tp

!"
LCP$ $ TP1.$Erreurs$incertitudes$
TP1.%Erreurs%et%incertitudes%
Objectif$:$Apprendre$quelques$règles$de$base$pour$estimer$les$incertitudes$expérimentales$et$valoriser$
ainsi$les$mesures$effectuées$au$laboratoire.$
$
#$" %&'()*+," -.$/$)00," 123-)3+,00,4,3-" $/,1" 5,(" $%%.26)4$-)23(7" 83," 5,(" .$)(23(",3" ,(-" *+," -2+-,"
4,(+.," 59+3," :.$35,+." *+,0123*+," ,(-" 3;1,(($).,4,3-" ,3-$1&;," 59,..,+.7" <0" ,(-" )4%2(()=0,"
59,>>,1-+,."5,("4,(+.,(".):2+.,+(,4,3-",6$1-,(7"
?2+."%.,35.,"123(1),31,"5+"5,:.;"59$%%.26)4$-)23"$/,1"0,*+,0"23"-.$/$)00,@"23">$)-"09,(-)4$-)23"5,("
,..,+.("*+)"%,+/,3-"$/2).";-;"1244)(,("5$3("0,("5)/,.(,("4,(+.,(",-"23"1$01+0,"0,+.("123(;*+,31,("
5$3("0,(".;(+0-$-("2=-,3+(7"A,1)"123(-)-+,"0,"1$01+0"59,..,+.@"2+"1$01+0"59)31,.-)-+5,7"
"
"
1.#Erreurs#
"
B,023"0,"(,3(":;3;.$0"5+"42-@"+3,"erreur#,(-"-2+C2+.(",3".,0$-)23"$/,1"*+,0*+,"1&2(,"5,"C+(-,"2+"5,"
/.$)@"2+"*+)",(-"123()5;.;"1244,"-,07"<0",3",(-"5,"4D4,",3"%&'()*+,7"
"
1.1#Erreur#absolue#
?$."5;>)3)-)23"09erreur#absolue#59+3,":.$35,+."4,(+.;,",(-"09;1$.-"*+)"(;%$.,"0$"/$0,+.",6%;.)4,3-$0,"
5,"0$"/$0,+."*+,"0923"123()5E.,"1244,"/.$),7"
?.,323("%$.",6,4%0,"0$"/)-,((,"5,"0$"0+4)E.,"5$3("0,"/)5,"F""
"
#$"/$0,+."123()5;.;,"$1-+,00,4,3-"1244,"/.$),",(-"F"c0$=$299$792$km$sI1$
"
#2.("59+3,"4,(+.,@"+3",6%;.)4,3-$-,+."-.2+/,F"c$=$305$000$km$sI1,$
"
23"5)-"*+,"09erreur#absolue#5,"(23".;(+0-$-",(-"F"" Δc$=$c$–$c0$=$5208$km$sI1$
"
"
1.2#Erreur#relative#
?$."5;>)3)-)23"09erreur#relative#,(-"0,"*+2-),3-"5,"09,..,+."$=(20+,"G"0$"/$0,+."/.$),"F"
H$3("32-.,",6,4%0,@"09,..,+.".,0$-)/,",(-"F" " " $$=$ $=$0,0174$ ≈ 1,7$%$
"
#9,..,+.".,0$-)/,"39$"%$("59+3)-;"I",00,"32+(")35)*+,"0$"*+$0)-;"J09,6$1-)-+5,K"5+".;(+0-$-"2=-,3+7"
L00,"(9,6%.)4,":;3;.$0,4,3-",3"M"J%2+."1,3-K7"
N3" /2)-" 10$).,4,3-" *+9)0" 39,(-" %2(()=0," 5," %$.0,." 59,..,+." *+," ()" 0923" $" G" 5)(%2()-)23" +3," valeur$ de$
référence"*+,"0923"%,+-"123()5;.,."1244,"/.$),7"
"
"
"
2.#Incertitudes#
#
#2.("5,"0$"%0+%$.-"5,(" 4,(+.,("%&'()*+,(@"23"3,"%2((E5," %$("5,"/$0,+."5," .;>;.,31,@"1244,"1,00,"
523-"32+("/,323("5,"%$.0,.7"#2.(*+923"4,(+.,"0$"5)(-$31,"5,"5,+6"%2)3-(@"2+"09)3-,./$00,"5,"-,4%("
*+)"(;%$.,"5,+6";/;3,4,3-(@"2+"0$"4$((,"59+3"2=C,-@"23"3,"($)-"%$("*+,00,",(-"0$"/$0,+.",6$1-,"5,"0$"
:.$35,+." 4,(+.;,7" N3" 3," 5)(%2(," *+," 5," 0$" /$0,+." ,6%;.)4,3-$0,7" O;$342)3(@" %$." +3," 1.)-)*+,"
2=C,1-)/,"5,("42',3("+-)0)(;("%2+.">$).,"0$"4,(+.,@"23"%,+-"(,">$).,"+3,")5;,"5,"09P",..,+."Q"4$6)4$0,"
*+923"%,+-"$/2)."1244)(,@"P",..,+."Q"*+,"0923"$%%,00,"5,">$R23"%0+("$%%.2%.);,"incertitude7"
"

S"
LCP$ $ TP1.$Erreurs$incertitudes$
T+$35"32+(",>>,1-+23("+3,"4,(+.,@"5,+6"-'%,("5U,..,+.(",3-.,3-",3"C,+"F"
VLes$erreurs$systématiques$F",00,("(23-"5+,("0,"%0+("(2+/,3-"G"+3,")4%,.>,1-)23"5,"0U$%%$.,)00$:,"2+"
5," 0$" -,1&3)*+," 5," 4,(+.,7" L00,(" $:)((,3-" -2+C2+.(" 5$3(" 0," 4D4," (,3(" ,-" 0,+." $4%0)-+5," ,(-"
123(-$3-,7"
V"Les$erreurs$aléatoires$F":;3;.$0,4,3-@",00,("%.2/),33,3-"5,("1$.$1-;.)(-)*+,("5,"0U$%%$.,)00$:,@"5,"0$"
-,1&3)*+," +-)0)(;,@" ,-" 5," 0U)3-,./,3-)23" 5+" 4$3)%+0$-,+.7" L00,(" (23-" ,(-)4;,(" (2)-" ,3" 124%$.$3-"
(-$-)(-)*+,4,3-" 0,(" .;(+0-$-(" 5U,6%;.),31,(" (2):3,+(,4,3-" .;%;-;,(@" (2)-" ,3" ,>>,1-+$3-" +3" 1$01+0"
5U)31,.-)-+5,7"
H,+6"4;-&25,("(23-"5231"+-)0)(;,("%2+.";/$0+,."0,("erreurs$aléatoires$F"
0 Méthode$ par$ calcul$ d'incertitude$ :$ 83," 4$3)E.," ()4%0," 5U,(-)4,." 0U)31,.-)-+5," (+." 0$"
/$0,+."5U+3,":.$35,+."%&'()*+,",(-"5U+-)0)(,."1,"*+U23"$%%,00,"+3"calcul$d'incertitude7"A,"
calcul$ 3U,(-" %2(()=0," *+," ()" 1,--," :.$35,+." ,(-" 0);,@" %$." +3," 02)" 1233+,@" G" 5U$+-.,("
:.$35,+.("523-"32+("$/23("5;CG"+3,",(-)4$-)23"(+."0,+.(")31,.-)-+5,(7"
0 Méthode$ statistique$ :$ ,00," ,(-" 0$" 4;-&25," 0$" %0+(" .):2+.,+(," 5U;/$0+$-)23" 5,(" ,..,+.("
$0;$-2).,(@"4$)(",00,",6):,"5,".;%;-,."+3":.$35"324=.,"5,">2)("0$"4$3)%+0$-)237"
"
"
2.1#L’incertitude#absolue#
#9)35)1$-)23"124%0E-,"5+".;(+0-$-"59+3,"4,(+.,"%&'()*+,"G"124%2.-,"0$"/$0,+."*+923",(-)4,"0$"%0+("
%.2=$=0,"x0",-"09)3-,./$00,"G"09)3-;.),+."5+*+,0"23",(-"G"%,+"%.E("1,.-$)3"*+,"(,"()-+,"0$"/.$),"/$0,+.7"
#$"/$0,+."0$"%0+("%.2=$=0,",(-",3":;3;.$0"0,"1,3-.,"5,"1,-")3-,./$00,7"
#$"5,4)V023:+,+."5,"1,0+)V1)",(-"$%%,0;,"incertitude#absolue#5,"0$"4,(+.,"Δx7""
W)3()" 09)31,.-)-+5," $=(20+," ,(-" l’erreur#maximale"*+," 0923" ,(-" (+(1,%-)=0," 5," 1244,--.," 5$3("
09;/$0+$-)23"5,"x7"
x$I$Δx$≤$x0$≤$x$+$Δx$
"
B2+("+3,">2.4,"1235,3(;,@"0,".;(+0-$-"5,"0$"4,(+.,"(9;1.)-"F"
"
G$=$x$±$Δx$
"
#9)31,.-)-+5,"$=(20+,"52)-"123-,3)."+3"(,+0"1&)>>.,"5)>>;.,3-"5,"X7"
$
$
Exemples:$
1)"#$"023:+,+."L"59+3"2=C,-",(-"F"" L$=$15,3$±$0,1$cm.$
A,0$"():3)>),"*+9$/,1"+3,")31,.-)-+5,"$=(20+,"ΔL$=$0,1$cm@"0$"/$0,+.",6$1-,",(-"124%.)(,",3-.,"15,2$cm"
,-"15,4$cm."
"
2)"#$"-,4%;.$-+.,"θ"59+3"021$0",(-"F" θ$=$22$±$1$°C."
<1)"09)31,.-)-+5,"$=(20+,"Δθ$=$1$°C@"19,(-VGV5).,"*+,"0923":$.$3-)-"*+,"0$"-,4%;.$-+.,"39,(-"%$("
)3>;.),+.,"G"21°C"3)"(+%;.),+.,"G"23$°C."
"
#
Remarque:$
#2.(*+923" 4,(+.," +3," :.$35,+."%&'()*+,"J023:+,+.@" -,4%(@" 4$((,@" -,4%;.$-+.,@" YK"23" %,+-"
123()5;.,." Z" %2+."()4%0)>),." V" *+," 09)31,.-)-+5," $=(20+," 12..,(%235" G" la# moitié# de# la# plus# petite#
graduation"5,"09)3(-.+4,3-"5,"4,(+.,"+-)0)(;@$G"1235)-)23"5U,3"5)(-)3:+,."-.E("10$).,4,3-"0,(".,%E.,(7"
Par$exemple$dans$le$cas$d’une$règle$précise$à$1$mm,$l’incertitude$absolue$est$:$ Δx$=$0,5$mm.$
B)"0U,6%;.)4,3-$-,+.",(-"%0$1;"5$3("5,"4$+/$)(,("1235)-)23("%2+.",>>,1-+,."($"4,(+.,@"0U)31,.-)-+5,"
%,+-"D-.,";:$0,"G"la#plus#petite#division#ou#plus7"<0"52)-"$02.("(,"(,./)."5,"(23"=23"(,3("%2+."0U;/$0+,.7$
Par$exemple,$les$graduations$d’un$thermomètre$précis$à$0,1°C$sont$difficilement$lisibles,$l’incertitude$
absolue$est$:$ Δθ$=$0,1$°C."

["
LCP$ $ TP1.$Erreurs$incertitudes$
2.2#L’incertitude#relative#
#9)31,.-)-+5,"$=(20+,@"02.(*+9,00,",(-"123()5;.;,"(,+0,@"39)35)*+,".),3"(+."0$"*+$0)-;"5,"0$"4,(+.,7"
?2+."C+:,."5,"1,--,"*+$0)-;@")0">$+-"124%$.,."09)31,.-)-+5,"$=(20+,"G"0$":.$35,+."4,(+.;,7"#,".$%%2.-"
5,"1,(":.$35,+.(",(-"$%%,0;"incertitude#relative@",00,"%,.4,-"59,(-)4,."la#précision"5+".;(+0-$-.#
"
Incertitude#relative#F" " " "
"
A244," %2+." 09,..,+." .,0$-)/,@" 09)31,.-)-+5," .,0$-)/," ,(-" +3" 324=.," ($3(" 5)4,3()23" J,-" ($3(" +3)-;K@"
%.$-)*+,4,3-"-2+C2+.("=,$+12+%"%0+("%,-)-"*+,"!@"*+,"0923",6%.)4,":;3;.$0,4,3-",3"M"J100$." K7"
"
"
"
"
3.#Calcul#d’incertitude#
L3"%&'()*+,",6%;.)4,3-$0,@"0,(":.$35,+.("*+,"0923"4,(+.,"(23-":;3;.$0,4,3-"+-)0)(;,("%2+."5;5+).,"
5,(".;(+0-$-("%$."5,("1$01+0(7"<0",(-"$02.(")3-;.,(($3-"5,"($/2)."5,"*+,00,"4$3)E.,"0,(")31,.-)-+5,("5,("
4,(+.,("(,".;%,.1+-,3-"(+."0,(")31,.-)-+5,("5,(".;(+0-$-(7"
#
3.1#Addition#et#soustraction#
B+%%2(23("*+,"0$":.$35,+."1&,.1&;,"\"(2)-"0$"(244,"5,"S"4,(+.,("W",-"]"F" " \"^"W"_"]"
"
H$3("1,"1$("09)31,.-)-+5,"(+."0,".;(+0-$-",(-"F" " " " " "
"
<0",3",(-"5,"4D4,"%2+."F"\"^"W"Z"]"
"
L’incertitude#absolue#sur#une#somme#ou#une#différence#est#la#somme$des$incertitudes$absolues#de#
chaque#terme.#
#
#
Exemple$:$
3)"83".;1)%),3-"$"+3,"4$((,"m$=$50$±$1$g7"\,4%0)"59,$+@"($"4$((,"/$+-"F"M$=$200$±$1$g7"
#$"4$((,"59,$+"*+9)0"123-),3-",(-"5231"F"" meau$=$M$I$m$
L3"$%%0)*+$3-"0$".E:0,"1)V5,((+("F"" " Δmeau$=$ΔM$+$Δm$=$1$+$1$=$2$g."
H92`"F"""""""meau$=$150$±$2$g"
"
"
3.2#Multiplication#et#division#
B+%%2(23("4$)3-,3$3-"*+,"0$":.$35,+."1&,.1&;,"\"(2)-"0,".;(+0-$-"5+"1$01+0"(+)/$3-"F"\"^" "
2`"W@"]",-"A"(23-"5,(":.$35,+.("*+,"0923"4,(+.,7"
"
H$3("1,"1$("09)31,.-)-+5,".,0$-)/,"(+."0,".;(+0-$-",(-"F""
"
"
L’incertitude# relative# sur# un# produit# ou# un# quotient# est# la$ somme$ des$ incertitudes$relatives#de#
chaque#terme."
a\"^"aW"_"a]"
"^" "_" "_" "

b"
LCP$ $ TP1.$Erreurs$incertitudes$
Exemple$:$
4)#?2+."5;-,.4)3,."0$"(+.>$1,"S#5U+3".,1-$3:0,@"23"4,(+.,"(,("5,+6"1c-;("F"x#J023:+,+.K",-"y#
J0$.:,+.K7"N3"-.2+/,"F"x$=$24,6$±$0,1$cm",-"y$=$8,3$±$0,1$cm."
#U$%%0)1$-)23"5).,1-,"5,"S$=$x_y"1235+)-"G"0$"/$0,+."F"S$=$204,18$cm27"
B)"0U23"123(,./,"1,--,"/$0,+."-,00,"*+U,00,",(-@"1,0$"/,+-"5).,"*+,"0$"(+.>$1,"S",(-"1233+,"$/,1"
+3,")31,.-)-+5,"5,"0,01$cm27"
N.@"09)31,.-)-+5,".,0$-)/,",(-"F" " $=$ $+$ "@" 592`"F" " $=$S_( $+$ )$=$3,29$cm2"
N3"52)-"$..235)."G"F" " $=$3$cm2""""J09)31,.-)-+5,"52)-"123-,3)."+3"(,+0"1&)>>.,"5)>>;.,3-"5,"XK"
d)3$0,4,3-"F" " S$=$204$±$3$cm2"
"
"
3.3#Chiffres#significatifs#
H$3(" 0," 1$(" 2`" 0U)31,.-)-+5," (+." +3," :.$35,+." )3-,.4;5)$).," 3U,(-" %$(" explicitement$ 5233;,@" 0,("
(1),3-)>)*+,(" $54,--,3-" 0," 3)/,$+" 5+" 5,.3),." 1&)>>.," ():3)>)1$-)>" 1244," 2.5.," 5," :.$35,+." 5,"
0U)31,.-)-+5,7"
"
Exemple$
5)#B)"(+."+3,"4$((,"m$+-)0)(;,",3"0$=2.$-2).,"23"-.2+/,")3(1.)-"23,0$g@"$02.("am$="e"X@!:"
B)"L$=$1,37$m"" $02.("L$=$1,37$±$0.01$m7"
B)"M$=$3500$kg""$02.("M$=$3500$±$1$kg7"
"
#
#
#
#
4.#Méthode#statistique#
B)" 0923" .;%E-," %0+(),+.(" >2)(" 5," (+)-,@" ,-" 5$3(" 0,(" 4D4,(" 1235)-)23(@" 0$" 4,(+.," 59+3," :.$35,+."
%&'()*+,"f@"0,("324=.,("g)"*+,"0923"2=-),3-"(23-",3":;3;.$0"0;:E.,4,3-"5)>>;.,3-(7"
B2+/,3-"23"$52%-,"%2+."/$0,+."$%%.21&;,"0$"42',33,"$.)-&4;-)*+,"5,("5)>>;.,3-("g)"F"
g4"^" "
n",(-"0,"324=.,"5,"4,(+.,(",>>,1-+;,(7"
g4"3,".,%.;(,3-,"%$("+3,"/$0,+.",6$1-,"5,"0$":.$35,+."%&'()*+,"f@"4$)("+3,"valeur#moyenne."
"
#9)31,.-)-+5,"$=(20+,",(-"F"" " "
ag"^"4$6" "
"
"
"
Remarque$:$
B)"0U+3,"5,("/$0,+.(",(-"-.E(";02):3;,"5,("$+-.,(@"1,--,"/$0,+."52)-"D-.,"rejetée@",-",00,"3,"52)-"%$("
)3-,./,3)."5$3("0,"1$01+0"5,"g4"3)"5,"(23")31,.-)-+5,7"
"
"

g"
LCP$ $ TP1.$Erreurs$incertitudes$
Exemple$
6)#B)6";-+5)$3-("(,"(23-".,0$';("%2+."4,(+.,."0,"5)$4E-.,"H"5U+3"5)(*+,"124%$1-"JCDK@")0("
)3(1.)/,3-"0,+.(".;(+0-$-("5$3("0,"-$=0,$+"(+)/$3-F"
"
L-+5)$3-"!"
L-+5)$3-"S"
L-+5)$3-"["
L-+5)$3-"b"
L-+5)$3-"g"
L-+5)$3-"h"
120.5$mm$
119.0$mm$
119.7$mm$
118.9$mm$
120.0$mm$
128.3mm$
"
#$"/$0,+."!Si@["44"52)-"D-.,".,C,-;,7"
#,"1$01+0"5,"0$"/$0,+."42',33,"5,"H"G"%$.-)."5,("g"$+-.,("4,(+.,("5233," 119,62$mmI"
#U)31,.-)-+5,"$=(20+,",(-"F""
ΔD$=$max{|+0.88|,|I0.62|,|+0.08|,|I0.72|,|+0.38|}$=$0.88$mm7"
H92`"F"D$=$119.6$±$0.9$mm7"
#,"5)$4E-.,",6$1-"5+"AH"$%%$.-),3-"G"0U)3-,./$00,"F"D$ [118,7$;$120,5]$mm" "
#U)31,.-)-+5,".,0$-)/,"(+."0$"4,(+.,"5+"5)$4E-.,"5+"AH",(-"F"$$=$7.5$‰$
"
"
"
"
"
5.#Incertitude#sur#le#graphe#
"
<0",(-"12+.$3-"5U;-+5),.":.$%&)*+,4,3-"+3,"%.2%.);-;@",3">231-)23"5U+3"%$.$4E-.,@"%2+.",3"déduire$
2+"vérifier$+3,"loi$0,(".,0)$3-7"Les$rectangles$d'incertitudes$J2+"0,("barres$d'erreursK"52)/,3-"D-.,"
%2.-;("(+."0,":.$%&,"%2+."C+:,."5,"0$"/$0)5)-;"5,"0U)3-,.%.;-$-)237"
B2)-"+3"%2)3-",6%;.)4,3-$0"5;>)3)"%$."0,("122.5233;,("F"j"$>>,1-;"5,"0U)31,.-)-+5,"e"aj@",-"k"$>>,1-;"5,"
0U)31,.-)-+5,"e"ak7"#,"-.$1;"5,"1,"%2)3-"(+."+3":.$%&,"12..,(%235"$+"(1&;4$"(+)/$3-"F"
"
"
"
"
"
"
"
"
"
#$" l23," :.)(," 12..,(%235" G" 0U$).," 5U)31,.-)-+5," 5+" %2)3-" ,6%;.)4,3-$07" L00," %,+-" (," .;5+).," G" +3,"
()4%0,"=$..,"()"0U+3,"5,(")31,.-)-+5,(",(-"-.E(">$)=0,"J23"%$.0,.$"$02.("5,"barres$d'erreursK7"
83,">2)("4)(" 0,(".,1-$3:0,("5U,..,+.@"23" -.$1,"4$3+,00,4,3-"0$" 4,)00,+.,"12+.=," passant$au$ mieux$
dans$tous$les$rectangles$d'incertitude7"
"
"
Tracé$du$graphe$
?2+."2=-,3).@"5$3("+3"-,4%(".$)(233$=0,@"+3":.$%&,",6%02)-$=0,"2`")0",(-"$)(;"5U$3$0'(,."+3"
%&;324E3,@")0",(-".,1244$35;"5,"(+)/.,"0,("3,+>";-$%,("(+)/$3-,("F"
"
!V"\;$0)(,."$/,1"(2)3"0,("5)>>;.,3-,("4,(+.,(",-".,%2.-,."0,(".;(+0-$-("5$3("+3"-$=0,$+7"
SV"L(-)4,."0,(")31,.-)-+5,("ax$,-"ay$%2+."1&$*+,"12+%0,"Jx@yK"5+"-$=0,$+7"
[V"A&2)()."123/,3$=0,4,3-"0U2.):)3,"5,("$6,("J)0"3U,(-"%$(")35)(%,3($=0,"*+,"0U2.):)3,"5,("$6,("
12..,(%235,"G"x$=$0$,-"y$=$0K7"
bV"O244,."0,("$6,(",3")35)*+$3-"0,("+3)-;("5,"x$,-"5,"y7"