MATHHX B
\(\newcommand{\footnotename}{footnote}\)
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\(\let \LWRref \ref \)
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\( \newcommand {\multicolumn }[3]{#3}\)
\(\require {textcomp}\)
\(\require {colortbl}\)
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\(\newcommand {\LWRsiunitxENDTWO }{}\)
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\(\def \LWRsiunitxprintdecimalsub #1.#2.#3\LWRsiunitxEND {\LWRsiunitxprintdecimalsubtwo #1,,\LWRsiunitxENDTWO \ifblank {#2}{}{{\LWRsiunitxdecimal }\LWRsiunitxprintdecimalsubtwo
#2,,\LWRsiunitxENDTWO }}\)
\(\newcommand {\LWRsiunitxprintdecimal }[1]{\LWRsiunitxprintdecimalsub #1...\LWRsiunitxEND }\)
\(\def \LWRsiunitxnumplus #1+#2+#3\LWRsiunitxEND {\ifblank {#2}{\LWRsiunitxprintdecimal {#1}}{\ifblank {#1}{\LWRsiunitxprintdecimal {#2}}{\LWRsiunitxprintdecimal {#1}\unicode
{x02B}\LWRsiunitxprintdecimal {#2}}}\LWRsiunitxdistribunit }\)
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{#2}\LWRsiunitxdistribunit }}\)
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{#2}\LWRsiunitxdistribunit }}\)
\(\def \LWRsiunitxnumpm #1+-#2+-#3\LWRsiunitxEND {\ifblank {#2}{\LWRsiunitxnumpmmacro #1\pm \pm \pm \LWRsiunitxEND }{\LWRsiunitxprintdecimal {#1}\unicode {x0B1}\LWRsiunitxprintdecimal
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\(\newcommand {\num }[2][]{\LWRsiunitxnumx #2xxxxx\LWRsiunitxEND }\)
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\(\newcommand {\SI }[2][]{\ifnextchar [{\LWRsiunitxSIopt {#2}}{\LWRsiunitxSI {#2}}}\)
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\(\newcommand {\SIrange }[4][]{\num {#2}\,#4\ \LWRsiunitxrangephrase \ \num {#3}\,#4}\)
\(\newcommand {\tablenum }[2][]{\mathrm {#2}}\)
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\(\newcommand {\kelvin }{\mathrm {K}}\)
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\(\newcommand {\metre }{\mathrm {m}}\)
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\(\newcommand {\second }{\mathrm {s}}\)
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\(\newcommand {\degreeCelsius }{\unicode {x2103}}\)
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\(\newcommand {\joule }{\mathrm {J}}\)
\(\newcommand {\katal }{\mathrm {kat}}\)
\(\newcommand {\lumen }{\mathrm {lm}}\)
\(\newcommand {\lux }{\mathrm {lx}}\)
\(\newcommand {\newton }{\mathrm {N}}\)
\(\newcommand {\ohm }{\mathrm {\Omega }}\)
\(\newcommand {\pascal }{\mathrm {Pa}}\)
\(\newcommand {\radian }{\mathrm {rad}}\)
\(\newcommand {\siemens }{\mathrm {S}}\)
\(\newcommand {\sievert }{\mathrm {Sv}}\)
\(\newcommand {\steradian }{\mathrm {sr}}\)
\(\newcommand {\tesla }{\mathrm {T}}\)
\(\newcommand {\volt }{\mathrm {V}}\)
\(\newcommand {\watt }{\mathrm {W}}\)
\(\newcommand {\weber }{\mathrm {Wb}}\)
\(\newcommand {\day }{\mathrm {d}}\)
\(\newcommand {\degree }{\mathrm {^\circ }}\)
\(\newcommand {\hectare }{\mathrm {ha}}\)
\(\newcommand {\hour }{\mathrm {h}}\)
\(\newcommand {\litre }{\mathrm {l}}\)
\(\newcommand {\liter }{\mathrm {L}}\)
\(\newcommand {\arcminute }{^\prime }\)
\(\newcommand {\minute }{\mathrm {min}}\)
\(\newcommand {\arcsecond }{^{\prime \prime }}\)
\(\newcommand {\tonne }{\mathrm {t}}\)
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\(\newcommand {\bohr }{\mathit {a}_0}\)
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\(\newcommand {\planckbar }{\mathit {\unicode {x210F}}}\)
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\(\newcommand {\decibel }{\mathrm {dB}}\)
\(\newcommand {\knot }{\mathrm {kn}}\)
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\(\newcommand {\nauticalmile }{\mathrm {M}}\)
\(\newcommand {\neper }{\mathrm {Np}}\)
\(\newcommand {\yocto }{\mathrm {y}}\)
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\(\newcommand {\atto }{\mathrm {a}}\)
\(\newcommand {\femto }{\mathrm {f}}\)
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\(\newcommand {\micro }{\mathrm {\unicode {x00B5}}}\)
\(\newcommand {\milli }{\mathrm {m}}\)
\(\newcommand {\centi }{\mathrm {c}}\)
\(\newcommand {\deci }{\mathrm {d}}\)
\(\newcommand {\deca }{\mathrm {da}}\)
\(\newcommand {\hecto }{\mathrm {h}}\)
\(\newcommand {\kilo }{\mathrm {k}}\)
\(\newcommand {\mega }{\mathrm {M}}\)
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\(\newcommand {\of }[1]{_{\mathrm {#1}}}\)
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\(\newcommand {\celsius }{\unicode {x2103}}\)
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\(\newcommand {\mg }{\milli \gram }\)
\(\newcommand {\g }{\gram }\)
\(\newcommand {\kg }{\kilo \gram }\)
\(\newcommand {\amu }{\mathrm {u}}\)
\(\newcommand {\nm }{\nano \metre }\)
\(\newcommand {\um }{\micro \metre }\)
\(\newcommand {\mm }{\milli \metre }\)
\(\newcommand {\cm }{\centi \metre }\)
\(\newcommand {\dm }{\deci \metre }\)
\(\newcommand {\m }{\metre }\)
\(\newcommand {\km }{\kilo \metre }\)
\(\newcommand {\as }{\atto \second }\)
\(\newcommand {\fs }{\femto \second }\)
\(\newcommand {\ps }{\pico \second }\)
\(\newcommand {\ns }{\nano \second }\)
\(\newcommand {\us }{\micro \second }\)
\(\newcommand {\ms }{\milli \second }\)
\(\newcommand {\s }{\second }\)
\(\newcommand {\fmol }{\femto \mol }\)
\(\newcommand {\pmol }{\pico \mol }\)
\(\newcommand {\nmol }{\nano \mol }\)
\(\newcommand {\umol }{\micro \mol }\)
\(\newcommand {\mmol }{\milli \mol }\)
\(\newcommand {\mol }{\mol }\)
\(\newcommand {\kmol }{\kilo \mol }\)
\(\newcommand {\pA }{\pico \ampere }\)
\(\newcommand {\nA }{\nano \ampere }\)
\(\newcommand {\uA }{\micro \ampere }\)
\(\newcommand {\mA }{\milli \ampere }\)
\(\newcommand {\A }{\ampere }\)
\(\newcommand {\kA }{\kilo \ampere }\)
\(\newcommand {\ul }{\micro \litre }\)
\(\newcommand {\ml }{\milli \litre }\)
\(\newcommand {\l }{\litre }\)
\(\newcommand {\hl }{\hecto \litre }\)
\(\newcommand {\uL }{\micro \liter }\)
\(\newcommand {\mL }{\milli \liter }\)
\(\newcommand {\L }{\liter }\)
\(\newcommand {\hL }{\hecto \liter }\)
\(\newcommand {\mHz }{\milli \hertz }\)
\(\newcommand {\Hz }{\hertz }\)
\(\newcommand {\kHz }{\kilo \hertz }\)
\(\newcommand {\MHz }{\mega \hertz }\)
\(\newcommand {\GHz }{\giga \hertz }\)
\(\newcommand {\THz }{\tera \hertz }\)
\(\newcommand {\mN }{\milli \newton }\)
\(\newcommand {\N }{\newton }\)
\(\newcommand {\kN }{\kilo \newton }\)
\(\newcommand {\MN }{\mega \newton }\)
\(\newcommand {\Pa }{\pascal }\)
\(\newcommand {\kPa }{\kilo \pascal }\)
\(\newcommand {\MPa }{\mega \pascal }\)
\(\newcommand {\GPa }{\giga \pascal }\)
\(\newcommand {\mohm }{\milli \ohm }\)
\(\newcommand {\kohm }{\kilo \ohm }\)
\(\newcommand {\Mohm }{\mega \ohm }\)
\(\newcommand {\pV }{\pico \volt }\)
\(\newcommand {\nV }{\nano \volt }\)
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\(\newcommand {\V }{\volt }\)
\(\newcommand {\kV }{\kilo \volt }\)
\(\newcommand {\W }{\watt }\)
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\(\newcommand {\kW }{\kilo \watt }\)
\(\newcommand {\MW }{\mega \watt }\)
\(\newcommand {\GW }{\giga \watt }\)
\(\newcommand {\J }{\joule }\)
\(\newcommand {\uJ }{\micro \joule }\)
\(\newcommand {\mJ }{\milli \joule }\)
\(\newcommand {\kJ }{\kilo \joule }\)
\(\newcommand {\eV }{\electronvolt }\)
\(\newcommand {\meV }{\milli \electronvolt }\)
\(\newcommand {\keV }{\kilo \electronvolt }\)
\(\newcommand {\MeV }{\mega \electronvolt }\)
\(\newcommand {\GeV }{\giga \electronvolt }\)
\(\newcommand {\TeV }{\tera \electronvolt }\)
\(\newcommand {\kWh }{\kilo \watt \hour }\)
\(\newcommand {\F }{\farad }\)
\(\newcommand {\fF }{\femto \farad }\)
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\(\newcommand {\K }{\mathrm {K}}\)
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\(\newcommand {\kibi }{\mathrm {Ki}}\)
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\(\let \complexnum \num \)
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\(\newcommand {\LWRbooktabscmidrulenoparen }[1]{}\)
\(\newcommand {\cmidrule }[1][]{\ifnextchar (\LWRbooktabscmidruleparen \LWRbooktabscmidrulenoparen }\)
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\(\def \LWRsiunitxrangephrase { \protect \mbox {to (numerical range)} }\)
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5.2 Indekstal
Indekstal bruges til at få overblik over en udvikling. De er især nyttige hvis man vil sammenligne udviklinger.
Betragt følgende to tilskuerudviklinger:
.
Sæson
08/09
9/10
10/11
11 /12
12/13
13/14
Tilskuergennemsnit
16908
14372
12849
12600
9166
15931
Tilskuerudvikling for Brøndby IF
.
Sæson
08/09
9/10
10/11
11 /12
12/13
13/14
Tilskuergennemsnit
7777
7107
7212
7422
5800
8061
Tilskuerudvikling for FCM
Det er svært direkte at sammenligne de to udviklinger. Vi kan se at de begge starter godt, er i krise i 12/13-sæsonen, og slutter godt. Hvem har mon haft den bedste udvikling?
Det finder vi ud af ved at lave indekstal. Man laver indekstal ved først at vælge et basisår. Det er typisk det første år i udviklingen. Basisåret er udgangspunktet for sammenligningen og derfor sætter man værdien til \(100\) (som i
\(100\%\)). Derefter udregner man de andre indekstal ved at finde ud af hvor mange procent den pågældende værdi udgør af værdien i basisåret.
Selvom indekstal egentligt er procentværdier, skrives de uden procenttegn.
Eksempel 5.2.1
Vi beregner indekstallet for BIF for 11/12 sæsonen med 08/09 som basisår. Tilskuertallet for 11/12 er 12600 og tilskuertallet for basisåret er 16908. Derfor kan vi udregne:
\[I=\frac {12600}{16908}=0{,}7452=75\%.\]
Dvs. indekstallet \(I\) er \(75\). Vi laver et nyt skema:
.
Sæson
08/09
9/10
10/11
11 /12
12/13
13/14
Indekstal
100
75
Øvelse 5.2.1
Nu er det din tur.
Løsning 5.2.1
a)
.
Sæson
08/09
9/10
10/11
11 /12
12/13
13/14
Indekstal
100
85
76
75
54
94
Procentpoint
Kigger vi på indekstallene for tilskuertallene for BIF, kan vi se at fra sæsonen 12/13 til 13/14 er indekstallet vokset med:
\[94-54=40.\]
Vi siger at indekstallet er vokset med 40 procentpoint. Procentpoint er altså forskellen mellem to indekstal og det er noget andet end den procentvise stigning. Vil vi have væksten i procent skal vi regne:
\[\frac {94-54}{54}\cdot 100\%=74\%.\]
Vi ser at stigningen på \(40\) procentpoint faktisk er en stigning på \(74\%\).
Øvelse 5.2.2
Her ses indekstallene for FCM:
.
Sæson
08/09
9/10
10/11
11 /12
12/13
13/14
Indekstal
100
91
93
95
75
104
Øvelse 5.2.3
Se på indekstallene for FCM i sæsonen 12/13 og 13/14.
Beregning af værdier ud fra indekstal
Ligesom man kan udregne indekstal ud fra nogle værdier kan man også regne baglæns hvis man har indekstallene, men ikke værdierne. Man skal dog have mindst en værdi før man kan finde resten.
Antag at en vare koster \(250\) kr. i 2005 som er basisåret og antag at indekstallet for varens pris i 2011 er \(130\). Vi kan så regne varens pris i 2011 ved at tage \(130\%\) af \(250\):
\[250\cdot 1{,}3=325.\]
Varens pris i år 2011 var altså \(325\) kr.
Øvelse 5.2.4
Nedenunder ses tilskuertallene for SønderjyskE:
.
Sæson
08/09
9/10
10/11
11 /12
12/13
13/14
Tilskuergennemsnit
3420
3419
3283
?
3005
?
Indekstal
100
?
96
96
?
102
Løsning 5.2.4
a)
.
Sæson
08/09
9/10
10/11
11 /12
12/13
13/14
Tilskuergennemsnit
3420
3419
3283
3283
3005
3488
Indekstal
100
100
96
96
88
102