The Museum of HP Calculators The Museum of HP Calculators
This program is Copyright © 1976 by Hewlett-Packard and is used here by permission. This program was originally published in the HP-67 Stat Pac 1.
This program is supplied without representation or warranty of any kind. Hewlett-Packard Company and The Museum of HP Calculators therefore assume no responsibility and shall have no liability, consequential or otherwise, of any kind arising from the use of this program material or any part thereof.
| Normal And Inverse Normal Distribution 1 | |||||
| Shift | |||||
| Label | Start | ||||
| Key | A | B | C | D | E |
| Normal And Inverse Normal Distribution 2 | |||||
| Shift | |||||
| Label | P? | x→f(x) | x→Q(x) | Q(x)→x | |
| Key | A | B | C | D | E |
This program evaluates the standard normal density function f(x) and the normal integral Q(x) for given x. If Q is given, x can also be found. The standard normal distribution has mean 0 and standard deviation 1.
1. Standard normal density
f(x) = (1/√(2π))e(-x2/2)
2. Normal integral
Q(x) = (1/√(2π))∫(x..infinity)e(-t2/2)dt
Polynomial approximation is used to compute Q(x) for given x.
Define R = f(x) (b1t + b2t2 + b3t3 + b4t4 + b5t5) + E(X)
where |E(X)| < 7.5 x 10-8
t = 1/(1 + r|x|) , r = 0.2316419
b1 = .319381530, b2 =-.356563782
b3 = 1.781477937, b4 =-1.821255978
b5 = 1.330274429
R if x ≥ O
Then Q(x) = or
1-R if x < 0
3. Inverse normal
For a given Q > 0, x can be found such that
Q = (1/√(2π))*∫(x..infinity)e(-t2/2)dt.
The following rational approximation is used:
c0 + c1t + c2t2
Define y = t - --------------------- + E(Q)
1 + d1t + d2t2 + d3t3
where |E(Q)| < 4.5 x 10-4
√(ln(1/Q2)) if 0 < Q ≤ 0.5
t = or
√(ln(1/(1-Q)2)) if 0.5 < Q < 01
c0 = 2.515517 d1 = 1.432788
cl = 0.802853 d2 = 0.189269
c2 = 0.010328 d3 = 0.001308
y if 0 < Q ≤ 0.5
Then x = or
-y if 0.5 < Q < 01
Abramowitz and Stegun, Handbook of Mathematical Functions, National Bureau of Standards 1970.
| Step | Instructions | Input Data/Units | Keys | Output Data/Units |
| 1 | Load side 1 and side 2 of card 1 | |||
| 2 | Initialize | A | 0.00 | |
| 3 | Load side 1 and side 2 of card 2 | |||
| 4 | To set print mode* | B | 1.00 | |
| Optional: Step 5 | ||||
| 5 | Input x to compute f(x) | x | C | f(x) |
| 6 | Input x to compute Q(x) | x | D | Q(x) |
| For a new case of x, go to 5 or 6 | ||||
| 7 | Input Q(x) to compute x | Q(x) | E | x |
| For a new case of Q(x), go to 7 | ||||
| *Note: to clear print mode press--> | 0 STO A STO B |
Find f(x) and Q(x) for x = 1.18 and x = -2.28.
Keystrokes Outputs
Load side 1 and side 2 of card 1
A 0.00
Load side 1 and side 2 of card 2
B 1.00
1.18 C 1.18 ***
0.20 *** (f(1.18))
1.18 D 1.18 ***
0.12 *** (Q(1.18))
2.28 CHS D -2.28 ***
0.99 *** (Q(-2.28))
2.28 CHS C -2.28 ***
0.03 *** (f(-2.28))
Given Q = 0.12 and Q = 0.95, find x.
(If you have run through Example 1, then you can proceed; otherwise you have to load programs as described in Example 1).
Keystrokes Outputs
0.12 E 0.12 ***
1.18 *** (x)
.95 E 0.95 ***
-1.65 *** (x)
LINE KEYS 001 *LBL A 002 . 003 2 004 3 005 1 006 6 007 4 008 1 009 9 010 STO 3 011 1 012 . 013 3 014 3 015 0 016 2 017 7 018 4 019 4 020 2 Store constants for normal distribution. 021 9 022 STO 4 023 1 024 . 025 8 026 2 027 1 028 2 029 5 030 5 031 9 032 7 033 8 034 CHS 035 STO 5 036 1 037 . 038 7 039 8 040 1 041 4 042 7 043 7 044 9 045 3 046 7 047 STO 6 048 . 049 3 050 5 051 6 052 5 053 6 054 3 055 7 056 8 057 2 058 CHS 059 STO 7 060 . 061 3 062 1 063 9 064 3 065 8 066 1 067 5 068 3 069 STO 8 070 P⇔S 071 2 072 . 073 5 074 1 075 5 076 5 077 1 078 7 079 STO 1 080 . 081 8 082 0 083 2 084 8 085 5 086 3 087 STO 2 088 . 089 0 090 1 091 0 092 3 093 2 094 8 095 STO 3 Strore constants for inverse N.D. 096 1 097 . 098 4 099 3 100 2 101 7 102 8 103 8 104 STO 4 105 . 106 1 107 8 108 9 109 2 110 6 111 9 112 STO 5 113 . 114 0 115 0 116 1 117 3 118 0 119 8 120 STO 6 121 P⇔S 122 0 123 STO A 124 STO B 125 RTN
LINE KEYS 001 *LBL B 002 1 Store 1 in RA for print. 003 STO A 004 RTN 005 *LBL C 006 GSB 9 007 STO 1 008 ENTER↑ 009 × Input x and compute f(x) 010 2 011 ÷ 012 CHS 013 eX 014 π 015 2 016 × 017 √x 018 ÷ 019 STO 2 020 GSB 9 021 GSB 6 022 GSB 3 023 RCL 2 024 RTN 025 *LBL D 026 GSB 9 027 STO 1 028 GSB 5 029 GSB C 030 RCL 1 031 X<0? 032 GTO 1 Input x and calculate Q(x) 033 SF 0 034 *LBL c 035 1 036 RCL 1 037 RCL 3 038 × 039 + 040 1/X 041 ENTER↑ 042 ENTER↑ 043 ENTER↑ 044 RCL 4 045 × 046 RCL 5 047 + 048 × 049 RCL 6 050 + 051 × 052 RCL 7 053 + 054 × 055 RCL 8 056 + 057 × 058 RCL 2 059 × 060 F0? 061 GSB 9 062 F0? 063 GSB 6 064 RTN 065 *LBL 1 066 CF 0 067 RCL 1 068 CHS 069 STO 1 070 GSB c 071 1 072 X⇔Y 073 - 074 STO 9 075 GSB 9 076 GSB 6 077 RCL 9 078 RTN 079 *LBL E 080 GSB 9 081 X<0? Input Q(x) and calculate x. 082 GTO 0 083 1 084 X≤Y? 085 GTO 0 086 R↓ 087 . 088 5 089 X⇔Y 090 X>Y? 091 GSB 8 092 ENTER↑ 093 × 094 1/X 095 LN 096 √x 097 P⇔S 098 STO 7 099 RCL 3 100 × 101 RCL 2 102 + 103 RCL 7 104 × 105 RCL 1 106 + 107 RCL 7 108 RCL 6 109 × 110 RCL 5 111 + 112 RCL 7 113 × 114 RCL 4 115 + 116 RCL 7 117 × 118 1 119 + 120 ÷ 121 RCL 7 122 X⇔Y 123 - 124 P⇔S 125 F1? 126 CHS 127 GSB 9 128 GSB 6 129 CF 1 130 RTN 131 *LBL 8 For (1-Q) 132 SF 1 133 1 134 - 135 CHS 136 RTN 137 *LBL 9 Subroutine to print. 138 RCL A 139 X>0? 140 GSB 7 141 R↓ 142 RTN 143 *LBL 7 144 R↓ 145 PRTX 146 R↑ 147 RTN 148 *LBL 6 Subroutine for space. 149 RCL A 150 X>0? 151 PRT SPC 152 R↓ 153 RTN 154 *LBL 5 155 RCL A Clear RA for calculating Q(x). 156 X>0? 157 GSB 4 158 R↓ 159 RTN 160 *LBL 4 161 STO B 162 CLX 163 STO A 164 RTN 165 *LBL 3 166 RCL B 167 X>0? 168 GSB 2 169 RTN 170 *LBL 2 Restore 1 to RA, 171 1 172 STO A 173 RTN
R1 x R2 f(x) R3 r R4 b5 R5 b4 R6 b3 R7 b2 R8 b1 R9 Used S1 C0 S2 C1 S3 C2 S4 d1 S5 d2 S6 d3 S7 t A 1 or 0 B 1 or 0
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