1),L3(3),L4(3)); END; #end // EPSILON2.pas //EPSILON2(epsilon,n,df,pcrit)/D.G.SCHRAUSSER/2022 //e.g.EPSILON2(0.38,100,99,0.95) //optimal effect size epsilon //optimal alpha t //1-p(alpha opt t) #cas EPSILON2(E,N,D,K):= BEGIN #t opt niv V=√(E^2*N)/2▸L6(2) P=1-STUDENT_CDF(D,V)▸L6(3) #e opt eff stke L=√((2*STUDENT_ICDF(D,K))^2/N)▸L6(1) "X>0 AND X<L AND Y<0 AND Y>-0.02"▸V1 RETURN(L6); END; #end // Schrausser, D. G. (2025). HP_Prime_MATH: Manual. https://www.academia.edu/130037807 37 EPSOLON3.pas //EPSILON3(100,110,15,25,0.99) #cas EPSILON3(M,N,S,D,K):= BEGIN F=STUDENT_ICDF(D,K) M+S*F▸L7(1) N-S*F▸L7(2) "X>E-F AND X≤E-F AND Y>0 AND Y<(G*(K*π)^(-1/2)*(1+((E-X)^2/K))^(-(K+1)/2))" ▸V4 "X>F AND X≤F AND Y>0 AND Y<(G*(K*π)^(-1/2)*(1+((X)^2/K))^(-(K+1)/2))"▸V5 RETURN(L7); END; #end // F FVTLG.pas //FVTLG(F,df1,df2)/D.G.SCHRAUSSER/2025 //e.g.FVTLG(2.8,10,5) #cas FVTLG(F,A,B):= BEGIN F▸X CAS(Gamma((A+B)/2))▸H CAS(Gamma(A/2))▸D CAS(Gamma(B/2))▸E CAS(H/(D*E))▸C CAS((X,Y)->Y=C*((A/B)^(A/2)*X^((A/2)-1)*(1+(A/B)*X)^(−(((A+B)/2)))) AND X>0 ▸V2) CAS((X,Y)->Y<C*((A/B)^(A/2)*X^((A/2)-1)*(1+(A/B)*X)^(−(((A+B)/2)))) AND Y>0 AND X<F AND X>0▸V1) FISHER_CDF(A,B,X)▸P STARTAPP("Erweiterte_Grafiken") STARTVIEW(1) P,[1-P] END; #end // FX.pas //FX_(cell count a,b,c,d)/D.G.SCHRAUSSER/2025 //e.g.FX_(1,2,3,1) //Exact hypergeometric 4-field test according to R. A. Fisher //(Fisher Exact Test): Hypergeometric probability p to cell a of the 4-field initial arrangement for all possible arrangements a //Exact significance levels p[exact1], p[exact2] #cas FX_(a,b,c,d):= BEGIN {}▸L1 1▸S 0▸X 0▸P20 0▸P21 0▸P3 a+b+c+d▸N a+b▸z1 c+d▸z2 a+c▸s1 b+d▸s2 P0= (z1!*z2!*s1!*s2!)/(N!*a!*b!*c!*d!); Schrausser, D. G. (2025). HP_Prime_MATH: Manual. https://www.academia.edu/130037807 38 P0▸L2(1) PRINT("P0-") PRINT(P0) PRINT("Pi-") // IF z1>s1 THEN max1=z1 ELSE max1=s1 END; IF z1>s2 THEN max2=z1 ELSE max2=s2 END; IF z2>s1 THEN max3=z2 ELSE max3=s1 END; IF z2>s2 THEN max4=z2 ELSE max4=s2 END; // FOR I FROM 0 TO max1 DO FOR J FROM 0 TO max2 DO FOR K FROM 0 TO max3 DO FOR L FROM 0 TO max4 DO a1=I+J; a2=K+L; b1=I+K; b2=J+L; // IF a1=z1 AND a2=z2 AND b1=s1 AND b2=s2 THEN IF I+J≠0 AND K+L≠0 AND I+K≠0 AND J+L≠0 THEN P10=(a1!*a2!*b1!*b2!)/(N!*I!*J!*K!*L!) X+1▸X P3=P10+P3 approx(P10)▸L1(X) IF approx(P10)<approx(P0) OR approx(P10)=approx(P0) THEN P20+P10▸P20 END; PRINT(P10) END; END; // END; END; END; END; PRINT("p--") PRINT(P3) // FOR I FROM 1 TO X DO IF L1(I)=L2(1) THEN 0▸S END; IF S=1 THEN L1(I)+P21▸P21 END; END; P21+P0▸P21 P22=1-(P21) //sums, P0, C, p[exact1], 1-p[exact1], p[exact2] z1,z2,s1,s2,N,[P0],X,P21,P22,[P20] END; #end // FX_.pas //FX_(cell count a,b,c,d)/D.G.SCHRAUSSER/2025 //e.g.FX_(1,2,3,1) Schrausser, D. G. (2025). HP_Prime_MATH: Manual. https://www.academia.edu/130037807 39 //Exact hypergeometric 4-field test according to R. A. Fisher //(Fisher Exact Test): Hypergeometric probability p to cell a of the 4-field initial arrangement for all possible arrangements a //Exact significance levels p[exact1], p[exact2] //(slow algorithm) #cas FX_(a,b,c,d):= BEGIN {}▸L1 1▸S 0▸X 0▸P20 0▸P21 0▸P3 a+b+c+d▸N a+b▸z1 c+d▸z2 a+c▸s1 b+d▸s2 P0= (z1!*z2!*s1!*s2!)/(N!*a!*b!*c!*d!); PRINT("P0-") PRINT(P0) PRINT("Pi-") // FOR I FROM 0 TO N DO FOR J FROM 0 TO N DO FOR K FROM 0 TO N DO FOR L FROM 0 TO N DO a1=I+J; a2=K+L; b1=I+K; b2=J+L; // IF a1=z1 AND a2=z2 AND b1=s1 AND b2=s2 THEN IF I+J≠0 AND K+L≠0 AND I+K≠0 AND J+L≠0 THEN P10=(a1!*a2!*b1!*b2!)/(N!*I!*J!*K!*L!) X+1▸X P3=P10+P3 approx(P10)▸L1(X) IF approx(P10)<approx(P0) OR approx(P10)=approx(P0) THEN P20+P10▸P20 END; PRINT(P10) END; END; // END; END; END; END; PRINT("p--") PRINT(P3) // FOR I FROM 1 TO X DO IF L1(I)=P0 THEN 0▸S END; Schrausser, D. G. (2025). HP_Prime_MATH: Manual. https://www.academia.edu/130037807 40 IF S=1 THEN L1(I)+P21▸P21 END; END; P21+P0▸P21 P22=1-(P21) // sums, P0, C, p[exact1], 1-p[exact1], p[exact2] z1,z2,s1,s2,N,[P0],X,P21,P22,[P20] END; #end // G GMVTLG.pas //GMVTLG(pA,r+1=n)/D.G.SCHRAUSSER/2025 //e.g.GMVTLG(1/6,10) #cas GMVTLG(P,N):= BEGIN MAKELIST(P*(1-P)^x,x,0,N-1)▸L1 MAKELIST(1-(1-P)^x,x,1,N)▸L2 STARTAPP("Statistiken_1_Var"); STARTVIEW(1); "L1"▸H1(1);6▸H1(3); "L2"▸H2(1);5▸H2(3); //P,p L1(N),L2(N) END; #end // I IC_M.pas //IC_M(n of variables k)/D.G.SCHRAUSSER/2025 //Intercorrelation matrix, Pearson correlation r //e.g.IC_M(5)[pCor] //M1:r //M2:det%(r²×100) //M3:t-value //M4:2-tailed p #cas IC_M(K):= BEGIN //L1(n)(k) provided {}▸L4 {}▸L5 {}▸L6 {}▸L7 {}▸L8 size(L1)▸N FOR I FROM 1 TO K DO FOR J FROM 1 TO K DO MAKELIST({L1(X,I),L1(X,J)},X,1,N)▸L4 approx(correlation(L4))▸R125▸L5(I,J) //r approx(pCor(R125,N)(3))▸L0;L0(1)▸L6(I,J) //p2 approx(pCor(R125,N)(1))▸L0;L0(1)▸L7(I,J) //t approx(R125^2*100)▸L8(I,J) //det END; END; Schrausser, D. G. (2025). HP_Prime_MATH: Manual. https://www.academia.edu/130037807 41 L5▸M1 L8▸M2 L7▸M3 L6▸M4 // END; #end // K KOR.pas //KOR()/D.G.SCHRAUSSER/2025 //Pearson corr/[pCor] #cas KOR():= BEGIN //L1()(2) provided SIZE(L1)(1)▸N N-2▸df covariance_correlation(L1)▸L0 L0(2)▸r L0(1)▸cv pCor(r,N)(2)▸p pCor(r,N)(3)▸p2 r^2*100▸D {}▸L0 // df,[cv,r,D],p,[p2] END; #end // M MCORR2.pas //MCORR2()/D.G.SCHRAUSSER/2022 //Multiple correlation R //[Statistiken_2_Var,Arbeitsblatt,Graph3D,FMCORR,MCORR] #cas MCORR2():= BEGIN //M1()(3) provided //C1C2 to S1 STARTAPP("Statistiken_2_Var"); STARTVIEW(−6) M2=TRN(M1) L7=M2(1);C1=L7 L7=M2(2);C2=L7 Do2VStats(S1) Corr▸L1(1) MeanX▸L3(1);sX▸L4(1) MeanY▸L3(2);sY▸L4(2) // L7=M2(1);C1=L7 L7=M2(3);C2=L7 Do2VStats(S1) Corr▸L1(2) MeanY▸L3(3);sY▸L4(3) // L7=M2(2);C1=L7 L7=M2(3);C2=L7 Schrausser, D. G. (2025). HP_Prime_MATH: Manual. https://www.academia.edu/130037807 48 approx(correlation(L4))▸r1 pCor(r1,N)(3)▸pr1 //ryz FOR I FROM 1 TO N DO L2(I)▸L5(1) L3(I)▸L5(2) L5▸L4(I) END; approx(correlation(L4))▸r2 pCor(r2,N)(3)▸pr2 //rxy.z rp=(r0-r1*r2) rp=rp/(sqrt(1-r1^2)*sqrt(1-r2^2)) rp=approx(rp) ZCor(rp,N)(1)▸L0 L0(1)*SQRT(N-2)▸zrp prp=NORMALD_CDF(zrp) IF prp>0.5 THEN prp=1-prp END; prp2=2*prp //pCor(rp,N)(3)▸p //df,rxy,p2,rxz,p2,ryz,p2,rxy.z,p2 df,[r0,pr0],[r1,pr1],[r2,pr2],[rp,prp] END; #end // PRM1.pas //PRM1(n perm)/D.G.SCHRAUSSER/2025 //e.g.PRM1(5)/permutation vector (p)n from L1 #cas PRM1(N):= BEGIN //L1(N) provided {}▸L0 {}▸L2 FOR A FROM 1 TO N DO {RANDOM(),L1(A)}▸L0(A) END; // sort(L0)▸L2 FOR A FROM 1 TO N DO L2(A)▸L8;L8(2)▸L2(A) END; // L2 END; #end // PRM2.pas //PRM2(elements n)/D.G.SCHRAUSSER/2025 //Complete permutation matrix (P)n of elements n to 1 class, //where P=n! //e.g.PRM2(3) #cas PRM2(n):= BEGIN MAKELIST(1,P,1,n+1)▸L1 P=PERM(n,n) 0▸L1(1) {}▸L2 0▸M1 1▸J 0▸I 0▸SW // WHILE I≠n AND L1(I)≤n DO FOR I FROM 1 TO n DO Schrausser, D. G. (2025). HP_Prime_MATH: Manual. https://www.academia.edu/130037807 49 IF I=1 THEN L1(1)+1▸L1(1) END; IF I=n AND L1(I)>n THEN BREAK END; IF L1(I)>n THEN 1▸L1(I);L1(I+1)+1▸L1(I+1) END; END;//I // FOR K FROM 1 TO n DO FOR L FROM K+1 TO n DO IF L1(K)=L1(L) THEN 1▸SW BREAK; END; END; END; IF SW=0 THEN SUPPRESS(L1,n+1)▸L2(J);J+1▸J END; 0▸SW END;//while // L2▸M1 IF n=2 THEN M1=[[1,2],[2,1]] M1▸L2 END; // P,M1 END; #end // PRM3.pas //PRM3(elements n, class m)/D.G.SCHRAUSSER/2025 //Complete permutation matrix w(P)n(km,kn-m) of n elements to class m, where P=n!/IIki!;n>=m //equivalent to combination without repetition Cn(m) //e.g.PRM3(6,3)[PRM3a] #cas PRM3(n,m):= BEGIN MAKELIST(1,P,1,n+1)▸L1 0▸L1(1) {}▸L2 0▸M1 1▸J 0▸I 0▸SW // WHILE I≠m AND L1(I)<n DO FOR I FROM 1 TO m DO IF I=1 THEN L1(1)+1▸L1(1) END; IF I=m AND L1(I)>n THEN BREAK END; IF L1(I)>n THEN 1▸L1(I) L1(I+1)+1▸L1(I+1) END; END;//I // FOR K FROM 1 TO m DO FOR L FROM K+1 TO m DO IF L1(K)=L1(L) OR L1(K)>L1(L) THEN //<--- 1▸SW END; END; END; IF SW=0 THEN Schrausser, D. G. (2025). HP_Prime_MATH: Manual. https://www.academia.edu/130037807 50 SUPPRESS(L1,n+1)▸L2(J) J+1▸J END; 0▸SW END;//while // //L2▸M1 PRM3a(n,m)// END; #end // PRM3a.pas //PRM3a(elements n,class m)/D.G.SCHRAUSSER/2025 //e.g.PRM3a(6,3) #cas PRM3a(N,M):= BEGIN {}▸L3 COMB(N,M)▸P MAKELIST(x+1-1,x,1,N)▸L1 // FOR J FROM 1 TO P DO FOR I FROM 1 TO M DO L2(J,I)▸L3(I) END; L3▸L4(J)▸M1 END; FOR I FROM 1 TO P DO L4(I)▸L5;DIFFERENCE(L5,L1)▸L7(I) END; FOR I FROM 1 TO P DO CONCAT(L4(I),L7(I))▸L8(I) END; L4▸L2;L8▸L3 {}▸L4 {}▸L8 {}▸L5 {}▸L6 {}▸L7 L3▸M2 P,M2 END; #end // PRM4.pas //PRM4(elements n, class m)/D.G.SCHRAUSSER/2025 //variation matrix w(V)n(m), where V=n^m;n>=m //e.g.PRM4(4,2)[PRM4a] #cas PRM4(n,m):= BEGIN MAKELIST(1,P,1,n+1)▸L1 0▸L1(1) {}▸L2 0▸M1 1▸J 0▸I 0▸SW // WHILE I≠m AND L1(I)<n DO FOR I FROM 1 TO m DO IF I=1 THEN L1(1)+1▸L1(1) END; IF I=m AND L1(I)>n THEN BREAK END; IF L1(I)>n THEN Schrausser, D. G. (2025). HP_Prime_MATH: Manual. https://www.academia.edu/130037807 51 1▸L1(I) L1(I+1)+1▸L1(I+1) END; END;//I // SUPPRESS(L1,n+1)▸L2(J);J+1▸J END;//while // //L2▸M1 PRM4a(n,m) END; #end // PRM4a.pas //PRM4a(elements n, class m)/D.G.SCHRAUSSER/2025 //e.g.PRM4a(4,3) #cas PRM4a(N,M):= BEGIN //L2 provided V=N^M X=M+1 FOR I FROM 1 TO V DO L2(I)▸L3 SUPPRESS(L3,X,N)▸L4(I) END; L4▸M1 V,M1 END; #end // PRM5.pas //PRM5(cases m)/D.G.SCHRAUSSER/2025 //variation matrix w(V)2(m) for paired 2 sample design PV_, //where V=2^m //e.g.PRM5(3) #cas PRM5(N):= BEGIN M1=0 2^N▸P X=−1 0▸Z P/2▸A // FOR I FROM 1 TO N DO FOR J FROM 1 TO P DO X▸M1(J,I) Z=Z+1 IF Z=A THEN X=X*−1;0▸Z; END; END; 0▸Z // A=A/2 // A=A*0.5 END; // M1▸L3 P,M1 END; #end // PRMDAT.pas //PRMDAT(rows n, cols k)/D.G.SCHRAUSSER/2025 //e.g.PRMDAT(720,6) Schrausser, D. G. (2025). HP_Prime_MATH: Manual. https://www.academia.edu/130037807 52 #cas PRMDAT(N,K):= BEGIN //L3 provided FOR J FROM 1 TO N DO FOR I FROM 1 TO K DO L3(M1(J,I))▸M2(J,I) END; END; END; #end // Q Q01_.pas //Q01_()/D.G.SCHRAUSSER/2025 //Statistical parameters 1.0 //[L1:Raw] //L2:Distribution //L3:z-value //L4:z´-value #cas Q01_():= BEGIN //L1 provided SORT(L1)▸L2 //distr SIZE(L1)▸N mean(L1)▸AM stddev(L1)▸SD stddevp(L1)▸SD1 variance(L1)▸VA VA1=VA*(N/(N-1)) //SD1^2 SEM=sqrt((VA1/N)) VQ=SD/AM QGM= N NTHROOT(product(L1)) QHM=N/Σ(1/L1) approx(MAKELIST(((L2(X)-AM)/SD),X,1,N))▸L3 //z approx(MAKELIST(((L2(X)-AM)/SD1),X,1,N))▸L4 //z´ // approx(N,[AM,SEM],SD,SD1,VA,VA1,VQ,[QGM,QHM]) END; #end // Q02_.pas //Q02_()/D.G.SCHRAUSSER/2025 //Statistical parameters 2.0 //[L1:Raw] //L2:Distribution //L3:z-value #cas Q02_():= BEGIN //L1 provided SORT(L1)▸L2 // SIZE(L1)▸N mean(L1)▸AM stddev(L1)▸SD stddevp(L1)▸SD1 approx(MAKELIST(((L2(X)-AM)/SD),X,1,N))▸L3 // Σ(L3.^3)/N▸A3 sqrt(6/N)▸SA3 Σ(L3.^4)/N-3▸A4 Schrausser, D. G. (2025). HP_Prime_MATH: Manual. https://www.academia.edu/130037807 53 2*SA3▸SA4 Σ((L1 .- AM) .^ 3)*N/((N-1)*(N-2)*SD1^3)▸A31 A41=((N-1)*(N-2)*(N-3)*SD1^4) EX1=Σ((L1 .- AM) .^ 4)*N*(N+1) EX2=Σ((L1 .- AM) .^ 2) EX2=3*EX2*EX2*(N-1) A41=(EX1-EX2)/A41 NORMALD_CDF(A3/SA3)▸PA3 P2A3=2*PA3 IF PA3>0.5 THEN P2A3=2*(1-PA3) END; NORMALD_CDF(A4/SA4)▸PA4 P2A4=2*PA4 IF PA4>0.5 THEN P2A4=2*(1-PA4) END; // approx(N,[A3,A31],A3/SA3,[P2A3],[A4,A41],A4/SA4,[P2A4]) END; #end // R rDiff.pas //rDiff(r1,n1,r2,n2)/D.G.SCHRAUSSER/2025 //e.g.rDiff(0.78,12,0.34,8)[ZCor] #cas rDiff(R1,N1,R2,N2):= BEGIN ZCor(R1,N1)(1)▸L2 L2(1)▸L1(1) ZCor(R2,N2)(1)▸L2 L2(1)▸L1(2) L1(1)-L1(2)▸L2(1) sqrt((1/(N1-3))+1/(N2-3))▸L2(2) L2(1)/L2(2)▸L2(3) NORMALD_CDF(L2(3))▸L2(4) 1-L2(4)▸L2(5) 2*L2(5)▸L2(6) IF L2(5)>0.5 THEN 2*L2(4)▸L2(6) END; //Zd,sZd,z,p,1-p,p2 [L2(1),L2(2)],[L2(3)],L2(4),L2(5),[L2(6)] END; #end // RHO.pas //RHO()/D.G.SCHRAUSSER/2025 //Spearman's rank correlation coefficient rho rs/[pCor] #cas RHO():= BEGIN //L1()(2) provided size(L1)▸N mean(L1)▸L3 MAKELIST((L1(I)(1)-L1(I)(2))^2,I,1,N)▸L2 Σ(L2)▸SUM RHO=1-((6*SUM)/(N*(N^2-1))) pCor(RHO,N)▸L4 //n,rho,r,p2rho approx(N,[RHO,correlation(L1)],L4(3)) END; #end // Schrausser, D. G. (2025). HP_Prime_MATH: Manual. https://www.academia.edu/130037807 54 RNK.pas //RNK()/D.G.SCHRAUSSER/2025 //L1 to ranking L3 #cas RNK():= BEGIN //L1 provided {}▸L2 {}▸L3 1▸R 0▸S1 0▸Q01 0▸V SIZE(L1)▸N SORT(L1)▸L2 FOR I FROM 2 TO N+1 DO R▸L3(I-1) IF L2(I-1)≠L2(I) THEN IF S1=1 THEN R=R+Q01 Q01=0;S1=0 1+V▸V END; R+1▸R ELSE Q01+1▸Q01;S1=1 END; END; Σ(L3)▸SR //n,ties,Rsum,meanR approx(N,V,SR,[SR/N]) END; #end // T TAU.pas //TAU()/D.G.SCHRAUSSER/2025 //Kendall's τ coefficient (tau-a) #cas TAU():= BEGIN //L1,L2 provided SIZE(L1)▸N FOR I FROM 1 TO N-1 DO ((SIGN(L1(I)-L1(I+1))))*((SIGN(L2(I)-L2(I+1))))▸L3(I) IF L3(I)=1 THEN 0▸L3(I) END; IF L3(I)=-1 THEN 1▸L3(I) END; 888▸M1(L1(I),L2(I)) END; 888▸M1(L1(N),L2(N)) //(N*(N-1)/2-2*ΣLIST(L3))/(N*(N-1)/2)▸tau_a approx(1-2*ΣLIST(L3)/(N*(N-1)/2))▸L4(1) approx((√(N*(N-1)))/(√(2*(2*N+5)))*3*L4(1))▸L4(2) NORMALD_CDF(L4(2))▸L4(3) 1-L4(3)▸L4(4) L4(4)*2▸L4(5) IF L4(3)<0.5 THEN 2*L4(3)▸L4(5) ((COMB(N,2)-ΣLIST(L3))-ΣLIST(L3))/√((1/18)*N*(N-1)*(2*N+5))▸L4(6) END; //taua,z,p,1-p,p2 Schrausser, D. G. (2025). HP_Prime_MATH: Manual. https://www.academia.edu/130037807 55 [L4(1)],[L4(2)],L4(3),L4(4),[L4(5)] END; #end // TKV.pas //TKV()/D.G.SCHRAUSSER/2025 //Variance p/[C2V,pCor] #cas TKV():= BEGIN //L1L2 provided SIZE(L1)▸N variance(L1)▸s21 variance(L2)▸s22 N-2▸df C2V(2) approx(correlation(L6))▸r pCor(r,N)(2)▸pr approx(((s21-s22)*sqrt(N-2))/(2*sqrt(s21*s22*(1-r^2))))▸t STUDENT_CDF(df,t)▸p 1-p▸p1 2*p▸p2 IF p2>1 THEN 2*(1-p)▸p2 END; df,[r],[pr],[t],p,p1,[p2] END; #end // TT_.pas //TT_(y)/D.G.SCHRAUSSER/2025 //One-sample t-test for test variable y //e.g TT_(5.3) #cas TT_(Y):= BEGIN //L1 provided size(L1)▸N N-1▸df mean(L1)▸x variance(L1)▸s2 approx((x-Y)/(sqrt(s2/(N-1))))▸t p=STUDENT_CDF(df,t) 1-p▸p1 2*p▸p2 IF p2>1 THEN 2*(1-p)▸p2 END; df,[t],p,p1,[p2] END; #end // TU_.pas //TU_()/D.G.SCHRAUSSER/2025 //t-test for unpaired samples #cas TU_():= BEGIN //L1L2 provided 0▸Sx1 0▸Sx2 SIZE(L1)▸n1 SIZE(L2)▸n2 n1+n2-2▸df mean(L1)▸x1 mean(L2)▸x2 FOR I FROM 1 TO n1 DO Schrausser, D. G. (2025). HP_Prime_MATH: Manual. https://www.academia.edu/130037807 56 Sx1=Sx1+(L1(I)-x1)^2 END; FOR I FROM 1 TO n2 DO Sx2=Sx2+(L2(I)-x2)^2 END; t=x1-x2 t=t/(√((Sx1+Sx2)/((n1-1)+(n2-1)))*√(1/n1+1/n2)) t=approx(t) p=STUDENT_CDF(df,t) p1=p IF p>0.5 THEN p1=1-p END; p2=2*p1 df,[t],p,p1,[p2] END; #end // TV_.pas //TV_()/D.G.SCHRAUSSER/2025 //t-test for paired samples #cas TV_():= BEGIN //L1()(2) provided size(L1)▸n n-1▸df FOR I FROM 1 TO n DO L1(I)(1)-L1(I)(2)▸L2(I) L2(I)^2▸L3(I) END; ΣLIST(L2)▸Sxd ΣLIST(L3)▸Sxd2 t=Sxd/n t = (t/((√((Sxd2-Sxd^2/n)/(n-1)))/(√n))) t=approx(t) p=STUDENT_CDF(df,t) p▸p1 IF p>0.5 THEN 1-p▸p1 END; 2*p1▸p2 // df,[t],p,p1,[p2] END; #end // tVTLG.pas //tVTLG(t,df)/D.G.SCHRAUSSER/2025 //e.g.tVTLG(2.65,8)[AdvancedGraphing] #cas tVTLG(T857,D187):= BEGIN G25478=Gamma((D187+1)/2)/Gamma(D187/2) P=∫(G25478*(D187*π)^(-1/2)*(1+(X^2/D187))^(-(D187+1)/2),X,−∞,T857) T857▸C D187▸D G25478▸G "Y=(G*(D*π)^(-1/2)*(1+(X^2/D))^(-(D+1)/2))"▸V1 "Y<(G*(D*π)^(-1/2)*(1+(X^2/D))^(-(D+1)/2)) AND Y>0 AND X<C"▸V2 "Y=(1/√(2*π))*e^((-1/2)*(X)^2)"▸V3 STARTAPP("Erweiterte_Grafiken"); STARTVIEW(1); IF P>0.5 THEN P=1-P END; P,[2*P] END; #end // Schrausser, D. G. (2025). HP_Prime_MATH: Manual. https://www.academia.edu/130037807 57 V VAR1.pas //VAR1(n var)/D.G.SCHRAUSSER/2025 //e.g.VAR1(5)/variation vector (v)n from L1 #cas VAR1(N):= BEGIN //L1(N) provided {}▸L2 FOR A FROM 1 TO N DO L1(RANDINT(1,N))▸L2(A) END; // L2 END; #end // VFC0.pas //VFC0(cell count a,b,c,d)/D.G.SCHRAUSSER/2025 //2×2 chi-squared test for independence //Observed frequencies abcd fb //Chi-squared, McNemar with 2-tailed p //e.g.VFC0(17,12,14,24) #cas VFC0(a,b,c,d):= BEGIN a+b+c+d▸N a+b▸z1;c+d▸z2 a+c▸s1;b+d▸s2 z1/z2▸Z01;s1/s2▸S01 VFX=(N*(a*d-b*c)^2)/((a+b)*(c+d)*(a+c)*(b+d)) VFC=1-CHISQUARE_CDF(1,VFX) MNX=(b-c)^2/(b+c) //McNemar Yates corr. IF b+c<30 AND b+c>20 THEN MNX=(ABS(b-c)-0.5)^2/(b+c) END pMNX=1-CHISQUARE_CDF(1,MNX) //n,mnchi2,mnp2,chi2,p2 N,[MNX],[pMNX],[VFX],[VFC] END; #end // VFCH.pas //VFCH(cell count a,b,c,d)/D.G.SCHRAUSSER/2025 //2×2 chi-squared test for independence //L0: Observed frequencies abcd fb //L1: Expected frequencies fe //L2,L3,L4: Probabilities p(A^B), p(B|A), p(A|B) //L5: Chi-squared,(w. Yates corr.) //L6: 2-tailed sig. p2 //Chi-square McNemar (w. Yates corr.) with p2 //e.g.VFCH(17,12,14,24) // #cas VFCH(a,b,c,d):= BEGIN a▸L0(1) b▸L0(2) c▸L0(3) d▸L0(4) a+b+c+d▸N Schrausser, D. G. (2025). HP_Prime_MATH: Manual. https://www.academia.edu/130037807 64 pTUX(n1,n2,x1,x2,s21,s22) STUDENT_CDF(D+F-2,((A-B)/(√((C*D+E*F)/(D-1+F1))*(√(1/D)+√(1/F))))) pTV_(L1) student_cdf(size(L1)-1,ΣLIST(MAKELIST(L1(A)- (L2(A)),A,1,size(L1)))/(size(L1))/(√((ΣLIST(MAKELIST((L1(A)- (L2(A)))^2,A,1,size(L1)))-ΣLIST(MAKELIST(L1(A)- (L2(A)),A,1,size(L1)))^2/(size(L1)))/(size(L1)- 1))*1/(√(size(L1)-1)))) pz4F(a,b,c,d)[e.g.pz4F(11,20,80,58)] NORMALD_CDF(((D- (A+B+C+D)*((D+B)*(C+D)/(A+B+C+D)^2))/(√((A+B+C+D)*(1- ((D+B)*(C+D)/(A+B+C+D)^2))-(A+B+C+D)*(A+B+C+D1)*((D+B)*(C+D)/(A+B+C+D)^2)*(((D+B)*(C+D)/(A+B+C+D)^2)-((D+B1)*(C+D-1)/(A+B+C+D-1)^2)))))) pzBN(a,b) NORMALD_CDF(((A-(A+B)/2)/(√((A+B)/4)))) R R2D(rad) X/π*180.000 rbis(L1,L2) ((mean(L1)- mean(L2))/stddev(CONCAT(L1,L2)))*SIZE(L1)*SIZE(L2)/((1/(√(2*π) ))*e^(- (NORMALD_ICDF((SIZE(L2)/SIZE(CONCAT(L1,L2))))^2)/2)*SIZE(CONCA T(L1,L2))^2) rbisR(L1,L2) (2/(SIZE(L1)+SIZE(L2)))*(mean(L1)-mean(L2)) RED(r) A^2*100 RND1(n) MAKELIST(RANDNORM,A,1,B) RND2(n) MAKELIST(RANDOM,A,1,B) Schrausser, D. G. (2025). HP_Prime_MATH: Manual. https://www.academia.edu/130037807 65 rpbis(L1,L2) (mean(L1)- mean(L2))/stddev(CONCAT(L1,L2))*√(SIZE(L1)*SIZE(L2)/(SIZE(CONC AT(L1,L2)))^2) rxy(x1,x2) approx(correlation(L1,L2)) rtet(b,c,a,d),rad COS((π/(1+√(B*C/(A*D))))) rxy_z(rxy,rxz,ryz) (A-B*C)/(√(1-B^2)*√(1-C^2)) ry_xz(rxy,rxz,ryz) (A-B*C)/(sqrt(1-B^2)) rZ(Z) (e^(2*A)-1)/(e^(2*A)+1) S SCR(n,k,R) 1.00-((A-3.00)/(A-B-2.00))*((1.00-C^2.00)+((2.00/(AB)))*(1.00-C^2.00)^2.00) SMG [SMG(A,B),SMG(sd,n)] √((A^2)*(B/(B-1))/B) SQR(x) A^2 srbis(L1,L2) √(SIZE(L1)*SIZE(L2))/((√SIZE(CONCAT(L1,L2))*SIZE(CONCAT(L1,L2) )*1/(√(2*π)))*e^(- (NORMALD_ICDF((SIZE(L2)/SIZE(CONCAT(L1,L2))))^2)/2)) srtet(b,c,a,d) √(((A+B)/(A+B+C+D))*((A+C)/(A+B+C+D))*((C+D)/(A+B+C+D))*((B+D) /(A+B+C+D))/(A+B+C+D))*(1/(((1/(√(2*π)))*e^(- (NORMALD_ICDF(((C+D)/(A+B+C+D)))^2)/2))*((1/(√(2*π)))*e^(- (NORMALD_ICDF(((B+D)/(A+B+C+D)))^2)/2)))) sumd2(L1) ΣLIST(MAKELIST((L1(A)-(L2(A)))^2,A,1,SIZE(L1))) CAS input Schrausser, D. G. (2025). HP_Prime_MATH: Manual. https://www.academia.edu/130037807 66 L4:=ΣLIST(L3:=MAKELIST((L1(x)-(L2(x)))^2,x,1,size(L1))) L4:=ΣLIST(L3:=MAKELIST(L1(x)-(L2(x)),x,1,size(L1))) L4:=ΣLIST(L3:=MAKELIST(L1(x)-(L2(x)),x,1,size(L1)))^2 sumx2(L1,L2) CAS input ΣLIST(L3:=approx(MAKELIST((L1(x)-mean(L1))^2,x,1,size(L1)))) T TEv(Ev,Av) 2^E/A^2 tr(r,n) (R*√(N-2))/(√(1-R^2)) TRW(L1,L2) (correlation(L1,L2)*√(SIZE(L1)-2))/(√(1-correlation(L1,L2)^2)) tTKV(L1,L2) ((variance(L1)-variance(L2))*sqrt(size(L1)- 2))/(2*sqrt(variance(L1)*variance(L2)*(1-correlation(L1,L2)))) tTT_(L1,y) (mean(L1)-A)/(√(stddev(L1)^2/(SIZE(L1)-1))) tTU_(L1,L2) (mean(L1)-mean(L2))/(sqrt((ΣLIST(MAKELIST((L1(x)- mean(L1))^2,x,1,size(L1)))+ ΣLIST(MAKELIST((L2(x)- mean(L2))^2,x,1,size(L2))))/ (size(L1)-1+size(L2)- 1))*(sqrt(1/size(L1))+sqrt(1/size(L2)))) tTUX(x1,x2,s21,n1,s22,n2) (A-B)/(√((C*D+E*F)/(D-1+F-1))*(√(1/D)+√(1/F))) tTV_(L1) ΣLIST(MAKELIST(L1(A)- (L2(A)),A,1,size(L1)))/(size(L1))/(√((ΣLIST(MAKELIST((L1(A)- (L2(A)))^2,A,1,size(L1)))-ΣLIST(MAKELIST(L1(A)- (L2(A)),A,1,size(L1)))^2/(size(L1)))/(size(L1)- 1))*1/(√(size(L1)-1))) U U_1(L1,L2) SIZE(L1)*SIZE(L2)+(((SIZE(L1))^2+SIZE(L1))/2)-ΣLIST(L1) Schrausser, D. G. (2025). HP_Prime_MATH: Manual. https://www.academia.edu/130037807 67 U_2(L1,L2) SIZE(L1)*SIZE(L2)+(((SIZE(L2))^2+SIZE(L2))/2)-ΣLIST(L2) X x2F(a,b) ((A-((A+B)/2))^2/((A+B)/2))+(B-((A+B)/2))^2/((A+B)/2) x4F(a,b,c,d) (A+B+C+D)*(A*D-B*C)^2/((A+B)*(C+D)*(A+C)*(B+D)) x4FY(a,b,c,d) Yates corr 4<fe<7 (A+B+C+D)*(ABS(A*D-B*C)- ((A+B+C+D)/2))^2/((A+B)*(C+D)*(A+C)*(B+D)) xMN(b,c) (A-B)^2/(A+B) xMNY(b,c) Yates corr 20<b+c<30 (ABS(A-B)-0.5)^2/(A+B) xPHI(a,d,b,c) (((A*D-B*C)/(√((A+C)*(B+D)*(A+B)*(C+D)))))^2*(A+B+C+D) Z z4F(a,b,c,d) (D-(A+B+C+D)*((D+B)*(C+D)/(A+B+C+D)^2))/(√((A+B+C+D)*(1- ((D+B)*(C+D)/(A+B+C+D)^2))-(A+B+C+D)*(A+B+C+D1)*((D+B)*(C+D)/(A+B+C+D)^2)*(((D+B)*(C+D)/(A+B+C+D)^2)-((D+B1)*(C+D-1)/(A+B+C+D-1)^2)))) zBN(a,b) (A-(A+B)/2)/(√((A+B)/4)) Zr(r) 0.5*LN((1+A)/(1-A)) zrbis(L1,L2) (((mean(L1)- mean(L2))/stddev(CONCAT(L1,L2)))*SIZE(L1)*SIZE(L2)/((1/(√(2*π) ))*e^(- (NORMALD_ICDF((SIZE(L2)/SIZE(CONCAT(L1,L2))))^2)/2)*SIZE(CONCA T(L1,L2))^2))/(√(SIZE(L1)*SIZE(L2))/((√SIZE(CONCAT(L1,L2))*SIZ Schrausser, D. G. (2025). HP_Prime_MATH: Manual. https://www.academia.edu/130037807 68 E(CONCAT(L1,L2))*1/(√(2*π)))*e^(- (NORMALD_ICDF((SIZE(L2)/SIZE(CONCAT(L1,L2))))^2)/2))) zrbisR(L1,L2) (size(L1)*size(L2)+(((size(L1))^2+size(L1))/2)-ΣLIST(L1)- size(L1)*size(L2)/2)/(sqrt(size(L1)*size(L2)*(size(L1)+size(L2 )+1)/12)) zrr(r1,r2,n1,n2) (0.5*LN((1+A)/(1-A))-0.5*LN((1+B)/(1-B)))/(√(1/(C-3)+1/(D-3))) zrxy_z(rxy_z,n) 0.5*LN(((1+A)/(1-A)))*√(B-2) ZWERT(x1,x,s) (A-B)/C zVAL(L1),e.g.L2:=zVAL(L1) approx(MAKELIST(((L1(X)-mean(L1))/stddev(L1)),X,1,SIZE(L1))) zVALp(L1),e.g.L3:=zVAL(L1) approx(MAKELIST(((L1(X)-mean(L1))/stddevp(L1)),X,1,SIZE(L1))) Application functions Function To select: F01.pas //F01()/D.G.SCHRAUSSER/2022 //Function: Equations 1.0 EXPORT F01() BEGIN "√(1-((X-W)/A)^2)*A+V"▸F1; "−√(1-((X-W)/A)^2)*A+V"▸F2; "√(1-((X-T)/B)^2)*B+U"▸F3; "-√(1-((X-T)/B)^2)*B+U"▸F4; "√(1-((X-R)/C)^2)*C+S"▸F5; "-√(1-((X-R)/C)^2)*C+S"▸F6; 200▸A; 150▸B; 344▸C; 1▸W; 450▸T; 1000▸R; "Function: Equations 1.0" END; // Schrausser, D. G. (2025). HP_Prime_MATH: Manual. https://www.academia.edu/130037807 69 F02.pas //F02()/D.G.SCHRAUSSER/2022 //Function: Equations 2.0 EXPORT F02() BEGIN "NORMALD_CDF(0,1,X)"▸F1; "NORMALD(0,1,X)"▸F2; "STUDENT(50,X)"▸F3; "STUDENT_CDF(50,X)"▸F4; "CHISQUARE(1,X)"▸F5; "CHISQUARE_CDF(1,X)"▸F6; "FISHER_CDF(25,3,X)"▸F7; "FISHER(25,3,X)"▸F8; "0"▸F9; "0"▸F0; "Function: Equations 2.0" END; // F03.pas //F03()/D.G.SCHRAUSSER/2025 //Function: Equations 3.0 //F3-7:Derivatives of the standard normal distribution function, f'(z)- f'''''(z) EXPORT F03() BEGIN "NORMALD_CDF(0,1,X)"▸F1; "NORMALD(0,1,X)"▸F2; "∂((1/√(2*π))*e^((-1/2)*X^2),X=X)"▸F3; "∂(∂((1/√(2*π))*e^((-1/2)*X^2),X),X)"▸F4; "∂(∂(∂((1/√(2*π))*e^((-1/2)*X^2),X),X),X)"▸F5; "∂(∂(∂(∂((1/√(2*π))*e^((-1/2)*X^2),X),X),X),X)"▸F6; "∂(∂(∂(∂(∂((1/√(2*π))*e^((-1/2)*X^2),X),X),X),X),X)"▸F7; "0"▸F8; "0"▸F9; "0"▸F0; "Function: Equations 3.0" END; // F04.pas //F04()/D.G.SCHRAUSSER/2025 //Function: Equations 4.0 //F3:Derivative of Gamma, f'(x) //F5-7:Derivatives of the exponential function, f(x)={f'(x)-f'''(x)...} EXPORT F04() BEGIN "CAS.Gamma(X)"▸F1; "(X)!"▸F2; "∂(Gamma(X),X=X)"▸F3; "EXP(X)"▸F4; "∂(e^X,X = X)"▸F5; "∂(∂(e^X,X=X),X=X)"▸F6; "∂(∂(∂(e^X,X=X),X=X),X=X)"▸F7; "0"▸F8; "0"▸F9; "0"▸F0; "Function: Equations 4.0" END; // Schrausser, D. G. (2025). HP_Prime_MATH: Manual. https://www.academia.edu/130037807 70 F05.pas //F05()/D.G.SCHRAUSSER/2025 //Function: Equations 5.0 //F2-5,F7-0:Derivatives of the circular function, f'(t)-f''''(t) EXPORT F05() BEGIN "√(1-X^2)"▸F1; "∂(√(1-X^2),X)"▸F2; "∂(∂(√(1-X^2),X),X)"▸F3; "∂(∂(∂(√(1-X^2),X),X),X)"▸F4; "∂(∂(∂(∂(√(1-X^2),X),X),X),X)"▸F5; "-√(1-X^2)"▸F6; "∂(-√(1-X^2),X)"▸F7; "∂(∂(-√(1-X^2),X),X)"▸F8; "∂(∂(∂(-√(1-X^2),X),X),X)"▸F9; "∂(∂(∂(∂(-√(1-X^2),X),X),X),X)"▸F0; "Function: Equations 5.0" END; // F06.pas //F06()/D.G.SCHRAUSSER/2025 //Function: Equations 6.0 //F3-6:Derivatives of Student's-t, f'(t)-f''''(t) EXPORT F06() BEGIN "STUDENT_CDF(D,X)"▸F1; "(Gamma(((D+1)/2))/Gamma((D/2)))*(D*π)^(-1/2)*(1+(X^2/D))^(-(D+1)/2)"▸F2; "∂(G*(D*π)^(-1/2)*(1+(X^2/D))^(-(D+1)/2),X)"▸F3; "∂(∂(G*(D*π)^(-1/2)*(1+(X^2/D))^(-(D+1)/2),X),X)"▸F4; "∂(∂(∂(G*(D*π)^(-1/2)*(1+(X^2/D))^(-(D+1)/2),X),X),X)"▸F5; "∂(∂(∂(∂(G*(D*π)^(-1/2)*(1+(X^2/D))^(-(D+1)/2),X),X),X),X)"▸F6; ""▸F7; ""▸F8; ""▸F9; ""▸F0; "Function: Equations 6.0" END; // F06_.pas //F06_(df)/D.G.SCHRAUSSER/2025 //Derivatives of Student’s-t //e.g.F06_(5)[F06] #cas F06_(DF):= BEGIN F06 DF▸D G=Gamma(((DF+1)/2))/Gamma((DF/2)) D,G END; #end // F07.pas //F07()/D.G.SCHRAUSSER/2025 //Function: Equations 7.0 //F3-6:Derivatives of chi², f'(t)-f''''(t) EXPORT F07() BEGIN "CHISQUARE_CDF(D,X)"▸F1; "(1/(2^(D/2)*G))*X^((D/2)-1)*e^(-X/2)"▸F2; Schrausser, D. G. (2025). HP_Prime_MATH: Manual. https://www.academia.edu/130037807 71 "∂((1/(2^(D/2)*G))*X^((D/2)-1)*e^(-X/2),X)"▸F3; "∂(∂((1/(2^(D/2)*G))*X^((D/2)-1)*e^(-X/2),X),X)"▸F4; "∂(∂(∂((1/(2^(D/2)*G))*X^((D/2)-1)*e^(-X/2),X),X),X)"▸F5; "∂(∂(∂(∂((1/(2^(D/2)*G))*X^((D/2)-1)*e^(-X/2),X),X),X),X)"▸F6; ""▸F7; ""▸F8; ""▸F9; ""▸F0; "Function: Equations 7.0" END; // F07_.pas //F07_(df)/D.G.SCHRAUSSER/2025 //Derivatives of chi² //e.g.F07_(1)[F07] #cas F07_(DF):= BEGIN F07 DF▸D G=Gamma(DF/2) D,G END; #end // Graph 3D To select: F01Z.pas //F01Z()/D.G.SCHRAUSSER/2025 //Graph 3D: Equations 1.0 //FZ1-2:Gamma //FZ4-9:Spherical functions //FZ0:Sine EXPORT F01Z() BEGIN "CAS.Gamma(Y)"▸FZ1; "X^(Y-1)*e^(-X)"▸FZ2; "(1/(2*π*√(1-R^2)))*e^((-1/(2*(1-R^2)))*(X^2-2*R*X*Y+Y^2))"▸FZ3; "√((1-X^2)+(1-Y^2))"▸FZ4; "−1*√((1-X^2)+(1-Y^2))"▸FZ5; "√(1-X^2-Y^2)"▸FZ6; "−1*√(1-X^2-Y^2)"▸FZ7; "√((X-X^2)+(Y-Y^2))"▸FZ8; "−1*√((X-X^2)+(Y-Y^2))"▸FZ9; "SIN(X)*SIN(Y)*1.5"▸FZ0; "Graph 3D: Equations 1.0" END; // F02Z.pas //F02Z()/D.G.SCHRAUSSER/2025 //Graph 3D: Equations 2.0 //Complex plane f(z)=z, //with z=|x+i| //where f(x,y=i)=√x²+y² EXPORT F02Z() BEGIN "√(X^2+Y^2)"▸FZ1; "(1/π)*e^(-(ABS(√(X^2+Y^2))^2))"▸FZ2; Schrausser, D. G. (2025). HP_Prime_MATH: Manual. https://www.academia.edu/130037807 72 "X^((√(X^2+Y^2))-1)*e^(-X)"▸FZ3; ""▸FZ4; ""▸FZ5; ""▸FZ6; ""▸FZ7; ""▸FZ8; ""▸FZ9; ""▸FZ0; "Graph 3D: Equations 2.0" END; // F03Z.pas //F03Z()/D.G.SCHRAUSSER/2025 //Graph 3D: Equations 3.0 //FZ1:Student's-t surface, f(t,df),x[-4,4],y[1,10],z[0,0.5] //FZ2:chi² surface, f(chi²,df), x[0,5],y[1,10],z[0,0.5] //FZ3-6:F space, f(F,df2);df1={1,5,9,13},x[0,5],y[0,20],z[0,5] EXPORT F03Z() BEGIN "(Gamma(((Y+1)/2))/Gamma((Y/2)))*(Y*π)^(-1/2)*(1+(X^2/Y))^(-(Y+1)/2)"▸FZ1; "(1/(2^(Y/2)*Gamma((Y/2))))*X^((Y/2)-1)*e^(-X/2)"▸FZ2; "(Gamma(((1+Y)/2))/(Gamma((1/2))*Gamma((1/2))))*(1/Y)^(1/2)*X^((1/2)- 1)*(1+(1/Y)*X)^(-(1+Y)/2)"▸FZ3; "(Gamma(((5+Y)/2))/(Gamma((5/2))*Gamma((5/2))))*(5/Y)^(5/2)*X^((5/2)- 1)*(1+(5/Y)*X)^(-(5+Y)/2)"▸FZ4; "(Gamma(((9+Y)/2))/(Gamma((9/2))*Gamma((9/2))))*(9/Y)^(9/2)*X^((9/2)- 1)*(1+(9/Y)*X)^(-(9+Y)/2)"▸FZ5; "(Gamma(((13+Y)/2))/(Gamma((13/2))*Gamma((13/2))))*(13/Y)^(13/2)*X^((13/2)- 1)*(1+(13/Y)*X)^(-(13+Y)/2)"▸FZ6; ""▸FZ7; ""▸FZ8; ""▸FZ9; ""▸FZ0; "Graph 3D: Equations 3.0" END; // Solve To select: E01.pas //E01()/D.G.SCHRAUSSER/2025 //Solve: Equations 1.0 //E1: Additive probability (P=p,X=pb,N=n; special addition theorem) //E2: Negative binomial probability (P=p,N=pnb,R=n,K=1,2..,I=0.0) //E3: Binomial probability (b,c) //E4: Standard normal distribution z(0,1) //E5: Effect size epsilon //E6: Linear regression y' //E7: Standard error of prediction y'+-C with zcrit //E8: p of correlation r with n (2-tailed sig p2=2*(1-p);p>0.5) EXPORT E01() BEGIN "X=1-(1-P)^N"▸E1; "P=Σ(((A+B)!/(I!*(A+B-I)!))*2^(-I)*2^(-(A+B-I)),I,0,A)"▸E3; "N=Σ(((K+I-1)!/(I!*(K-1)!))*P^K*(1-P)^I,I,0,R-K)"▸E2; "P=NORMALD_CDF(0,1,Z)"▸E4; "E=(A-X)/S"▸E5; "Y=A*X+B"▸E6; Schrausser, D. G. (2025). HP_Prime_MATH: Manual. https://www.academia.edu/130037807 73 "C=(Z*√(1-R^2))*S"▸E7; "P=STUDENT_CDF(N-2,(R*√(N-2))/√(1-R^2))"▸E8; ""▸E9; ""▸E0; "Solve: Equations 1.0" END; // E02.pas //E02()/D.G.SCHRAUSSER/2022 //Solve: Equations 2.0 //E1: Aperture value A for exposure value E and shutter speed T //E2: Exposure value E for illuminance lux L and ISO I //E3: Magnification M at focal length F //E4: Angle of view W at focal length F EXPORT E02() BEGIN "A=1/√(2^(-E)*T)"▸E1; "E=LN((L*I/250))*(LN(2))^(-1)"▸E2; "M=F/50"▸E3; "W=-0.95908335982/(1-1.00293098572*e^(0.000351450126836*F))"▸E4; ""▸E5; ""▸E6; ""▸E7; ""▸E8; ""▸E9; ""▸E0; "Solve: Equations 2.0" END; // E03.pas //E03()/D.G.SCHRAUSSER/2025 //Solve: Equations 3.0 //E1: Resolution R from R0, f0, f1 //E2: Exposure Value E from Tv, Av //E3: Aperture B from Av0, ISO0, ISO1 EXPORT E03() BEGIN "(A*B^2/F^2)=R"▸E1; "(LN(2))^(-1)*LN(T*A^2)=E"▸E2; "A*e^(0.5*LN(S^(-1)*I))=B"▸E3; ""▸E4; ""▸E5; ""▸E6; ""▸E7; ""▸E8; ""▸E9; ""▸E0; "Solve: Equations 3.0" END; // E04.pas //E04()/D.G.SCHRAUSSER/2025 //Solve: Equations 4.0 //E1: Astronomical unit A from meters M //E2: Parsec P from astronomical unit A //E3: Parsec P from parallax X in milliarcseconds mas //E4: Light-year L from parsec P //E5: Speed of light C from m/c M //E6: Luminosity distance P from distance modulus M //E7: Radius R at a given distance D with angular diameter V° Schrausser, D. G. (2025). HP_Prime_MATH: Manual. https://www.academia.edu/130037807 80 .,5.,6.,9.,10.],[2.,3.,4.,7.,8.,1.,5.,6.,9.,10.],[1.,2.,5.,7.,8.,3.,4.,6.,9.,10.],[1. ,3.,5.,7.,8.,2.,4.,6.,9.,10.],[2.,3.,5.,7.,8.,1.,4.,6.,9.,10.],[1.,4.,5.,7.,8.,2.,3., 6.,9.,10.],[2.,4.,5.,7.,8.,1.,3.,6.,9.,10.],[3.,4.,5.,7.,8.,1.,2.,6.,9.,10.],[1.,2.,6 .,7.,8.,3.,4.,5.,9.,10.],[1.,3.,6.,7.,8.,2.,4.,5.,9.,10.],[2.,3.,6.,7.,8.,1.,4.,5.,9. ,10.],[1.,4.,6.,7.,8.,2.,3.,5.,9.,10.],[2.,4.,6.,7.,8.,1.,3.,5.,9.,10.],[3.,4.,6.,7., 8.,1.,2.,5.,9.,10.],[1.,5.,6.,7.,8.,2.,3.,4.,9.,10.],[2.,5.,6.,7.,8.,1.,3.,4.,9.,10.] ,[3.,5.,6.,7.,8.,1.,2.,4.,9.,10.],[4.,5.,6.,7.,8.,1.,2.,3.,9.,10.],[1.,2.,3.,4.,9.,5. ,6.,7.,8.,10.],[1.,2.,3.,5.,9.,4.,6.,7.,8.,10.],[1.,2.,4.,5.,9.,3.,6.,7.,8.,10.],[1., 3.,4.,5.,9.,2.,6.,7.,8.,10.],[2.,3.,4.,5.,9.,1.,6.,7.,8.,10.],[1.,2.,3.,6.,9.,4.,5.,7 .,8.,10.],[1.,2.,4.,6.,9.,3.,5.,7.,8.,10.],[1.,3.,4.,6.,9.,2.,5.,7.,8.,10.],[2.,3.,4. ,6.,9.,1.,5.,7.,8.,10.],[1.,2.,5.,6.,9.,3.,4.,7.,8.,10.],[1.,3.,5.,6.,9.,2.,4.,7.,8., 10.],[2.,3.,5.,6.,9.,1.,4.,7.,8.,10.],[1.,4.,5.,6.,9.,2.,3.,7.,8.,10.],[2.,4.,5.,6.,9 .,1.,3.,7.,8.,10.],[3.,4.,5.,6.,9.,1.,2.,7.,8.,10.],[1.,2.,3.,7.,9.,4.,5.,6.,8.,10.], [1.,2.,4.,7.,9.,3.,5.,6.,8.,10.],[1.,3.,4.,7.,9.,2.,5.,6.,8.,10.],[2.,3.,4.,7.,9.,1., 5.,6.,8.,10.],[1.,2.,5.,7.,9.,3.,4.,6.,8.,10.],[1.,3.,5.,7.,9.,2.,4.,6.,8.,10.],[2.,3 .,5.,7.,9.,1.,4.,6.,8.,10.],[1.,4.,5.,7.,9.,2.,3.,6.,8.,10.],[2.,4.,5.,7.,9.,1.,3.,6. ,8.,10.],[3.,4.,5.,7.,9.,1.,2.,6.,8.,10.],[1.,2.,6.,7.,9.,3.,4.,5.,8.,10.],[1.,3.,6., 7.,9.,2.,4.,5.,8.,10.],[2.,3.,6.,7.,9.,1.,4.,5.,8.,10.],[1.,4.,6.,7.,9.,2.,3.,5.,8.,1 0.],[2.,4.,6.,7.,9.,1.,3.,5.,8.,10.],[3.,4.,6.,7.,9.,1.,2.,5.,8.,10.],[1.,5.,6.,7.,9. ,2.,3.,4.,8.,10.],[2.,5.,6.,7.,9.,1.,3.,4.,8.,10.],[3.,5.,6.,7.,9.,1.,2.,4.,8.,10.],[ 4.,5.,6.,7.,9.,1.,2.,3.,8.,10.],[1.,2.,3.,8.,9.,4.,5.,6.,7.,10.],[1.,2.,4.,8.,9.,3.,5 .,6.,7.,10.],[1.,3.,4.,8.,9.,2.,5.,6.,7.,10.],[2.,3.,4.,8.,9.,1.,5.,6.,7.,10.],[1.,2. ,5.,8.,9.,3.,4.,6.,7.,10.],[1.,3.,5.,8.,9.,2.,4.,6.,7.,10.],[2.,3.,5.,8.,9.,1.,4.,6., 7.,10.],[1.,4.,5.,8.,9.,2.,3.,6.,7.,10.],[2.,4.,5.,8.,9.,1.,3.,6.,7.,10.],[3.,4.,5.,8 .,9.,1.,2.,6.,7.,10.],[1.,2.,6.,8.,9.,3.,4.,5.,7.,10.],[1.,3.,6.,8.,9.,2.,4.,5.,7.,10 .],[2.,3.,6.,8.,9.,1.,4.,5.,7.,10.],[1.,4.,6.,8.,9.,2.,3.,5.,7.,10.],[2.,4.,6.,8.,9., 1.,3.,5.,7.,10.],[3.,4.,6.,8.,9.,1.,2.,5.,7.,10.],[1.,5.,6.,8.,9.,2.,3.,4.,7.,10.],[2 .,5.,6.,8.,9.,1.,3.,4.,7.,10.],[3.,5.,6.,8.,9.,1.,2.,4.,7.,10.],[4.,5.,6.,8.,9.,1.,2. ,3.,7.,10.],[1.,2.,7.,8.,9.,3.,4.,5.,6.,10.],[1.,3.,7.,8.,9.,2.,4.,5.,6.,10.],[2.,3., 7.,8.,9.,1.,4.,5.,6.,10.],[1.,4.,7.,8.,9.,2.,3.,5.,6.,10.],[2.,4.,7.,8.,9.,1.,3.,5.,6 .,10.],[3.,4.,7.,8.,9.,1.,2.,5.,6.,10.],[1.,5.,7.,8.,9.,2.,3.,4.,6.,10.],[2.,5.,7.,8. ,9.,1.,3.,4.,6.,10.],[3.,5.,7.,8.,9.,1.,2.,4.,6.,10.],[4.,5.,7.,8.,9.,1.,2.,3.,6.,10. ],[1.,6.,7.,8.,9.,2.,3.,4.,5.,10.],[2.,6.,7.,8.,9.,1.,3.,4.,5.,10.],[3.,6.,7.,8.,9.,1 .,2.,4.,5.,10.],[4.,6.,7.,8.,9.,1.,2.,3.,5.,10.],[5.,6.,7.,8.,9.,1.,2.,3.,4.,10.],[1. ,2.,3.,4.,10.,5.,6.,7.,8.,9.],[1.,2.,3.,5.,10.,4.,6.,7.,8.,9.],[1.,2.,4.,5.,10.,3.,6. ,7.,8.,9.],[1.,3.,4.,5.,10.,2.,6.,7.,8.,9.],[2.,3.,4.,5.,10.,1.,6.,7.,8.,9.],[1.,2.,3 .,6.,10.,4.,5.,7.,8.,9.],[1.,2.,4.,6.,10.,3.,5.,7.,8.,9.],[1.,3.,4.,6.,10.,2.,5.,7.,8 .,9.],[2.,3.,4.,6.,10.,1.,5.,7.,8.,9.],[1.,2.,5.,6.,10.,3.,4.,7.,8.,9.],[1.,3.,5.,6., 10.,2.,4.,7.,8.,9.],[2.,3.,5.,6.,10.,1.,4.,7.,8.,9.],[1.,4.,5.,6.,10.,2.,3.,7.,8.,9.] ,[2.,4.,5.,6.,10.,1.,3.,7.,8.,9.],[3.,4.,5.,6.,10.,1.,2.,7.,8.,9.],[1.,2.,3.,7.,10.,4 .,5.,6.,8.,9.],[1.,2.,4.,7.,10.,3.,5.,6.,8.,9.],[1.,3.,4.,7.,10.,2.,5.,6.,8.,9.],[2., 3.,4.,7.,10.,1.,5.,6.,8.,9.],[1.,2.,5.,7.,10.,3.,4.,6.,8.,9.],[1.,3.,5.,7.,10.,2.,4., 6.,8.,9.],[2.,3.,5.,7.,10.,1.,4.,6.,8.,9.],[1.,4.,5.,7.,10.,2.,3.,6.,8.,9.],[2.,4.,5. ,7.,10.,1.,3.,6.,8.,9.],[3.,4.,5.,7.,10.,1.,2.,6.,8.,9.],[1.,2.,6.,7.,10.,3.,4.,5.,8. ,9.],[1.,3.,6.,7.,10.,2.,4.,5.,8.,9.],[2.,3.,6.,7.,10.,1.,4.,5.,8.,9.],[1.,4.,6.,7.,1 0.,2.,3.,5.,8.,9.],[2.,4.,6.,7.,10.,1.,3.,5.,8.,9.],[3.,4.,6.,7.,10.,1.,2.,5.,8.,9.], [1.,5.,6.,7.,10.,2.,3.,4.,8.,9.],[2.,5.,6.,7.,10.,1.,3.,4.,8.,9.],[3.,5.,6.,7.,10.,1. ,2.,4.,8.,9.],[4.,5.,6.,7.,10.,1.,2.,3.,8.,9.],[1.,2.,3.,8.,10.,4.,5.,6.,7.,9.],[1.,2 .,4.,8.,10.,3.,5.,6.,7.,9.],[1.,3.,4.,8.,10.,2.,5.,6.,7.,9.],[2.,3.,4.,8.,10.,1.,5.,6 .,7.,9.],[1.,2.,5.,8.,10.,3.,4.,6.,7.,9.],[1.,3.,5.,8.,10.,2.,4.,6.,7.,9.],[2.,3.,5., 8.,10.,1.,4.,6.,7.,9.],[1.,4.,5.,8.,10.,2.,3.,6.,7.,9.],[2.,4.,5.,8.,10.,1.,3.,6.,7., 9.],[3.,4.,5.,8.,10.,1.,2.,6.,7.,9.],[1.,2.,6.,8.,10.,3.,4.,5.,7.,9.],[1.,3.,6.,8.,10 .,2.,4.,5.,7.,9.],[2.,3.,6.,8.,10.,1.,4.,5.,7.,9.],[1.,4.,6.,8.,10.,2.,3.,5.,7.,9.],[ 2.,4.,6.,8.,10.,1.,3.,5.,7.,9.],[3.,4.,6.,8.,10.,1.,2.,5.,7.,9.],[1.,5.,6.,8.,10.,2., 3.,4.,7.,9.],[2.,5.,6.,8.,10.,1.,3.,4.,7.,9.],[3.,5.,6.,8.,10.,1.,2.,4.,7.,9.],[4.,5. ,6.,8.,10.,1.,2.,3.,7.,9.],[1.,2.,7.,8.,10.,3.,4.,5.,6.,9.],[1.,3.,7.,8.,10.,2.,4.,5. ,6.,9.],[2.,3.,7.,8.,10.,1.,4.,5.,6.,9.],[1.,4.,7.,8.,10.,2.,3.,5.,6.,9.],[2.,4.,7.,8 .,10.,1.,3.,5.,6.,9.],[3.,4.,7.,8.,10.,1.,2.,5.,6.,9.],[1.,5.,7.,8.,10.,2.,3.,4.,6.,9 .],[2.,5.,7.,8.,10.,1.,3.,4.,6.,9.],[3.,5.,7.,8.,10.,1.,2.,4.,6.,9.],[4.,5.,7.,8.,10. ,1.,2.,3.,6.,9.],[1.,6.,7.,8.,10.,2.,3.,4.,5.,9.],[2.,6.,7.,8.,10.,1.,3.,4.,5.,9.],[3 .,6.,7.,8.,10.,1.,2.,4.,5.,9.],[4.,6.,7.,8.,10.,1.,2.,3.,5.,9.],[5.,6.,7.,8.,10.,1.,2 .,3.,4.,9.],[1.,2.,3.,9.,10.,4.,5.,6.,7.,8.],[1.,2.,4.,9.,10.,3.,5.,6.,7.,8.],[1.,3., 4.,9.,10.,2.,5.,6.,7.,8.],[2.,3.,4.,9.,10.,1.,5.,6.,7.,8.],[1.,2.,5.,9.,10.,3.,4.,6., 7.,8.],[1.,3.,5.,9.,10.,2.,4.,6.,7.,8.],[2.,3.,5.,9.,10.,1.,4.,6.,7.,8.],[1.,4.,5.,9. ,10.,2.,3.,6.,7.,8.],[2.,4.,5.,9.,10.,1.,3.,6.,7.,8.],[3.,4.,5.,9.,10.,1.,2.,6.,7.,8. ],[1.,2.,6.,9.,10.,3.,4.,5.,7.,8.],[1.,3.,6.,9.,10.,2.,4.,5.,7.,8.],[2.,3.,6.,9.,10., 1.,4.,5.,7.,8.],[1.,4.,6.,9.,10.,2.,3.,5.,7.,8.],[2.,4.,6.,9.,10.,1.,3.,5.,7.,8.],[3. ,4.,6.,9.,10.,1.,2.,5.,7.,8.],[1.,5.,6.,9.,10.,2.,3.,4.,7.,8.],[2.,5.,6.,9.,10.,1.,3. ,4.,7.,8.],[3.,5.,6.,9.,10.,1.,2.,4.,7.,8.],[4.,5.,6.,9.,10.,1.,2.,3.,7.,8.],[1.,2.,7 .,9.,10.,3.,4.,5.,6.,8.],[1.,3.,7.,9.,10.,2.,4.,5.,6.,8.],[2.,3.,7.,9.,10.,1.,4.,5.,6 .,8.],[1.,4.,7.,9.,10.,2.,3.,5.,6.,8.],[2.,4.,7.,9.,10.,1.,3.,5.,6.,8.],[3.,4.,7.,9., 10.,1.,2.,5.,6.,8.],[1.,5.,7.,9.,10.,2.,3.,4.,6.,8.],[2.,5.,7.,9.,10.,1.,3.,4.,6.,8.] ,[3.,5.,7.,9.,10.,1.,2.,4.,6.,8.],[4.,5.,7.,9.,10.,1.,2.,3.,6.,8.],[1.,6.,7.,9.,10.,2 .,3.,4.,5.,8.],[2.,6.,7.,9.,10.,1.,3.,4.,5.,8.],[3.,6.,7.,9.,10.,1.,2.,4.,5.,8.],[4., 6.,7.,9.,10.,1.,2.,3.,5.,8.],[5.,6.,7.,9.,10.,1.,2.,3.,4.,8.],[1.,2.,8.,9.,10.,3.,4., 5.,6.,7.],[1.,3.,8.,9.,10.,2.,4.,5.,6.,7.],[2.,3.,8.,9.,10.,1.,4.,5.,6.,7.],[1.,4.,8. ,9.,10.,2.,3.,5.,6.,7.],[2.,4.,8.,9.,10.,1.,3.,5.,6.,7.],[3.,4.,8.,9.,10.,1.,2.,5.,6. ,7.],[1.,5.,8.,9.,10.,2.,3.,4.,6.,7.],[2.,5.,8.,9.,10.,1.,3.,4.,6.,7.],[3.,5.,8.,9.,1 Schrausser, D. G. (2025). HP_Prime_MATH: Manual. https://www.academia.edu/130037807 81 0.,1.,2.,4.,6.,7.],[4.,5.,8.,9.,10.,1.,2.,3.,6.,7.],[1.,6.,8.,9.,10.,2.,3.,4.,5.,7.], [2.,6.,8.,9.,10.,1.,3.,4.,5.,7.],[3.,6.,8.,9.,10.,1.,2.,4.,5.,7.],[4.,6.,8.,9.,10.,1. ,2.,3.,5.,7.],[5.,6.,8.,9.,10.,1.,2.,3.,4.,7.],[1.,7.,8.,9.,10.,2.,3.,4.,5.,6.],[2.,7 .,8.,9.,10.,1.,3.,4.,5.,6.],[3.,7.,8.,9.,10.,1.,2.,4.,5.,6.],[4.,7.,8.,9.,10.,1.,2.,3 .,5.,6.],[5.,7.,8.,9.,10.,1.,2.,3.,4.,6.],[6.,7.,8.,9.,10.,1.,2.,3.,4.,5.]] L1={{1,2,3,4,5,6,7,8,9,10},{1,2,3,4,6,5,7,8,9,10},{1,2,3,5,6,4,7,8,9,10},{1,2,4,5,6,3 ,7,8,9,10},{1,3,4,5,6,2,7,8,9,10},{2,3,4,5,6,1,7,8,9,10},{1,2,3,4,7,5,6,8,9,10},{1,2, 3,5,7,4,6,8,9,10},{1,2,4,5,7,3,6,8,9,10},{1,3,4,5,7,2,6,8,9,10},{2,3,4,5,7,1,6,8,9,10 },{1,2,3,6,7,4,5,8,9,10},{1,2,4,6,7,3,5,8,9,10},{1,3,4,6,7,2,5,8,9,10},{2,3,4,6,7,1,5 ,8,9,10},{1,2,5,6,7,3,4,8,9,10},{1,3,5,6,7,2,4,8,9,10},{2,3,5,6,7,1,4,8,9,10},{1,4,5, 6,7,2,3,8,9,10},{2,4,5,6,7,1,3,8,9,10},{3,4,5,6,7,1,2,8,9,10},{1,2,3,4,8,5,6,7,9,10}, {1,2,3,5,8,4,6,7,9,10},{1,2,4,5,8,3,6,7,9,10},{1,3,4,5,8,2,6,7,9,10},{2,3,4,5,8,1,6,7 ,9,10},{1,2,3,6,8,4,5,7,9,10},{1,2,4,6,8,3,5,7,9,10},{1,3,4,6,8,2,5,7,9,10},{2,3,4,6, 8,1,5,7,9,10},{1,2,5,6,8,3,4,7,9,10},{1,3,5,6,8,2,4,7,9,10},{2,3,5,6,8,1,4,7,9,10},{1 ,4,5,6,8,2,3,7,9,10},{2,4,5,6,8,1,3,7,9,10},{3,4,5,6,8,1,2,7,9,10},{1,2,3,7,8,4,5,6,9 ,10},{1,2,4,7,8,3,5,6,9,10},{1,3,4,7,8,2,5,6,9,10},{2,3,4,7,8,1,5,6,9,10},{1,2,5,7,8, 3,4,6,9,10},{1,3,5,7,8,2,4,6,9,10},{2,3,5,7,8,1,4,6,9,10},{1,4,5,7,8,2,3,6,9,10},{2,4 ,5,7,8,1,3,6,9,10},{3,4,5,7,8,1,2,6,9,10},{1,2,6,7,8,3,4,5,9,10},{1,3,6,7,8,2,4,5,9,1 0},{2,3,6,7,8,1,4,5,9,10},{1,4,6,7,8,2,3,5,9,10},{2,4,6,7,8,1,3,5,9,10},{3,4,6,7,8,1, 2,5,9,10},{1,5,6,7,8,2,3,4,9,10},{2,5,6,7,8,1,3,4,9,10},{3,5,6,7,8,1,2,4,9,10},{4,5,6 ,7,8,1,2,3,9,10},{1,2,3,4,9,5,6,7,8,10},{1,2,3,5,9,4,6,7,8,10},{1,2,4,5,9,3,6,7,8,10} ,{1,3,4,5,9,2,6,7,8,10},{2,3,4,5,9,1,6,7,8,10},{1,2,3,6,9,4,5,7,8,10},{1,2,4,6,9,3,5, 7,8,10},{1,3,4,6,9,2,5,7,8,10},{2,3,4,6,9,1,5,7,8,10},{1,2,5,6,9,3,4,7,8,10},{1,3,5,6 ,9,2,4,7,8,10},{2,3,5,6,9,1,4,7,8,10},{1,4,5,6,9,2,3,7,8,10},{2,4,5,6,9,1,3,7,8,10},{ 3,4,5,6,9,1,2,7,8,10},{1,2,3,7,9,4,5,6,8,10},{1,2,4,7,9,3,5,6,8,10},{1,3,4,7,9,2,5,6, 8,10},{2,3,4,7,9,1,5,6,8,10},{1,2,5,7,9,3,4,6,8,10},{1,3,5,7,9,2,4,6,8,10},{2,3,5,7,9 ,1,4,6,8,10},{1,4,5,7,9,2,3,6,8,10},{2,4,5,7,9,1,3,6,8,10},{3,4,5,7,9,1,2,6,8,10},{1, 2,6,7,9,3,4,5,8,10},{1,3,6,7,9,2,4,5,8,10},{2,3,6,7,9,1,4,5,8,10},{1,4,6,7,9,2,3,5,8, 10},{2,4,6,7,9,1,3,5,8,10},{3,4,6,7,9,1,2,5,8,10},{1,5,6,7,9,2,3,4,8,10},{2,5,6,7,9,1 ,3,4,8,10},{3,5,6,7,9,1,2,4,8,10},{4,5,6,7,9,1,2,3,8,10},{1,2,3,8,9,4,5,6,7,10},{1,2, 4,8,9,3,5,6,7,10},{1,3,4,8,9,2,5,6,7,10},{2,3,4,8,9,1,5,6,7,10},{1,2,5,8,9,3,4,6,7,10 },{1,3,5,8,9,2,4,6,7,10},{2,3,5,8,9,1,4,6,7,10},{1,4,5,8,9,2,3,6,7,10},{2,4,5,8,9,1,3 ,6,7,10},{3,4,5,8,9,1,2,6,7,10},{1,2,6,8,9,3,4,5,7,10},{1,3,6,8,9,2,4,5,7,10},{2,3,6, 8,9,1,4,5,7,10},{1,4,6,8,9,2,3,5,7,10},{2,4,6,8,9,1,3,5,7,10},{3,4,6,8,9,1,2,5,7,10}, {1,5,6,8,9,2,3,4,7,10},{2,5,6,8,9,1,3,4,7,10},{3,5,6,8,9,1,2,4,7,10},{4,5,6,8,9,1,2,3 ,7,10},{1,2,7,8,9,3,4,5,6,10},{1,3,7,8,9,2,4,5,6,10},{2,3,7,8,9,1,4,5,6,10},{1,4,7,8, 9,2,3,5,6,10},{2,4,7,8,9,1,3,5,6,10},{3,4,7,8,9,1,2,5,6,10},{1,5,7,8,9,2,3,4,6,10},{2 ,5,7,8,9,1,3,4,6,10},{3,5,7,8,9,1,2,4,6,10},{4,5,7,8,9,1,2,3,6,10},{1,6,7,8,9,2,3,4,5 ,10},{2,6,7,8,9,1,3,4,5,10},{3,6,7,8,9,1,2,4,5,10},{4,6,7,8,9,1,2,3,5,10},{5,6,7,8,9, 1,2,3,4,10},{1,2,3,4,10,5,6,7,8,9},{1,2,3,5,10,4,6,7,8,9},{1,2,4,5,10,3,6,7,8,9},{1,3 ,4,5,10,2,6,7,8,9},{2,3,4,5,10,1,6,7,8,9},{1,2,3,6,10,4,5,7,8,9},{1,2,4,6,10,3,5,7,8, 9},{1,3,4,6,10,2,5,7,8,9},{2,3,4,6,10,1,5,7,8,9},{1,2,5,6,10,3,4,7,8,9},{1,3,5,6,10,2 ,4,7,8,9},{2,3,5,6,10,1,4,7,8,9},{1,4,5,6,10,2,3,7,8,9},{2,4,5,6,10,1,3,7,8,9},{3,4,5 ,6,10,1,2,7,8,9},{1,2,3,7,10,4,5,6,8,9},{1,2,4,7,10,3,5,6,8,9},{1,3,4,7,10,2,5,6,8,9} ,{2,3,4,7,10,1,5,6,8,9},{1,2,5,7,10,3,4,6,8,9},{1,3,5,7,10,2,4,6,8,9},{2,3,5,7,10,1,4 ,6,8,9},{1,4,5,7,10,2,3,6,8,9},{2,4,5,7,10,1,3,6,8,9},{3,4,5,7,10,1,2,6,8,9},{1,2,6,7 ,10,3,4,5,8,9},{1,3,6,7,10,2,4,5,8,9},{2,3,6,7,10,1,4,5,8,9},{1,4,6,7,10,2,3,5,8,9},{ 2,4,6,7,10,1,3,5,8,9},{3,4,6,7,10,1,2,5,8,9},{1,5,6,7,10,2,3,4,8,9},{2,5,6,7,10,1,3,4 ,8,9},{3,5,6,7,10,1,2,4,8,9},{4,5,6,7,10,1,2,3,8,9},{1,2,3,8,10,4,5,6,7,9},{1,2,4,8,1 0,3,5,6,7,9},{1,3,4,8,10,2,5,6,7,9},{2,3,4,8,10,1,5,6,7,9},{1,2,5,8,10,3,4,6,7,9},{1, 3,5,8,10,2,4,6,7,9},{2,3,5,8,10,1,4,6,7,9},{1,4,5,8,10,2,3,6,7,9},{2,4,5,8,10,1,3,6,7 ,9},{3,4,5,8,10,1,2,6,7,9},{1,2,6,8,10,3,4,5,7,9},{1,3,6,8,10,2,4,5,7,9},{2,3,6,8,10, 1,4,5,7,9},{1,4,6,8,10,2,3,5,7,9},{2,4,6,8,10,1,3,5,7,9},{3,4,6,8,10,1,2,5,7,9},{1,5, 6,8,10,2,3,4,7,9},{2,5,6,8,10,1,3,4,7,9},{3,5,6,8,10,1,2,4,7,9},{4,5,6,8,10,1,2,3,7,9 },{1,2,7,8,10,3,4,5,6,9},{1,3,7,8,10,2,4,5,6,9},{2,3,7,8,10,1,4,5,6,9},{1,4,7,8,10,2, 3,5,6,9},{2,4,7,8,10,1,3,5,6,9},{3,4,7,8,10,1,2,5,6,9},{1,5,7,8,10,2,3,4,6,9},{2,5,7, 8,10,1,3,4,6,9},{3,5,7,8,10,1,2,4,6,9},{4,5,7,8,10,1,2,3,6,9},{1,6,7,8,10,2,3,4,5,9}, {2,6,7,8,10,1,3,4,5,9},{3,6,7,8,10,1,2,4,5,9},{4,6,7,8,10,1,2,3,5,9},{5,6,7,8,10,1,2, 3,4,9},{1,2,3,9,10,4,5,6,7,8},{1,2,4,9,10,3,5,6,7,8},{1,3,4,9,10,2,5,6,7,8},{2,3,4,9, 10,1,5,6,7,8},{1,2,5,9,10,3,4,6,7,8},{1,3,5,9,10,2,4,6,7,8},{2,3,5,9,10,1,4,6,7,8},{1 ,4,5,9,10,2,3,6,7,8},{2,4,5,9,10,1,3,6,7,8},{3,4,5,9,10,1,2,6,7,8},{1,2,6,9,10,3,4,5, 7,8},{1,3,6,9,10,2,4,5,7,8},{2,3,6,9,10,1,4,5,7,8},{1,4,6,9,10,2,3,5,7,8},{2,4,6,9,10 ,1,3,5,7,8},{3,4,6,9,10,1,2,5,7,8},{1,5,6,9,10,2,3,4,7,8},{2,5,6,9,10,1,3,4,7,8},{3,5 ,6,9,10,1,2,4,7,8},{4,5,6,9,10,1,2,3,7,8},{1,2,7,9,10,3,4,5,6,8},{1,3,7,9,10,2,4,5,6, 8},{2,3,7,9,10,1,4,5,6,8},{1,4,7,9,10,2,3,5,6,8},{2,4,7,9,10,1,3,5,6,8},{3,4,7,9,10,1 ,2,5,6,8},{1,5,7,9,10,2,3,4,6,8},{2,5,7,9,10,1,3,4,6,8},{3,5,7,9,10,1,2,4,6,8},{4,5,7 ,9,10,1,2,3,6,8},{1,6,7,9,10,2,3,4,5,8},{2,6,7,9,10,1,3,4,5,8},{3,6,7,9,10,1,2,4,5,8} ,{4,6,7,9,10,1,2,3,5,8},{5,6,7,9,10,1,2,3,4,8},{1,2,8,9,10,3,4,5,6,7},{1,3,8,9,10,2,4 ,5,6,7},{2,3,8,9,10,1,4,5,6,7},{1,4,8,9,10,2,3,5,6,7},{2,4,8,9,10,1,3,5,6,7},{3,4,8,9 ,10,1,2,5,6,7},{1,5,8,9,10,2,3,4,6,7},{2,5,8,9,10,1,3,4,6,7},{3,5,8,9,10,1,2,4,6,7},{ 4,5,8,9,10,1,2,3,6,7},{1,6,8,9,10,2,3,4,5,7},{2,6,8,9,10,1,3,4,5,7},{3,6,8,9,10,1,2,4 ,5,7},{4,6,8,9,10,1,2,3,5,7},{5,6,8,9,10,1,2,3,4,7},{1,7,8,9,10,2,3,4,5,6},{2,7,8,9,1 0,1,3,4,5,6},{3,7,8,9,10,1,2,4,5,6},{4,7,8,9,10,1,2,3,5,6},{5,7,8,9,10,1,2,3,4,6},{6, 7,8,9,10,1,2,3,4,5}} Schrausser, D. G. (2025). HP_Prime_MATH: Manual. https://www.academia.edu/130037807 82 wP12_2_10.dat M1=[[1,2,3,4,5,6,7,8,9,10,11,12],[1,3,2,4,5,6,7,8,9,10,11,12],[2,3,1,4,5,6,7,8,9,10,1 1,12],[1,4,2,3,5,6,7,8,9,10,11,12],[2,4,1,3,5,6,7,8,9,10,11,12],[3,4,1,2,5,6,7,8,9,10 ,11,12],[1,5,2,3,4,6,7,8,9,10,11,12],[2,5,1,3,4,6,7,8,9,10,11,12],[3,5,1,2,4,6,7,8,9, 10,11,12],[4,5,1,2,3,6,7,8,9,10,11,12],[1,6,2,3,4,5,7,8,9,10,11,12],[2,6,1,3,4,5,7,8, 9,10,11,12],[3,6,1,2,4,5,7,8,9,10,11,12],[4,6,1,2,3,5,7,8,9,10,11,12],[5,6,1,2,3,4,7, 8,9,10,11,12],[1,7,2,3,4,5,6,8,9,10,11,12],[2,7,1,3,4,5,6,8,9,10,11,12],[3,7,1,2,4,5, 6,8,9,10,11,12],[4,7,1,2,3,5,6,8,9,10,11,12],[5,7,1,2,3,4,6,8,9,10,11,12],[6,7,1,2,3, 4,5,8,9,10,11,12],[1,8,2,3,4,5,6,7,9,10,11,12],[2,8,1,3,4,5,6,7,9,10,11,12],[3,8,1,2, 4,5,6,7,9,10,11,12],[4,8,1,2,3,5,6,7,9,10,11,12],[5,8,1,2,3,4,6,7,9,10,11,12],[6,8,1, 2,3,4,5,7,9,10,11,12],[7,8,1,2,3,4,5,6,9,10,11,12],[1,9,2,3,4,5,6,7,8,10,11,12],[2,9, 1,3,4,5,6,7,8,10,11,12],[3,9,1,2,4,5,6,7,8,10,11,12],[4,9,1,2,3,5,6,7,8,10,11,12],[5, 9,1,2,3,4,6,7,8,10,11,12],[6,9,1,2,3,4,5,7,8,10,11,12],[7,9,1,2,3,4,5,6,8,10,11,12],[ 8,9,1,2,3,4,5,6,7,10,11,12],[1,10,2,3,4,5,6,7,8,9,11,12],[2,10,1,3,4,5,6,7,8,9,11,12] ,[3,10,1,2,4,5,6,7,8,9,11,12],[4,10,1,2,3,5,6,7,8,9,11,12],[5,10,1,2,3,4,6,7,8,9,11,1 2],[6,10,1,2,3,4,5,7,8,9,11,12],[7,10,1,2,3,4,5,6,8,9,11,12],[8,10,1,2,3,4,5,6,7,9,11 ,12],[9,10,1,2,3,4,5,6,7,8,11,12],[1,11,2,3,4,5,6,7,8,9,10,12],[2,11,1,3,4,5,6,7,8,9, 10,12],[3,11,1,2,4,5,6,7,8,9,10,12],[4,11,1,2,3,5,6,7,8,9,10,12],[5,11,1,2,3,4,6,7,8, 9,10,12],[6,11,1,2,3,4,5,7,8,9,10,12],[7,11,1,2,3,4,5,6,8,9,10,12],[8,11,1,2,3,4,5,6, 7,9,10,12],[9,11,1,2,3,4,5,6,7,8,10,12],[10,11,1,2,3,4,5,6,7,8,9,12],[1,12,2,3,4,5,6, 7,8,9,10,11],[2,12,1,3,4,5,6,7,8,9,10,11],[3,12,1,2,4,5,6,7,8,9,10,11],[4,12,1,2,3,5, 6,7,8,9,10,11],[5,12,1,2,3,4,6,7,8,9,10,11],[6,12,1,2,3,4,5,7,8,9,10,11],[7,12,1,2,3, 4,5,6,8,9,10,11],[8,12,1,2,3,4,5,6,7,9,10,11],[9,12,1,2,3,4,5,6,7,8,10,11],[10,12,1,2 ,3,4,5,6,7,8,9,11],[11,12,1,2,3,4,5,6,7,8,9,10]] L1={{1,2,3,4,5,6,7,8,9,10,11,12},{1,3,2,4,5,6,7,8,9,10,11,12},{2,3,1,4,5,6,7,8,9,10,1 1,12},{1,4,2,3,5,6,7,8,9,10,11,12},{2,4,1,3,5,6,7,8,9,10,11,12},{3,4,1,2,5,6,7,8,9,10 ,11,12},{1,5,2,3,4,6,7,8,9,10,11,12},{2,5,1,3,4,6,7,8,9,10,11,12},{3,5,1,2,4,6,7,8,9, 10,11,12},{4,5,1,2,3,6,7,8,9,10,11,12},{1,6,2,3,4,5,7,8,9,10,11,12},{2,6,1,3,4,5,7,8, 9,10,11,12},{3,6,1,2,4,5,7,8,9,10,11,12},{4,6,1,2,3,5,7,8,9,10,11,12},{5,6,1,2,3,4,7, 8,9,10,11,12},{1,7,2,3,4,5,6,8,9,10,11,12},{2,7,1,3,4,5,6,8,9,10,11,12},{3,7,1,2,4,5, 6,8,9,10,11,12},{4,7,1,2,3,5,6,8,9,10,11,12},{5,7,1,2,3,4,6,8,9,10,11,12},{6,7,1,2,3, 4,5,8,9,10,11,12},{1,8,2,3,4,5,6,7,9,10,11,12},{2,8,1,3,4,5,6,7,9,10,11,12},{3,8,1,2, 4,5,6,7,9,10,11,12},{4,8,1,2,3,5,6,7,9,10,11,12},{5,8,1,2,3,4,6,7,9,10,11,12},{6,8,1, 2,3,4,5,7,9,10,11,12},{7,8,1,2,3,4,5,6,9,10,11,12},{1,9,2,3,4,5,6,7,8,10,11,12},{2,9, 1,3,4,5,6,7,8,10,11,12},{3,9,1,2,4,5,6,7,8,10,11,12},{4,9,1,2,3,5,6,7,8,10,11,12},{5, 9,1,2,3,4,6,7,8,10,11,12},{6,9,1,2,3,4,5,7,8,10,11,12},{7,9,1,2,3,4,5,6,8,10,11,12},{ 8,9,1,2,3,4,5,6,7,10,11,12},{1,10,2,3,4,5,6,7,8,9,11,12},{2,10,1,3,4,5,6,7,8,9,11,12} ,{3,10,1,2,4,5,6,7,8,9,11,12},{4,10,1,2,3,5,6,7,8,9,11,12},{5,10,1,2,3,4,6,7,8,9,11,1 2},{6,10,1,2,3,4,5,7,8,9,11,12},{7,10,1,2,3,4,5,6,8,9,11,12},{8,10,1,2,3,4,5,6,7,9,11 ,12},{9,10,1,2,3,4,5,6,7,8,11,12},{1,11,2,3,4,5,6,7,8,9,10,12},{2,11,1,3,4,5,6,7,8,9, 10,12},{3,11,1,2,4,5,6,7,8,9,10,12},{4,11,1,2,3,5,6,7,8,9,10,12},{5,11,1,2,3,4,6,7,8, 9,10,12},{6,11,1,2,3,4,5,7,8,9,10,12},{7,11,1,2,3,4,5,6,8,9,10,12},{8,11,1,2,3,4,5,6, 7,9,10,12},{9,11,1,2,3,4,5,6,7,8,10,12},{10,11,1,2,3,4,5,6,7,8,9,12},{1,12,2,3,4,5,6, 7,8,9,10,11},{2,12,1,3,4,5,6,7,8,9,10,11},{3,12,1,2,4,5,6,7,8,9,10,11},{4,12,1,2,3,5, 6,7,8,9,10,11},{5,12,1,2,3,4,6,7,8,9,10,11},{6,12,1,2,3,4,5,7,8,9,10,11},{7,12,1,2,3, 4,5,6,8,9,10,11},{8,12,1,2,3,4,5,6,7,9,10,11},{9,12,1,2,3,4,5,6,7,8,10,11},{10,12,1,2 ,3,4,5,6,7,8,9,11},{11,12,1,2,3,4,5,6,7,8,9,10}} wV2_3.dat M1=[[-1,-1,-1],[-1,-1,1],[-1,1,-1],[-1,1,1],[1,-1,-1],[1,-1,1],[1,1,-1],[1,1,1]] L1={{-1,-1,-1},{-1,-1,1},{-1,1,-1},{-1,1,1},{1,-1,-1},{1,-1,1},{1,1,-1},{1,1,1}} wV2_9.dat M1=[[-1,-1,-1,-1,-1,-1,-1,-1,-1],[-1,-1,-1,-1,-1,-1,-1,-1,1],[-1,-1,-1,-1,-1,-1,- 1,1,-1],[-1,-1,-1,-1,-1,-1,-1,1,1],[-1,-1,-1,-1,-1,-1,1,-1,-1],[-1,-1,-1,-1,-1,-1,1,- 1,1],[-1,-1,-1,-1,-1,-1,1,1,-1],[-1,-1,-1,-1,-1,-1,1,1,1],[-1,-1,-1,-1,-1,1,-1,-1,- 1],[-1,-1,-1,-1,-1,1,-1,-1,1],[-1,-1,-1,-1,-1,1,-1,1,-1],[-1,-1,-1,-1,-1,1,-1,1,1],[- 1,-1,-1,-1,-1,1,1,-1,-1],[-1,-1,-1,-1,-1,1,1,-1,1],[-1,-1,-1,-1,-1,1,1,1,-1],[-1,-1,- 1,-1,-1,1,1,1,1],[-1,-1,-1,-1,1,-1,-1,-1,-1],[-1,-1,-1,-1,1,-1,-1,-1,1],[-1,-1,-1,- 1,1,-1,-1,1,-1],[-1,-1,-1,-1,1,-1,-1,1,1],[-1,-1,-1,-1,1,-1,1,-1,-1],[-1,-1,-1,-1,1,- 1,1,-1,1],[-1,-1,-1,-1,1,-1,1,1,-1],[-1,-1,-1,-1,1,-1,1,1,1],[-1,-1,-1,-1,1,1,-1,-1,- 1],[-1,-1,-1,-1,1,1,-1,-1,1],[-1,-1,-1,-1,1,1,-1,1,-1],[-1,-1,-1,-1,1,1,-1,1,1],[-1,- 1,-1,-1,1,1,1,-1,-1],[-1,-1,-1,-1,1,1,1,-1,1],[-1,-1,-1,-1,1,1,1,1,-1],[-1,-1,-1,- Schrausser, D. G. (2025). HP_Prime_MATH: Manual. https://www.academia.edu/130037807 83 1,1,1,1,1,1],[-1,-1,-1,1,-1,-1,-1,-1,-1],[-1,-1,-1,1,-1,-1,-1,-1,1],[-1,-1,-1,1,-1,- 1,-1,1,-1],[-1,-1,-1,1,-1,-1,-1,1,1],[-1,-1,-1,1,-1,-1,1,-1,-1],[-1,-1,-1,1,-1,-1,1,- 1,1],[-1,-1,-1,1,-1,-1,1,1,-1],[-1,-1,-1,1,-1,-1,1,1,1],[-1,-1,-1,1,-1,1,-1,-1,-1],[- 1,-1,-1,1,-1,1,-1,-1,1],[-1,-1,-1,1,-1,1,-1,1,-1],[-1,-1,-1,1,-1,1,-1,1,1],[-1,-1,- 1,1,-1,1,1,-1,-1],[-1,-1,-1,1,-1,1,1,-1,1],[-1,-1,-1,1,-1,1,1,1,-1],[-1,-1,-1,1,- 1,1,1,1,1],[-1,-1,-1,1,1,-1,-1,-1,-1],[-1,-1,-1,1,1,-1,-1,-1,1],[-1,-1,-1,1,1,-1,- 1,1,-1],[-1,-1,-1,1,1,-1,-1,1,1],[-1,-1,-1,1,1,-1,1,-1,-1],[-1,-1,-1,1,1,-1,1,- 1,1],[-1,-1,-1,1,1,-1,1,1,-1],[-1,-1,-1,1,1,-1,1,1,1],[-1,-1,-1,1,1,1,-1,-1,-1],[-1,- 1,-1,1,1,1,-1,-1,1],[-1,-1,-1,1,1,1,-1,1,-1],[-1,-1,-1,1,1,1,-1,1,1],[-1,-1,- 1,1,1,1,1,-1,-1],[-1,-1,-1,1,1,1,1,-1,1],[-1,-1,-1,1,1,1,1,1,-1],[-1,-1,- 1,1,1,1,1,1,1],[-1,-1,1,-1,-1,-1,-1,-1,-1],[-1,-1,1,-1,-1,-1,-1,-1,1],[-1,-1,1,-1,- 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G. (2025). 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