(50G) Percentiles according to Hyndman&Fan's 9 types + Tukey_inc _box plot +- HP Forums (https://www.hpmuseum.org/forum) +-- Forum: HP Software Libraries (https://www.hpmuseum.org/forum/forum-10.html) +--- Forum: General Software Library (https://www.hpmuseum.org/forum/forum-13.html) +--- Thread: (50G) Percentiles according to Hyndman&Fan's 9 types + Tukey_inc _box plot (/thread-23928.html)

(50G) Percentiles according to Hyndman&Fan's 9 types + Tukey_inc _box plot - Gil - 2025-08-13 19:23 Version 11b Input: 2 arguments - in stack level 2: list (unordered) of the elements/numbers  - in stack level 1: wished percentile; for 10%, 25%, etc., write here 10, 25, etc.   (and not .10, 0.25, etc.) or Input: 3 arguments - in stack level 3: list (unordered) of the elements/numbers  - in stack level 2: wished percentile; for 10%, 25%, etc., write here 10, 25, etc.   (and not .10, 0.25, etc.) - in stack level 1: list of integers >=1 corresponding to the frequencies of the numbers in the first list  (or in stack level 1: list of wi, 0<wi<1, corresponding to the weights of the numbers in the first list)  Output a) recap summary for checking  and b) results according to up to 17 different methods a) recap summary for checking - P% - ordered list of numbers  (&, if frequencies or weights, ordered list of frequencies/weights and cumulative list of frequencies/weights)  - n - mean b) results according to up to 17 different methods - type method or principal softwares  - k (rank k, k>= 1 and k<=n, in the ordered list: 1, for the first value, 2 for the second one, 3 for the 3rd one; 3.25 = 3rd value +25%[value4 - value3] (but k is <1 and >0 in case of weights)  - x_k, corresponding value, always lin. interpolated as above with k=3.25 (but no interpolation in case of weights)  * if weights, only one single result for k & also only one single result for the corresponding x_k value, then always according to just method type R4, with k=np=(Swi)p  * if no weights, always 15 methods —a) b), c) & d) — for quantiles/percentiles + possible 2 other methods — e) & f) — for Q1, Q2 and Q3: a) 9 Hyndman&Fan's types (same types in R statistical software) b) 4 variations of type R7 (floor, ceil, midpoint and nearest point)  c) Mendenhall&Sincich method d) Cunnane method  * and also, always if no weights, 2 others methods, e) & f), for quartiles Q1, Q2 & Q3, but only when P%=25%, P=0.5 or 75%: e) Moore&McCabe = for Q1, Q2 & Q3  used in both HP Prime & TI[83] calculators (≠quantile/percentiles type R1 used in HP calculators)  f) Tukey's inclusive method for boxplots Q1, Q2 & Q3 - and, lastly, but only if no weights and only when P%=50%,  g) the "usual median" with k=(n+1)/2, with then repeated results of definition R6 calculated previously  Goal Comparison of possible different results for same percentile input, according to  - 15 methods for P%≠25%, P≠50% or 75% - or 17 methods when P%=25%, P=50% or 75% with reference to some (statistical) softwares or calculators (R, SAS, Minitab, SPSS, S(plus), Mathematica, TI 83 & HP Prime.  *if weights, only one single method calculated: R4.  Correspondences A) 9 types in R according to 9 Hyndman&Fan's types: R1 —>SAS3 —>pHP   *if p=.25/.5 /.75, R1 does not correspond to 2nd method described below in E) or in F) with quartiles Q1, Q2 and Q3 (MM Qi TI HP)  R2 —>SAS5 (SAS by default)  R3 —>SAS2 R4 —>old defaultPython<=1.9 —>SAS1 —>weighted method  R5 —>Hazen —>Mathematica —>old XLS.exc ! R6 —>Weibull/Gumbel1939 —>SAS4 —>XLSexc            —>Minitab —>SPSS R7 (default in R) —>Python>=1.1—>XLSinc        —>S(plus)    Added old NumPy:    7b floor    7c ceil    7d midpoint    7e next point R8 (Johnson&Kotz 1970)  R9 (Blom 1958)  B) 1 Mendenhall&Sincich method;       no correspondence  C) 5 SAS PCTDEF: PCTDEF=1 —>R4 —>old defaultPython<=1.9 PCTDEF=2 —>R3 —>SAS2 PCTDEF=3 —>R1 —>pTI  —>pHP PCTDEF=4 —>R6 —>Weibull/Gumbel1939 —>XLSexc                                —>Minitab —>SPSS PCTDEF=5 (default in SAS) —>R2 * missing in SAS the R types 5/7/8/9 D) 2 XLS methods: excluding —>R6 >Weibull/Gumbel1939 —>SAS4                     —>Minitab —>SPSS including  (by default) —>R7 >Python>=1.1                                          —>S(plus) (*old XLS.exc —> R5 —>Hazen —>Mathematica)  E) 2 TI[83] or HP[Prime] methods: e1) 1st method: "for percentiles, only for HP Prime" —>R1 —>SAS3 * if p=.25/.5/.75, that 1st method "HP for percentiles" (R1) does not correspond to 2nd method below "TI HP for quartiles Q1, Q2 & Q3" (MM)!   e2) 2nd method: "for quartiles Q1, Q2 & Q3, for both calculators TI & HP" —>Moore&McCabe (MM) —>Tukey_exclusive (Te)  * that 2nd method "TI HP for quartiles Q1, Q2 & Q3" (MM) does not correspond to 1st method above "HP for percentile" (R1) when p=0.25, 0.5 or 0.75! F) 1 Mathematica method —>R5 —>Hazen                                                 —>old XLS.exc ! G) 1 Hazen method —>R5 —> Matemática                                                —>old XLS.exc ! H) Minitab —>R6 —>SAS4 —>Weibull/Gumbel1939                     —>XLSexc —>SPSS I) SPSS —>R6 —>SAS4 —>Weibull/Gumbel1939                —>XLSexc —>Minitab J) Weibull/Gumbel1939 —>R6 —>SAS4 —>XLSexc                                            —>Minitab K) Python>=1.1 —>R7 —>XLSinc —>S(plus) L) 1 Moore&McCabe (MM) method: for Q1 Q2 & Q3 TI HP calculators ≠ method HP for percentiles!  (see both methods e1) & e2) mentioned in E) above)  M) 2 Tukey methods for Q1, Q2 & Q3 used for boxplots :  exclusive (Te) —> MM inclusive: no correspondence  Code  \<< "2/3 Arg    Version11 2 Arg:   a1) original     unordered list of#     {#1 #2  #n}   a2) percentile (ex:     7 for 7%       <not 0.07>     25 or 75 for     1st/3rd quartile     50 for median) or 3 Arg: .  a1 {#1#n} a2 (P%)   & a3 {f1fn} = list   freq corresp. to a1   fi integers \>= 1 or .  a1 {#1#n} a2 (P%)   & a3 {w1wn} = list   weights corr. to a1   0<wi<1   NB with wi, k & x_k       only with       type R4 below 9 Hyndman&Fan's (HF/R)   types \->R; & 1517 methods(types)   (with weights, only   1 type: R4 below) https://www. rdocumentation.org/ packages/stats/ versions/3.6.2/ topics/quantile 1=inv empirical   nearest upper rank   \->SAS3   \->pHP (only)     (if p=.25/.5 /.75     \=/ MM Q1/2/3 HP TI) 2=like 1, but avg   inv at discont.   best for Eric   Langford   for population   \->SAS5=SAS default 3=nearest even integ   \->SAS2 4=lin interp   <old defaultPython   \<=1.9> \->SAS1   only method avail.     with weights! 5=lin interp   \->Hines&Montgomery   (HM)   \->Hazen (H)   \->Mathematica (M)   \->Wolfram (W)   \->Hog&Ledolter (HL)   <=old XLS.exc !> 6=lin inter   Popular method (P)   Weibull/Gumbel 1939   exclude median   \->Python? (Py)   \->SAS4   \->Minitab (MT)   \->XLSe[xc]   \->q_SPSS (\=/Q_SPSS) 7=best? in gener. use   lin interp   for sample   always incl median   default in R     & Python\>=1.1       (\->defP)   \->vanFreund&Perles     (V)   \->XLSi[nc]   \->S(plus) 7b floor 7c ceiling 7d mid point 7e next point 8=lin interp   \->Johnson&Kotz 1970   median unbiased 9=lin interp   normal unbiased   \->Blom 1958 1 Mendenhall&Sincich   (MS)   . (wrong formula by     Wolfram shown but     not calc  here)   . correct calc here 1 Cunnane   \->SciPy.mquantiles   (SciPy.mq) 1 Moore&McCabe (MM)   med excl for Q1&Q3   \->TI[83] \->HP[Prime] 1 Tukey: avg   for Q1&3 in boxplot   median inc, but if     n=even, median exc   \->Q_SPSS 5 SAS options:   1\->HF4   2\->3   3\->1   4\->6   5(default)\->2   missing: HF 5/7/8/9 2 XLS: interp   .exc (n\>=5) \->R6     but when n=5 or 6     it excl min/max!   .inc:best? (n\>=4)\->R7 1 vanFreund&Perles(V)   \->R7 1 Mathematica   \->R5 1 Wolfram (W)   \->R5 2 HP (1 TI)   .only HP   p \->R1; if p=.25/.5   .75\=/Q1/2/3 in HP/TI   .both TI/HP,   but only for Q1/2/3   (\=/pHP .25/.5/.75)   \->MM 2 SPSS   .p \->R6   .Qi \->Tukey 1 Minitab (MT)   \->R6 1 S(plus)   \->R7 " DROP -105 CF DUP TYPE 5 \=/   IF   THEN 0   END PICK3 SIZE 0 0 0 0 0 MAXR \->NUM \-> l p lf s lo lf\GS n k j fmin   \<<     \<< \->NUM DUP FP 0 == { R\->I } IFT 'k' STO       IF fmin 1 \>=       THEN         CASE k 1 <           THEN k 4 RND "\->1" + 1 'k' STO           END k n >           THEN k 4 RND "\->" + n DUP FP 0 == { R\->I } IFT + n 'k' STO           END k         END       ELSE k       END "k" fmin 1 < { "=pn" + } IFT \->TAG       IF lf 0 \=/       THEN 1 s         FOR i lf\GS i GET k - 0 \>=           IF           THEN i             IF i 1 > fmin 1 \>= AND             THEN k lf\GS i 1 - GET - 1 <               IF               THEN k FP + 1 -               END             END s 'i' STO           END         NEXT 'k' STO       END k FP lo k IP GETI UNROT GET OVER - ROT * + "x_k" \->TAG     \>> \-> xk     \<< p DUP FP 0 == { R\->I } IFT "P%" \->TAG .01 'p' STO* l SORT "lo" \->TAG DUP 'lo' STO       IF lf 0 \=/       THEN { } 1 s         FOR i lf l lo i GET POS GET DUP fmin MIN 'fmin' STO +         NEXT 'lf' STO lf HEAD DUP 1 \->LIST SWAP 2 s         FOR i lf i GET + DUP ROT SWAP + SWAP         NEXT 'n' STO DUP 'lf\GS' STO fmin 1 \>= "f\GS" "wi\GS" IFTE \->TAG lf fmin 1 \>= "fo" "wi.o" IFTE \->TAG SWAP OVER 4 PICK * \GSLIST n / \->NUM "mean" \->TAG n DUP FP 0 == { R\->I } IFT "n" fmin 1 < { "=\GSwi" + } IFT \->TAG SWAP       ELSE s 'n' STO DUP \GSLIST n / \->NUM "mean" \->TAG n "n" \->TAG SWAP       END       IF fmin 1 \>=       THEN "R1=SAS3 pHP"         CASE p .25 ==           THEN "\=/Q1HP" +           END p .5 ==           THEN "\=/Q2HP" +           END p .75 ==           THEN "\=/Q3HP" +           END         END 'n*p' \->NUM CEIL xk EVAL "R2=SAS5 Langford" 'n*p' \->NUM DUP 'k' STO         IF FP 0 ==         THEN k .5 +         ELSE k CEIL         END xk EVAL "R3=SAS2" 'n*p-.5' \->NUM DUP 'k' STO FLOOR 'j' STO j         IF j 2 MOD NOT k j == AND NOT         THEN 1 +         END xk EVAL       END "R4=SAS1" fmin 1 <       IF       THEN "Only " SWAP +       END 'n*p' xk EVAL       IF fmin 1 \>=       THEN "R5=p&Q:M/H/HM/W" 'n*p+.5' xk EVAL "6=XLSeSAS4MTpSPSS P Py" '(n+1)*p' xk EVAL "7=XLSidefP/R S+V" '(n-1)*p+1' xk EVAL "7b (n-1)p+1 down" '(n-1)*p+1' FLOOR xk EVAL "7c (n-1)p+1 ceil" '(n-1)*p+1' CEIL xk EVAL "7d(n-1)p+1 midPt" '(n-1)*p+1' DUP FLOOR SWAP CEIL + 2 / xk EVAL "7e(n-1)p+1 nxtPt" '(n-1)*p+1' DUP FP .5 \<= { FLOOR } { CEIL } IFTE xk EVAL "R8=Johnson&Kotz" '(n+1/3)*p+1/3' 5 RND xk EVAL "R9=Blom" '(n+1/4)*p+3/8' xk EVAL "Mendenhall&S(MS)" "Wolfram wrong eq for MS:" 'n*p' \->NUM DUP FP .5 \=/         IF         THEN 0 RND         END DROP2 "Correct eq for MM:" DROP '(n+1)*p'         CASE p .5 >           THEN FLOOR           END p .5 <           THEN CEIL           END         END xk EVAL "Cunnane SciPy.mquantiles" '(n+.2)*p+.4' xk EVAL         IF p .25 == p .5 == OR p .75 == OR         THEN "MMexc Q"           CASE p .25 ==             THEN "1"             END p .5 ==             THEN "2"             END p .75 ==             THEN "3"             END           END + "(\=/p)TI HP" + 'n*p+.5+(p-.5)*(n MOD 2)' xk EVAL         END p .25 == p .5 == OR p .75 == OR         IF         THEN "TukeyincBox QSPSS" 'n*p+.5+(.5-p)*(n MOD 2)' xk EVAL         END p .5 ==         IF         THEN "Usual med (Q2) R6" '(n+1)*p' xk EVAL         END       END     \>>   \>> \>> Tests/Checking No guarantee of the real algorithms implementation for the different methods/types or softwares mentioned.  12 Examples { { { 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 } :n: 100 :mean: 50.5 :P%: 25 "R1=SAS3 pHP‹Q1HP" :k: 25 :x_k: 25 "R2=SAS5 Langford" :k: 25.5 :x_k: 25.5 "R3=SAS2" :k: 25 :x_k: 25 "R4=SAS1" :k: 25 :x_k: 25. "R5=Mathematica H" :k: 25.5 :x_k: 25.5 "R6XLSeSAS4MTSPSS" :k: 25.25 :x_k: 25.25 "R7XLSi S Python" :k: 25.75 :x_k: 25.75 "R8" :k: 25.41667 :x_k: 25.41667 "R9" :k: 25.4375 :x_k: 25.4375 "Mendenhall&S.(MS)" :k: 25 :x_k: 25. "MM: Q1(‹p) TI HP" :k: 25.5 :x_k: 25.5 "Tukeyinc_boxplot" :k: 25.5 :x_k: 25.5 }  { { 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 } :n: 101 :mean: 51. :P%: 25 "R1=SAS3 pHP‹Q1HP" :k: 26 :x_k: 26 "R2=SAS5 Langford" :k: 26 :x_k: 26 "R3=SAS2" :k: 25 :x_k: 25 "R4=SAS1" :k: 25.25 :x_k: 25.25 "R5=Mathematica H" :k: 25.75 :x_k: 25.75 "R6XLSeSAS4MTSPSS" :k: 25.5 :x_k: 25.5 "R7XLSi S Python" :k: 26 :x_k: 26. "R8" :k: 25.66667 :x_k: 25.66667 "R9" :k: 25.6875 :x_k: 25.6875 "Mendenhall&S.(MS)" :k: 26 :x_k: 26. "MM: Q1(‹p) TI HP" :k: 25.5 :x_k: 25.5 "Tukeyinc_boxplot" :k: 26 :x_k: 26. }  { { 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 } :n: 102 :mean: 51.5 :P%: 25 "R1=SAS3 pHP‹Q1HP" :k: 26 :x_k: 26 "R2=SAS5 Langford" :k: 26 :x_k: 26 "R3=SAS2" :k: 26 :x_k: 26 "R4=SAS1" :k: 25.5 :x_k: 25.5 "R5=Mathematica H" :k: 26 :x_k: 26. "R6XLSeSAS4MTSPSS" :k: 25.75 :x_k: 25.75 "R7XLSi S Python" :k: 26.25 :x_k: 26.25 "R8" :k: 25.91667 :x_k: 25.91667 "R9" :k: 25.9375 :x_k: 25.9375 "Mendenhall&S.(MS)" :k: 26 :x_k: 26. "MM: Q1(‹p) TI HP" :k: 25.5 :x_k: 25.5 "Tukeyinc_boxplot" :k: 26 :x_k: 26. }  { { 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 } :n: 103 :mean: 52. :P%: 25 "R1=SAS3 pHP‹Q1HP" :k: 26 :x_k: 26 "R2=SAS5 Langford" :k: 26 :x_k: 26 "R3=SAS2" :k: 26 :x_k: 26 "R4=SAS1" :k: 25.75 :x_k: 25.75 "R5=Mathematica H" :k: 26.25 :x_k: 26.25 "R6XLSeSAS4MTSPSS" :k: 26 :x_k: 26. "R7XLSi S Python" :k: 26.5 :x_k: 26.5 "R8" :k: 26.16667 :x_k: 26.16667 "R9" :k: 26.1875 :x_k: 26.1875 "Mendenhall&S.(MS)" :k: 26 :x_k: 26. "MM: Q1(‹p) TI HP" :k: 26 :x_k: 26. "Tukeyinc_boxplot" :k: 26.5 :x_k: 26.5 }  { { 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 } :n: 100 :mean: 50.5 :P%: 50 "R1=SAS3 pHP‹Q2HP" :k: 50 :x_k: 50 "R2=SAS5 Langford" :k: 50.5 :x_k: 50.5 "R3=SAS2" :k: 50 :x_k: 50 "R4=SAS1" :k: 50 :x_k: 50. "R5=Mathematica H" :k: 50.5 :x_k: 50.5 "R6XLSeSAS4MTSPSS" :k: 50.5 :x_k: 50.5 "R7XLSi S Python" :k: 50.5 :x_k: 50.5 "R8" :k: 50.5 :x_k: 50.5 "R9" :k: 50.5 :x_k: 50.5 "Mendenhall&S.(MS)" :k: 50 :x_k: 50. "MM: Q3(‹p) TI HP" :k: 50.5 :x_k: 50.5 "Tukeyinc_boxplot" :k: 50.5 :x_k: 50.5 "R6:usual med (Q2)" :k: 50.5 :x_k: 50.5 }  { { 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 } :n: 101 :mean: 51. :P%: 50 "R1=SAS3 pHP‹Q2HP" :k: 51 :x_k: 51 "R2=SAS5 Langford" :k: 51 :x_k: 51 "R3=SAS2" :k: 50 :x_k: 50 "R4=SAS1" :k: 50.5 :x_k: 50.5 "R5=Mathematica H" :k: 51 :x_k: 51. "R6XLSeSAS4MTSPSS" :k: 51 :x_k: 51. "R7XLSi S Python" :k: 51 :x_k: 51. "R8" :k: 51 :x_k: 51. "R9" :k: 51 :x_k: 51. "Mendenhall&S.(MS)" :k: 50.5 :x_k: 50.5 "MM: Q3(‹p) TI HP" :k: 51 :x_k: 51. "Tukeyinc_boxplot" :k: 51.5 :x_k: 51.5 "R6:usual med (Q2)" :k: 51. :x_k: 51. }  { { 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 } :n: 102 :mean: 51.5 :P%: 50 "R1=SAS3 pHP‹Q2HP" :k: 51 :x_k: 51 "R2=SAS5 Langford" :k: 51.5 :x_k: 51.5 "R3=SAS2" :k: 51 :x_k: 51 "R4=SAS1" :k: 51 :x_k: 51. "R5=Mathematica H" :k: 51.5 :x_k: 51.5 "R6XLSeSAS4MTSPSS" :k: 51.5 :x_k: 51.5 "R7XLSi S Python" :k: 51.5 :x_k: 51.5 "R8" :k: 51.5 :x_k: 51.5 "R9" :k: 51.5 :x_k: 51.5 "Mendenhall&S.(MS)" :k: 51 :x_k: 51. "MM: Q3(‹p) TI HP" :k: 51.5 :x_k: 51.5 "Tukeyinc_boxplot" :k: 51.5 :x_k: 51.5 "R6:usual med (Q2)" :k: 51.5 :x_k: 51.5 }  { { 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 } :n: 103 :mean: 52. :P%: 50 "R1=SAS3 pHP‹Q2HP" :k: 52 :x_k: 52 "R2=SAS5 Langford" :k: 52 :x_k: 52 "R3=SAS2" :k: 52 :x_k: 52 "R4=SAS1" :k: 51.5 :x_k: 51.5 "R5=Mathematica H" :k: 52 :x_k: 52. "R6XLSeSAS4MTSPSS" :k: 52 :x_k: 52. "R7XLSi S Python" :k: 52 :x_k: 52. "R8" :k: 52 :x_k: 52. "R9" :k: 52 :x_k: 52. "Mendenhall&S.(MS)" :k: 51.5 :x_k: 51.5 "MM: Q3(‹p) TI HP" :k: 52 :x_k: 52. "Tukeyinc_boxplot" :k: 52.5 :x_k: 52.5 "R6:usual med (Q2)" :k: 52. :x_k: 52. }  { { 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 } :n: 100 :mean: 50.5 :P%: 75 "R1=SAS3 pHP‹Q3HP" :k: 75 :x_k: 75 "R2=SAS5 Langford" :k: 75.5 :x_k: 75.5 "R3=SAS2" :k: 75 :x_k: 75 "R4=SAS1" :k: 75 :x_k: 75. "R5=Mathematica H" :k: 75.5 :x_k: 75.5 "R6XLSeSAS4MTSPSS" :k: 75.75 :x_k: 75.75 "R7XLSi S Python" :k: 75.25 :x_k: 75.25 "R8" :k: 75.58333 :x_k: 75.58333 "R9" :k: 75.5625 :x_k: 75.5625 "Mendenhall&S.(MS)" :k: 75 :x_k: 75. "MM: Q3(‹p) TI HP" :k: 75.5 :x_k: 75.5 "Tukeyinc_boxplot" :k: 75.5 :x_k: 75.5 }  { { 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 } :n: 101 :mean: 51. :P%: 75 "R1=SAS3 pHP‹Q3HP" :k: 76 :x_k: 76 "R2=SAS5 Langford" :k: 76 :x_k: 76 "R3=SAS2" :k: 76 :x_k: 76 "R4=SAS1" :k: 75.75 :x_k: 75.75 "R5=Mathematica H" :k: 76.25 :x_k: 76.25 "R6XLSeSAS4MTSPSS" :k: 76.5 :x_k: 76.5 "R7XLSi S Python" :k: 76 :x_k: 76. "R8" :k: 76.33333 :x_k: 76.33333 "R9" :k: 76.3125 :x_k: 76.3125 "Mendenhall&S.(MS)" :k: 75 :x_k: 75. "MM: Q3(‹p) TI HP" :k: 76.5 :x_k: 76.5 "Tukeyinc_boxplot" :k: 77 :x_k: 77. }  { { 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 } :n: 102 :mean: 51.5 :P%: 75 "R1=SAS3 pHP‹Q3HP" :k: 77 :x_k: 77 "R2=SAS5 Langford" :k: 77 :x_k: 77 "R3=SAS2" :k: 76 :x_k: 76 "R4=SAS1" :k: 76.5 :x_k: 76.5 "R5=Mathematica H" :k: 77 :x_k: 77. "R6XLSeSAS4MTSPSS" :k: 77.25 :x_k: 77.25 "R7XLSi S Python" :k: 76.75 :x_k: 76.75 "R8" :k: 77.08333 :x_k: 77.08333 "R9" :k: 77.0625 :x_k: 77.0625 "Mendenhall&S.(MS)" :k: 76 :x_k: 76. "MM: Q3(‹p) TI HP" :k: 77.5 :x_k: 77.5 "Tukeyinc_boxplot" :k: 77 :x_k: 77. }  { { 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 } :n: 103 :mean: 52. :P%: 75 "R1=SAS3 pHP‹Q3HP" :k: 78 :x_k: 78 "R2=SAS5 Langford" :k: 78 :x_k: 78 "R3=SAS2" :k: 77 :x_k: 77 "R4=SAS1" :k: 77.25 :x_k: 77.25 "R5=Mathematica H" :k: 77.75 :x_k: 77.75 "R6XLSeSAS4MTSPSS" :k: 78 :x_k: 78. "R7XLSi S Python" :k: 77.5 :x_k: 77.5 "R8" :k: 77.83333 :x_k: 77.83333 "R9" :k: 77.8125 :x_k: 77.8125 "Mendenhall&S.(MS)" :k: 77 :x_k: 77. "MM: Q3(‹p) TI HP" :k: 78 :x_k: 78. "Tukeyinc_boxplot" :k: 78.5 :x_k: 78.5 } } RE: (50G) Percentiles according to 17 methods (Hyndman&Fan, Tukey, etc. ) - Gil - 2025-08-29 16:07 Version 11b Now you can add a list of frequencies fi, fi integer >=1 Example  1) Enter the following list of numbers in stack level 3: { 50 20 30 10 40 }  2) Enter 25 in stack level 2 for 1st Quartile  3) Enter the frequencies in stack level 1: { 30 40 10 20 10 }  and run the program. You can add also weights wi, 0<wi<1. Example  1) Enter the following list of numbers in stack level 3: { 50 20 30 10 40 }  2) Enter 25 in stack level 2 for 1st Quartile  3) Enter the weights in stack level 1: { 30 40 10 20 10 } 110./ ie { .272727272727 .363636363636 9.09090909091E-2 .181818181818 9.09090909091E-2 } or other values like {.2 .4 0.2 0.1 0.1} and execute the program. Tukey's method In the previous versions 9/10, there was an error in the formula that was fixed up now. Note To save memory, you can delete the huge explanation/comment at the program beginning before the first instruction DROP.