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1) "+,%9*9 "# #0%& ) (%94"2 "#)+! +"+-
х
2
( )
x x
$"*+," ' 16 +";
)!*"+%39 +4 ";$"*+, х #0%& ) -
2)
(%94"2
= 1/nHхi , (35)
14 хi – #0%& i-1 "#$"3;
n –
*"+% (%94"2;
3) )!*"+%39 7,0 +41 ,)4"*+, 1 ,% "3
#0%& "#$"2 ' ; $0%
σ = 1
1
1
n
−
∑
−
i
; (36)
4) '4%39 %"*" $%&!5 (%94"2 (10(!5
' 16 +2)
| хi – х| A 3σ. (37)
+%" )+) )!' %3+3, #0%& хi (+!)9, х " σ )!*"+%39 + );
5) ++*"!)9 7,0 +41 ,)4"*+, 1 ,% -
"3 +41 ";$"*+, 1 #*"3 ' ; $0%:
σ
= σ/n; (38)
х
6) '4%39 '"4%8 +& #0%& ) "#$"2 -
$%& $0 +'4%"9. " *"+% #0%& ) "#$"2
n > 50
4%3 ' )," = 2 '"4%8 +" "+' % ,""2
"+ (χ2). +%" n < 50, "+' % + +) 2 ,""2,
, $40$!2 8.207-76;
'4%39 4 )"%&! 1"7! ε +%0*2 2 ' -
7)
16 +" #0%& "#$"2 ' ; $0%
ε = tq·σх , (39)
14 tq – , =;;"7", '4%3$!2 ' (%"7$ +'4-
%"3 &94 ' #4 2 4 )"%& 2 ) 3 +" р " *"+%0 (%94"2 n;
71
8) +%" 1"' # $%& $ #, +'4%"3 ) 3-
2 2
( )
х x
-
+" #0%& "#$"3 ,% 3+3, ' )"40 1"+ 1$$
)!4)"1+3 1"' # +"$$"* +" #, +'4%"3 ) 3 +" #0%& "#$"3 4 2 "# +4!5 '' ,+"$"09:"5 ;0,7"2, '")4!5 ) (%. 3, 4 )"%&! 1"7! ' 16 +" #0%& "#$"3 '4%39 ' ; $0%
ε = α⋅σx , (40)
14 α – % 1 , =;;"7" tq, (+3 "# (%. 3.
+%" ' )"40 1"+ 1$$! % '4' % 8"&, * #,
+'4%"3 ) 3 +" #0%& "#$"3 3)%3+3 +"$$"*!$ "%" +%" ' 45 4" " 4 "# '' ,+"$"09:"5 ;0,7"2, 0+)%")9 '4%!, # , !$" #*"
"#$3$ 2 )%"*"! $ 8 ,#&+3 " '" ,, $ #,
+'4%"3 ) 3 +" #0%& "#$"3.
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АППРОКСИМИРУЮЩИЕ ФУНКЦИИ ЗАКОНА
РАСПРЕДЕЛЕНИЯ ВЕРОЯТНОСТИ РЕЗУЛЬТАТА
ИЗМЕРЕНИЯ
7, +41 ,)4"*+, 1 ,% "3 +41
";$"*+, 1 #*"3 '4%3+3 ' ; $0%
σ
= 1/n
⋅
x
Σ − ; (41)
i
72
2
i
θ
∑
2
x i
= + θ θ
∑
2 2
i x
σ σ
9) '4%39 1"7! I "+,%9* 2 +"+$"*+, 2
' 16 +". +%" "#)+ , * ' 16 +& #0%& "#$"3 '4%3+3 34 $ + +)%39:"5 "+,%9*!5 +"+$"*+,"5 ' 16 +2, ,843 "# , !5 "$ +) " 4 )"%&! 1"7!, '" "#)+!5 #, 5 +'4%"3 "5 1"7! +0$$ 2 ' 16 +" 5 43 ' ; $0%
m
I = k
⋅
, (42)
1
i=
14 m – *"+% "+,%9*!5 +"+$"*+,"5 + +)%39:"5
' 16 +2 #0%& "#$"3;
k – , =;;"7", '""$$!2 )!$ 1,1 '" 4 )"-
%& 2 ) 3 +" р = 0,95;
10) '4%39 + 6" I/σx. +%" = + 6"
$&6 0,8, "+,%9*!$" ' 16 +3$" '(19
" ) ,*+) 1"7! ' 16 +" #0%& "#$"2 '""$9 E = ε.
+%" I/σx > 8, '(19 +%0*2 2 ' 16 +&9 "
+*"9, * E = I.
+%" 0,8 < I/σx < 8, '" '4%"" 1"7 ' 16 +" ∆
+%40 0*"!)& " +%0*209, " +"+$"*+,09 + +)%39:";
11) '4%39 1"7! ' 16 +" #0%& "#$-
"2 ' ; $0%
∆ = ± k
σ
, (43)
J
14
m
( )/ 1/3
k
ε σ
m
1/ 3
= θ +
Σ
( )
∑
1
i
=
, (44)
1
i
=
; (45)
73
12) '4+)%39 #0%& "#$"3 " ' 16 +" 4%3
(
)
+ ∆
max min
x x
l
−
+%0*3 +"$$"*!5 4 )"%&!5 1"7 ) ; $
x
'" 4 )"%& 2 ) 3 +" р = … " *"+% (%94"2
n = ... .
" ( %&6 $ *"+% (%94"2 #'"+& #0%& ) "#$"2 + )"+3 +%"6, $ 1 $ #4, 2 " $% 1%34 2. %3
'"4"3 "$ ( %&62 , $', +" " 1%34 +" + "+3 ,
#!)$!2 статистический ряд. +& 4"'# (%94"2
#*"2 х 4%3 ")%! "%" #34!, .. ' ) 4"+3
#4%" 34 =,+'"$%&!5 4!5 "$&61
4 "( %&61 x
x
min
l ")% ), 4% ' 4+*"!)9
max
, %"*+) #*"2 ml, '"5 43:"5+3 ,84!2 l-2 ")%.
*"+% 4%3 (: *"+% (%94"2 n " 5 43 *+ 0, + )+)09:09 4 $0 ")%0:
Pl = m
/ n. (46)
l
0$$ *+ )+5 ")% ) 4 %8 (!& ) 4""7.
"+% ")% ), , ! +%40 +10''" )& +"+"*+,"2 $"%, 4 %8 (!& +%"6, $ ( %&6"$. %" ")% 10''" )"3 4 %8 (!& ( %&6 ' 16 +"
,01%"3 '" #'"+" (%94"3.
%"*+) ")% ) $ 8 (!& +%409::
%"*+) "#$"2 n %"*+) ")% ) l
40–100 7–9
100–500 8–12
500–1000 10–16
"+% ")% ) $ 8 ' 4+*"& ' ; $0% 4-
8++:
l = 1 + 3,32 lg n. (47)
%" ")% )!*"+%3+3 ' ; $0%
h
= . (48)
74
"+"*+,"2 34 *+ ; $%3+3 1;"*+," ) )"4
( )
i x i
x M P
, #!)$ 2 гистограммы, , 3 + "+3 +%409:"$
(# $. +" (+7"++ ,%4!)9 ")%!, ,84 $ "#
")% ) ,, "5 + )"" + "+3 '3$ 01 %&",, '% :4&
, 1 ) *+ 4 1 ")%. %3 ' + "3 1"+ 1$$! *+ 0 ,84 1 ")% 08 #4%"& 1
4%"0 " ' %0* *"+% )#3& ) ,*+) )!+ ! '3$ 01 %&",. +%0* )!5 ' 4%" ")% ) )!+ ! '3$ 01 %&", ) ' ' 7" %&! + )+)09:"$ *+ $. #
+' + ( ' + "3 1"+ 1$$! +%40, * ' %3 '% :4& ) 4""7. *)"4 , '" 0)%"*"" *"+% '! )
$ 8 )!("& )+ ( % " ( % $%," ")%!; '" = $
1"+ 1$$ (04 )+ ( % '"(%"8&+3 , , 2 ,") 2,
1"*")9:2 '% :4&, )09 4""7. ,")3 '4+)%3 + ( 2 1;", '% +" +'4%"3 )%"*"! х, "
' )"40 $ 8 '"(%"8 +04"& #, +'4%"3
' %0*!5 "#$"2.
" ( %&6 $ , %"*+) '! ) )!*"+%" 5,"+", (%94"3 + )"+3 *#$ 1 $ #4,"$, " $ 8
'"$"& +%409:"2 '"$: ) +' %&# )&+3 $" 8 #34$", , ! (!% +,%++";"7" ) +"+"*+,"2 $"% 4%3 ' + "3 +"+"*+, 1 34 "%" 1"+ 1$$!,
" +*"& '"(%"8 #*" +%0*2 2 )%"*"! ) ,84 $
#34 ' + 3!$ " )!$ +4$0 #*"9, , )!+0' ) %" «'4+)"%3» ")%. 14 *"+% )! 5,"+"," (040 )!8&+3 +%409:"$" '"(%"8!$"
; $0%$":
х
= Mx = ΣхiP
i
σ
= −
∑
i
l
1
=
, (49)
i
, (50)
14 хi – «'4+)"%&» li-1 ")%;
P
– *+ li-1 ")%;
i
l – *"+% ")% ).
75
+% 1 , ,, ' %0*! *"+% )! 5,"+"," "#$-
( ) / ,
i x xi
M M M
"2, '5 43 , +%409:$0 ='0 $$"*+, 2 (( ," – ' 740 ' )," 1"' #! )"4 ;0,7"" +'4%"3. %3 ' )," + 1%+"3 $840 '4' %1$!$ $%&!$ " =$'""*+,"$ +'4%"$ '"$3+3 критерий
Пирсона (
2
χχχχ
).
" "+' %&# )"" ,""3 "+ ) ,*+) $! +5 84"3 "*+, 1 " =$'""*+, 1 +'4%"2 '"$3+3 *"+% Н, , )!*"+%3+3 ' ')"%0
H =
∑
i
=
l
1
2
−
(51)
14 l – *"+% ")% ) #(""3 (+$. ' + " 1"+ -
1$$!);
Mxi – $$"*+, 8"4" #*"3 "#$3$ 2 )%"-
*"! ) li-$ ")%.
!*"+%3 ' = $0 ')"%0 $ +5 84"3 Н +&
+%0*23 )%"*", "$9:3
2
χ
+'4%" + *"+% $ +'2 +) ( 4! r = l – S – 1. 4+& S – *"+% )!*"+%3$!5 '$ ) "*+, 1 +'4%"3.
,%9*"%&!$ =' $ ' 740! ' )," 1"' #! '
,""9 "+ 3)%3+3 +)" )!*"+% 2 $! +5 84"3 , ,)"%9
+'3$" +) ( 4! –
2
χ
+'4%"3 ' 0 )9 α = 1 – q + k
2
χ
. 4+& q – 0 )& #*"$ +" () -
α
,r
3 +& 6"(,") – '4%3 $,+"$%& #*" $!
+5 84"3, , : $ 8 +*"& +%0*2!$. (!*
#*" q '""$+3 )!$ 0,05–0,1. +%" ) #0%&
+)"3 ,#!)+3, * $ +5 84"3 Н ') +5 4"
,)"%3
2
χ
, 4%+3 #,%9*", * + )"2 -
α
,r
)1& '"309 1"' #0.
),0 + 1%+"3 $840 $%&!$ #, $ +'4%"3 " =,+'"$%&!$" 4!$" '" *"+% (%94"2
10 < n < 50
, $40+3 )+" + "+' %&# )"$ составного
критерия.
76
Критерий 1. 4!$ (%94"2
, , ...,
n
x x x
2
1/ ( )
n x x
∑
2
α
1 2
)!*"+%3-
9 )%"*"0 d ' ; $0%
d = Σ|xi – x| / n
∗
14
σ
=
−
i
– +$:3 7, +41 ,)4-
"*+, 1 ,% "3.
"' # + 1%+0+3 + 4!$" (%94"2, +%" d
< d < d
, 14 d
q/2
1–q/2
" d
– ' 7! *," +'4%"3 +-
q/2
∗
σ
, (52)
1–q/2
<
"+",", , ! 5 43 "# (%"7 ' n, q/2 " 1–q/2; q – )!("$!2 # 0 )& #*"$ +" ,""3.
Критерий 2. "+% (%94"2, ,% "3 , !5
+41 ";$"*+, 1 #*"3 ')!69 )%"*"0 σZ
α
/2
4 %8 (!& ( %&6 4 1 '" n ≤ 20 " ( % 4)05, +%"
,
20 < n < 50. 4+& Z
– )533 100⋅
α
/2
-' 73 *,
$" ) 2 ;0,7"" '%+; α – 4 )"%&3 ) 3 +&, '4%3$3 "# (%"7 ' n " ' )!( $0 0 )9
#*"$ +" ,""3 q.
)& #*"$ +" + +) 1 ,""3:
q = q1 + q2 . (53)
Обработка косвенных измерений
+)! "#$"3 – = "#$"3, '" , !5 "+, $ #*" y 5 43 + ) "#)+ 2 #)"+"$ +"
y = f(х1, х2, …, хn),
14 х1, х2, …, хn – #*"3, ' %0*! '" '3$!5 "#$"35.
)"40 ;0,7" %& 2 #)"+"$ +" , +)! "#$"3 4%3+3 %"2! " %"2!.
" %"2 2 #)"+"$ +" $840 10$$", , 14 "#)+! ;0,7"" " '4%&! ' 16 +" "#$"2 10$ ), 4%3 '4%"3 '4%& 2 ' 16 +" "#$"3
"+, $ 2 )%"*"! (;0,7"") "+' %�+3 #)"+"$ +&
77
i
x
∆
∂
∂
∂
∂
Ey =
∑
i
k
y
∂
, (54)
x
∂
i
1
=
14 k – *"+% "#$3$!5 10$ );
∂y,∂x – *+! ' "#) 4! "%" , =;;"7"! )%"3"3
10$ ) "+, $09 )%"*"0;
Exi – '4%&! ' 16 +" "#$"2 + )+)09-
:"5 10$ ).
" %"2 2 #)"+"$ +" $840 10$$" "+' %�+3 метод линеаризации. + + " ) $, * %"23
;0,7"3, +)3#!)9:3 "#$3$09 )%"*"0 + 10$$",
#%1+3 ) 34 2% . ++$ "$ +%0*2, , 14 0)"
"#$"2 '4+)%3 + ( 2 ;0,7"9 4)05 10$ ):
y = f(х1, х2).
+%" 7," #*"2 10$ ) х1 " (+ %9! ' 16 +" ="5 7 , ∆х1, ∆х2 "#)+!, ) #*" $ 8
(!& '4+)% 34 $ 2% :
f
+∆
x
1
∂
x
2
∂
f
+∆∆
2
∂
x
2
+
Rx
12
+
n
+∆
x
2
2
2
+
∆
x
2
, (55)
14 R
~
1
+
!2
– + *!2 *% #% 8"3 ;0,7"" ) 34.
n+1
~
;~(
2
∂
f
2
+∆
x
∂
1
2
x
1
1
f
∂
+
!
x
n
∂
1
xxfxxfy
2
∂
f
2
∂∂
x
+∆
1
f
+==
);()
2121
∂
x
1
xx
xx
21
f
∂
x
∂
2
21
n
∆
%3 6"3 #4*" (!* 1"*")9+3 ')!$" *%$" #% 8"3, 4%3 *1 4 '"3& n = 1:
~
14 R2 – + *!2 *% #% 8"3:
xxfy +∆
f
);(
+= , (56)
21
x
x
∂
1
78
f
+∆
1
x
∂
2
Rx
22
2
1 2 1 2
= ( ; )+ .
y f x x x x
∆ + ∆
∂
∂
∂
∂
1
f
R ∆
∂
=
2
2
∂
+%" R2 < 0,8
2
x
1
2
x
1
f
∂
x
∂
1
2
f
∂
+∆
2
xx
∂∂
21
∂
2
+
~
x
∂
xx
21
f
2
(#4+&
σσ
~
xx
21
2
2
1
f
∂
+∆∆
2
∂
σ
2
x
. (57)
2
2
x
2
2
2
,
σ
~
x
1
– 7-
~
x
2
," +42 ,)4"*+, 2 ' 16 +" +41 ";$"*+, 1 ), + *!$ *% $ R2 '(19, .. $ 4 %""#7"" 4 '0+"$.
,"$ (# $, ) = $ +%0* 7, #0%& , +) 1
"#$"3 $ 8 (!& #'"+ ) )"4
∂ ∂
ɶ
f f
x x
∂ ∂
1 2
(58)
(+ %93 ' 16 +& , +) 1 "#$"3
~
=−=∆
xxfyyy ∆
f
=
∂
x
1
f
+);(
21
∂
x
1
f
+∆
x
1
x
∂
2
f
– f(x1; x2) =
+∆
x
1
x
∆
2
x
2
x
∂
2
. (59)
4! 5 84"3 #0%& ) , +)!5 "#$"2
' 4 ( "#% 8! ) 2083-90 .
79
12. НОРМИРУЕМЫЕ МЕТРОЛОГИЧЕСКИЕ
ХАРАКТЕРИСТИКИ СРЕДСТВ ИЗМЕРЕНИЙ
,"+",", '4#*! 4%3 '4%"3
#0%& "#$"2
,"+"," ' 16 +2
,"+"," *0)+)"%& +" , )%"39:"$ )%"*"$
"$"*+," 5,"+","
%++! * +"
,"+"," +) 2+) , ,#!)9:"5 )%"3" #0%&! "#$"2 " "5 * +&, #!)9+3 $ % 1"*+,"$" 5,"+",$" . +%4" (!* '"+!)9 '0$ 0,#"3 $"%&!5 #*"2 5 "%" "!5 5,"+", " 4 '0+"$!5 ,% "2 "5. " +)4"3 '") 43 )
$") -5"*+, 2 4 ,0$7"" , "( % )8! "# "5 0,#!)9 +$"5 +4+)5.
+ )%" $"%&!5 #*"2 " 1"7 4 '0+,$!5 ,% "2 %&!5 $ % 1"*+,"5 5,"+",
"5 #*"2 #!)+3 нормированием метрологических
характеристик. $,%00 $ % 1"*+,"5 5,"+",
0+)%") + )+)09:"2 +4, ) , $ '"+!)9+3 +) 2+) , 0+% )"3 =,+'%07"", 8"$! ( !
" .'. (:"$ ) ' +$ $" )"3 $ % 1"*+,"5 5,"+", ' +)3: 8.009-2003, ) , $ $ % 1"*+," 5,"+"," , $40+3 )!("& "# *"+% '")4!5 "8 5,"+",.
Характеристики, предназначенные для определения результата измерений:
– $"%&3 +"*+,3 5,"+", ' (# )-
"3 (140" ) *3 5,"+",) "#$"%& 1
' (# )%3, ,8 "#$"%& 1 '"( + "$ ) 2 6,% 2;
– #*" 4 #* 2 "%" #*"3 $ 1 #* 2 $!;
– 7 4%"3 6,%! "#$"%& 1 '"( "%" $ -
1 #* 2 $!;
)"4! " '$! 7"; ) 1 , 4 7"; )!5 .
–
80
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