値 ± 不確定さ（誤差）

ᐁ㥺䝋䞀䝃䛴ฌ⌦䛮ㄏᕣ䛴ぜ✒䜈䜐
- 㞹Ẵ㏳ಘኬᏕ䚭ὰ஬ྚⶮ -
ῼᏽೋ䛱䛵䚮ᚪ䛠୘☔ᏽ䛛䟺ㄏᕣ䟻䛒ఔ䛌䚯
ῼᏽ⤎ᯕ
䚭䚭䚭䚭ೋ䚭±䚭୘☔ᏽ䛛䟺ㄏᕣ䟻
䠃䠀ㄏᕣ䛴⛸㢦
䠄䠀᭯ຝᩐᏊ
䠅䠀ᖲᆍ䛮ᵾ‵೩ᕣ
䠆䠀᭩ᑚ䠄஋Ἢ
䠇䠀ㄏᕣ䛴ఎᦑ
䛙䛴㈠ᩩ䛵䚮ᇱ♇⛁Ꮥᐁ㥺A-㞹Ẵ㏳
ಘኬᏕ-䛴䝊䜱䜽䝌䟺௿᮶ᨼ㞕௙ ⥽䟻
䜘ᇱ䛱䛝䛬ష⿿䛝䜄䛝䛥䚯
䠃䠀ㄏᕣ䛴⛸㢦䟺䠃䟻
1.1 ഃ↓ㄏᕣ (random errors)
䚭ྜྷୌ᮪௲䛭ị䜇䛥ῼᏽೋ䛭䜈䛶䜏䛪䛕
䚭䚭䚭โᚒ୘⬗䛰ንິ䚭䟺✭Ẵ䛴᥺䜏䛔䟻
䚭䚭䚭⤣ゝⓏ⌟㇗䚭䚭䟺ᰶቪን䟻
䚭䚭䚭᥺䜏䛔䚭䚭䚭䚭䚭䟺⇍㞟㡚䟻
䚭䚭䚭䚭䚭䚭┷䛴ೋ䛴䜄䜕䜐䛱ฦᕱ
䚭䚭䚭䚭䚭䚭ῼᏽᅂᩐ䜘ቌ䜊䛟䛙䛮䛭ᙫ㡢䜘㍇΅
1.2 ⣌⤣ㄏᕣ (systematic errors)
䚭≁ᏽ䛴ῼᏽᶭჹ䜊ῼᏽ᪁Ἢ䛭⏍䛞䜑
䚭䚭䚭ゝჹ䛴පᕣ䚮≤䛊
䚭䚭䚭ῼᏽ⩽䛴ㄖ䜅ཱི䜐䛴⒯
䚭䚭䚭よᯊ䛱⏕䛊䜑௫ᏽ䜊㎾జ
䚭䚭䚭䚭䚭䚭ᖲᆍೋ䛒┷䛴ೋ䛑䜏䛠䜒䜑
䚭䚭䚭䚭䚭䚭⿭ḿ䛒ྊ⬗
1.3 ಴ெㄏᕣ
䚭䛣䛴௙䚮ㅎ䚱
1
䠃䠀ㄏᕣ䛴⛸㢦䟺䠄䟻
1.4 ⤧ᑊㄏᕣ䛮┞ᑊㄏᕣ
䚭⤧ᑊㄏᕣ䚭䚭䚭䟺஢᝷䛛䜒䜑䟻┷䛴ೋ䛮ῼᏽೋ䛴ᕣ
䚭┞ᑊㄏᕣ䚭䚭䚭⤧ᑊㄏᕣ䜘┷䛴ೋ䟺ᐁ㝷䛱䛵ῼᏽೋ䟻䛭
䚭䚭䚭䚭䚭䚭䚭䚭䚭䚭๪䛩䛥ೋ
䚭䚭䚭䚭䚭䚭䚭䚭䚭䚭“⢥ᗐ”䛮䜈䛊䛌
ౚ㢗
䚭᮶ா䞀ኬ㜨㛣䛴㊝㞫䜘1m䛴༟న䜄䛭ῼ䜑䛴䛮䚮ྦ䛴㌗
䚭㛏䜘mm䛴༟న䜄䛭ῼ䜑䛴䛮䛭䛵䚮䛯䛧䜏䛒ῼᏽ䛴⢥ᗐ
䚭䛒㧏䛊䛑䠑
2䠀᭯ຝᩐᏊ䟺䠃䟻
2.1 ཚᐠ䛰ᩐೋ䛮ㄏᕣ䟺୘☔䛑䛛䟻䜘ྱ䜆ᩐೋ
䚭R = 2 r
R:䚭ළ䛴├ᙼ䚮䚭r:䚭ළ䛴༖ᙼ
䚭䚭䚭䚭2 䛵ཚᐠ䛰ᩐೋ
䚭༐ළᮈ䛴ᶋᕮ䜘∸ᕣ䛝䛭ῼ䜑
149.2 mm
䚭ῼᏽㄏᕣ䛵᭩ᑚ┘┊䜐䛴1/2䛮䛟䜑䛮䚮䚭(149.2 ± 0.5) mm
䚭᭯ຝᩐᏊ䠆᰾䚭䚭᭩ᚃ䛴2䛵୘☔䛑䛛䜘ఔ䛌䛒᭯ຝᩐᏊ䛱ථ䜒䜑䚯
2.2 ᭯ຝᩐᏊ䛴⾪䛝᪁䟺⣑ᮨ஥䟻
䚭᰾䜘⾪䛟䛥䜇䛴”0”䛵᭯ຝᩐᏊ䛮䛝䛰䛊䚯
ౚ䟻䚭0.00167
1.67 × 10-3䚭䚭䚭䚭䚭᭯ຝᩐᏊ3᰾
䚭䚭䚭6400䚭䚭䚭䚭䚭䚭䚭6.4 × 103䚭䚭䚭䚭䚭䚭䚭᭯ຝᩐᏊ2᰾
䚭䚭䚭0.012300䚭䚭䚭䚭1.2300 × 10-2䚭䚭䚭᭯ຝᩐᏊ5᰾
䚭䚭䚭100.䚭䚭䚭䚭䚭䚭䚭䚭䚭䚭䚭䚭䚭䚭䚭䚭䚭䚭᭯ຝᩐᏊ3᰾
2
2䠀᭯ຝᩐᏊ䟺䠄䟻
2.3 ✒䛮ၛ
䚭6.2834 × 57.3
(᭯ຝᩐᏊ䠇᰾䟻×䟺᭯ຝᩐᏊ䠅᰾䟻
䚭䚭䚭䚭䚭䚭䚭䚭䚭䚭䚭䚭䚭䚭䚭䚭䚭䚭᭩ᚃ䛴᰾䛵୘☔䛑䛛䜘ྱ䜆
6. 2 8 3 4
䚭䚭䚭䚭6. 2 8 3
×
5 7. 3
×
188502
5 7. 3
䚭18849
439838
43981
314170
31415
3 6 0. 0 3 8 8 2
3 6 0. 0 1 5 9 䚭䚭䚭䚭䚭
3.60 ×102
᭯ຝᩐᏊ䠅᰾
✒䛮ၛ䛴᭯ຝ᰾ᩐ䛵䚮᭯ຝ᰾ᩐ䛴ᑚ䛛䛊ᅄᏄ䛴᭯ຝ᰾ᩐ䛱䛰䜑䚯
✒䛮ၛ䛴┞ᑊㄏᕣ䛵䚮┞ᑊㄏᕣ䛴ᝇ䛊ᅄᏄ䛭Ử䜄䜑䚯
⢥ᗐ䛴఩䛊᪁䛱ྙ䜕䛡䛬ゝ⟤䛝䛬䜎䛊䚯䟺䠃᰾䛵ఴฦ䛱䟻
2䠀᭯ຝᩐᏊ䟺䠅䟻
2.4 ࿰䛮ᕣ
0. 0 5 5 4
䚭
0. 0 5
0. 8 1 5
0. 8 2
1. 0 0
1. 0 0
0. 1 0 7
0. 1 1
+ 3 1 7. 8
3 1 9. 7 7 7 4
319.8
䚭
+ 3 1 7. 8
3 1 9. 7 8
᭯ຝᩐᏊ4᰾
࿰䛮ᕣ䛴⤧ᑊㄏᕣ䛵䚮⤧ᑊㄏᕣ䛴ᝇ䛊さ⣪䛭Ử䜄䜑䚯
䛣䜒䛱ྙ䜕䛡䛬ゝ⟤䛝䛬䜎䛊䚯䟺䠃᰾䛵ఴฦ䛱䟻
1. 6 7 4 9
- 1. 6 7 2 6
ᕣ䛴ሔྙ䚭䚭䚭᰾ⴘ䛧䛟䜑䛙䛮䛒䛈䜑䚯
0. 0 0 2 3
3
3䠀ᖲᆍೋ䛮ᵾ‵೩ᕣ䚭䟺䠃䟻
Nᅂ䛴ῼᏽ䜘⾔䛌䚯䚭ῼᏽೋ x1, x2, x3, 䝿䝿䝿䝿䝿䝿, xN
3.1 ᖲᆍೋ
x=
x1 + x 2 + L + x N 1 n
= " xi
N
N i=1
3.2 ᖲᆍೋ䛴䜄䜕䜐䛴ฦᕱ䟺䛶䜏䛪䛓䟻䛴ᕮ䟺䠃䟻
N
!
"=
1
$ (x i # x)2
N i=1
(ᖲᆍ䠄஋ㄏᕣ)1/2
= ᵾ‵೩ᕣ
3.3 ᖲᆍೋ䛴䜄䜕䜐䛴ฦᕱ䟺䛶䜏䛪䛓䟻䛴ᕮ䟺䠄䟻
!
"x =
1 N
$ xi # x
N i=1
|ㄏᕣ|䛴ᖲᆍ
!
3䠀ᖲᆍೋ䛮ᵾ‵೩ᕣ 䟺䠄䟻
3.4 䜰䜪䜽ฦᕱ䚭䟺ഃ↓ㄏᕣ䜘ᑊ㇗䛮䛟䜑䟻
䚭䚭䚭䚭䚭ඖฦ䛰⢥ᗐ䜘䜈䛩䛬䚮ඖฦ䛱ኣᩐᅂ䛴ῼᏽ䜘䛟䜑䛮
䚭䚭䚭䚭䚭ῼᏽೋ䚭x䚭䛵䚭┷䛴ೋ䚭X䚭䛴䜄䜕䜐䛱䜰䜪䜽ฦᕱ䜘䛟䜑䚯
2
+
- %x$ X( /
1
p(x) =
exp,$'
* 0
2"# 0
. & 2# 0 ) 1
$
!
#
"#
x=
%=
!
p(x)dx =1
$
#
"#
$
x p(x)dx = X
#
"#
(x " X) 2 p(x)dx = % 0
X - σ 0 ~ X + σ0
68.3 %
X -2σ0 ~ X +2σ 0
95.5 %
X -3σ0 ~ X +3σ 0
99.7 %
X-3σ0 X-2σ0 X-σ0
X
X+σ0 X+2σ0 X+3σ0
x
4
4䠀᭩ᑚ䠄஋Ἢ䚭䟺䠃䟻
┷䛴ೋ X䛵 ᮅ▩䚮
Nᅂ䛴ῼᏽೋ䚭x1, x2, x3, 䝿䝿䝿䝿䝿䝿, xN䚭䛑䜏䚭X 䜘᥆ᏽ
┷䛴ೋ䜘X䛮䛝䛥䛮䛓䚮䚭䠃ᅂ┘䛴ῼᏽೋ䛒x1䛭䛈䜑☔⋙
p(x1, X)
䚭䚭䚭䚭䚭䚭䚭䚭䚭䚭䚭䚭䚭䚭䚭䠄ᅂ┘䛴ῼᏽೋ䛒x2䛭䛈䜑☔⋙
p(x 2 , X)
䠃ᅂ┘,䠄ᅂ┘䚮 䝿䝿䝿䝿䝿䝿, Nᅂ┘䛒
x1, x2, x3, 䝿䝿䝿䝿䝿䝿, xN䚭䛭䛈䜑☔⋙
!
P(x1!
,x 2 ,L, x N , X)
䚭䚭䚭䚭䚭䚭䚭䚭䚭䚭䚭䚭䚭䜘᭩ኬ䛱䛟䜑
X䜘
P(x1,x 2 ,L, x N , X)
᭩Ⰳ᥆ᏽೋ䚭䛮䛟䜑䚯
= p(x1, X) " p(x 2 , X) "L" p(x N , X)
N
-/ 1 N !
1
# ( x i " X )2
% 1 (N
2 / X 䛵䠄஋ṟᕣ䛴࿰䚭䚭䚭䚭䚭䚭䚭䚭䚭䚭䚭䜘
='
i=1
* exp.+ 2 , ( x i + X ) 2
/0 2$ i=1
/3 ᭩ᑚ䛱䛟䜑䚯
& 2#$ )
!
!
4䠀᭩ᑚ䠄஋Ἢ䚭䟺䠄䟻
N
S(x1,x 2 ,L, x N , X )=# (x i " X ) 2
i=1
䠄஋ṟᕣ䛴࿰䜘᭩ᑛ䛱䛟䜑 X 䜘 X’ 䛮䛟䜑
X '= x
┷䛴ೋ䛱ᑊ䛟䜑᭩Ⰳ᥆ᏽೋ䛵䚭䚭䚭䚭䚭䚭䛭䛈䜑䚯
%N
(
$S N
=# 2(x i " X) = 2''# x i " NX **
$X i=1
& i=1
)
% $S (
' * =0 +
& $X ) X '
X '=
1 N
# xi = x
N i=1
!
!
xi 䛵┷䛴ೋ䟺ᮅ▩䟻 X 䛴䜄䜕䜐䛱ฦᕱ
䛣䛴ᵾ‵೩ᕣ䚭σ䚭䟺Ἰ䠃䟻
X’ 䜈┷䛴ೋ䟺ᮅ▩䟻 X 䛴䜄䜕䜐䛱ฦᕱ
䛣䛴ᵾ‵೩ᕣ䚭σX’
1
ῼᏽᅂᩐN䜘ቌ䜊䛟䛮 X’ 䛴ฦᕱᕮ䛵
N
䛱΅ᑛ䚭䟺Ἰ䠄䟻
N
"=
N
1
1
$ (x i # X )2 = N #1 $ (x i # x)2
N i=1
i=1
2
,
. & X'#X ) 0
.
1
p(X', X ) =
exp-#(
+ 1
2%" X '
. ' 2" X ' * 2
.
/
"X' =
"
N
!
!
5
5䠀ㄏᕣ䛴ఎᦑ䚭䟺䠃䟻
y 䛵 x 䛴㛭ᩐ
y= f (x)
x 䛴ῼᏽೋ
x ± "x
y 䛴ೋ
y ± "y
y= f (x)
!
"y= f (x + "x)# f (x)
$ df '
= & ) "x
% dx ( x= x
!
ළ䛴༖ᙼ䜘ῼᏽ䛝䛥䛮䛙䜓䚮r = (5.00 ± 0.05) cm䛭䛈䛩䛥䚯ළ䛴㟻✒䜘ị䜇䜎䚯
!
S = 3.14 "25 = 78.5
᭯ຝᩐᏊ䠃᰾
#S = 2 " 3.14 " 5.00 " 0.05 = 1.57 $ 2
S="r2
#S =
dS
#r = 2" r#r
dr r
S = (79 ± 2)cm2
!
!
5䠀ㄏᕣ䛴ఎᦑ䚭䟺䠄䟻
z 䛵 x 䛮 y 䛴㛭ᩐ
z= f (x,y)
x 䛮 y 䛴ῼᏽೋ
x ± "x, y ± "y
z 䛴ೋ
z ± "z
z= f (x, y)
!
!
Δx, Δy䛑䜏⏍䛞䜑ㄏᕣ
!
ྙ䜕䛡䛬
"z f (x + "x, y)# f (x, y) % $f (
=
=' *
& $x ) x,y
"x
"x
೩᚜ฦ
"z f (x, y + "y)# f (x, y) % $f (
=
=' *
"y
"y
& $y ) x,y
$ #f '
$ #f '
("z) x = & ) "x, ("z) y = & ) "y
% #x ( x,y
% #y ( x,y
"z =
*$ #f '
,& ) "x/ 2+
+% #x ( x,y .
*$ #f '
,& ) "y/ 2
,+% #y ( x,y /.
!
6
5䠀ㄏᕣ䛴ఎᦑ䚭䟺䠅䟻
㛏᪁ᙟ䛴䠄㎰䜘ῼᏽ䛝䛬䚮x = (15.9 ± 0.2) cm, y = (7.6 ± 0.1) cm䜘ᚋ䛥䚯
㟻✒ S 䛮䛣䛴ㄏᕣ ΔS 䜘ị䜇䜎䚯
S=xy =15.9 " 7.6 = 120.84
#S
#S
= y,
=x
#x
#y
$S = (y$x) 2 + (x$y) 2
= (7.6 " 0.2) 2 + (15.9 " 0.1) 2
᭯ຝᩐᏊ䠃᰾
ว䜐୕䛘
= 2.3 + 2.5 = 4.8 = 2.2 % 3
S ± $S = (121± 3) cm2
"S = y"x + x"y = 3.1 # 4
!
S ± "S = (121± 4) cm2
䛮䛟䜑ሔྙ䜈䛈䜑䚯
!
䟺Ἰ䠃䟻σ䛴ぜ✒䜈䜐
1 N
$ (x i # X)2
N i=1
"=
N
N
# (x
!
i
䜘ゝ⟤䛝䛥䛊䛒䚮X䛵ᮅ▩
[
" X ) 2 = # (x i " x) + (x " X)
i=1
i=1
N
]
2
N
(1)
= # (x i " x) + 2(x " X)# (x i " x) + N(x " X) 2
2
i=1
i=1
ᖲᆍ䛴ᏽ⩇䜎䜐䚮➠䠄㡧 = 0
➠䠅㡧䛴ゝ⟤
!
2
$1 N
'2 1 * N
N(x " X) 2 = N & # x i " X ) = ,# (x i " X)/
.
% N i=1
( N + i=1
N
N N
*
1
1 N
= ,# (x i " X) 2 + # # (x i " X)(x j " X )/ 1 # (x i " X) 2
N ,+ i=1
/. N i=1
i=1 j0i
(2)
(2)ᘟ䠄⾔┘୯㛣ᘟ䛴➠䠄㡧䜘┤␆
N
N
(1), (2)䜎䜐
1 N
# (x i " X)2 = # (x i " x)2 + N # (x i " X)2
!
i=1
i=1
i=1
N
$
# (x
i=1
%=
i
" X) 2 =
N N
# (x i " x)2
N "1 i=1
1 N
# (x i " x)2
N "1 i=1
!
7
䟺Ἰ䠄䟻
x1, x 2 䛒䜰䜪䜽ฦᕱ
䜘䛝䛬䛊䜑䛮䛓䚮y = x 1 + x2 䛴ฦᕱ f (y)䜘⩻䛎䜑䠀
x2 = y - x1 䛭䛈䜑䛴䛭
f (y) =
$
#
"#
p(x1, X) p(y " x1, X)dx1
' 1
*
exp(" 2 [(x1 " X) 2 + (y " x1 " X) 2 ]+dx1
) 2&
,
20
- 3
6
y
/ 5 x1 " 8 2
- 1
1
20 #
2 2dx
/
=
exp
"
(y
"
2X)
exp
"
5
8
1
/. 4& 2
21 $"#
2%& 2
/ 5 & 82
4
7
/.
21
20
3 y " 2X 6
1
=
exp/"5
82
2% 2&
/. 4 2( 2& ) 7 21
=
!
1
2%& 2
$
#
"#
y 䛴ᖲᆍ䛵 2X䚮ᵾ‵೩ᕣ䛵
y
x = 䛴ᖲᆍ䛵 X䚮ᵾ‵೩ᕣ䛵
2
!
!
2"
䟺2σ䛭䛵䛰䛊䛙䛮䛱Ἰណ䟻
2"
"
=
2
2
!
8