MATHHX B

MATHHX B

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12.1 Ugrupperede observationer

Vi tager udgangspunkt i en undersøgelse, hvor vi har spurgt 10 unge om, hvor meget tid de bruger på sport om ugen. De har svaret

\[4,1,2,9,4,4,2,5,1,5\]

Denne liste kaldes observationssættet og hvert enkelt tal kaldes en observation.

Frekvenstabeller

Vi starter med at lave en frekvenstabel. Den ser således ud (forklaring følger bagefter):

.
Observation (\(x_i\)) Hyppighed (\(h_i\)) Frekvens (\(f_i\)) Summeret frekvens (\(F_i\))
\(1\) \(2\) \(0{,}2\) \(0{,}2\)
\(2\) \(2\) \(0{,}2\) se øv. 12.1.4
\(4\) se øv. 12.1.2 \(0{,}3\) \(0{,}7\)
\(5\) \(2\) \(0{,}2\) \(0{,}9\)
\(9\) \(1\) se øv. 12.1.3 \(1\)
Observation (\(x_i\))

Første søjle viser de forskellige observationer. Fordi der er fem forskellige observationer er der fem tal i denne søjle, selvom der er ti observationer i alt. Vi skriver observationerne i rækkefølge fra mindste til største. Vi ser at der står (\(x_i\)) efter ”Observationer”. Det betyder at den første observationen i søjlen betegnes \(x_1\), den næste \(x_2\) osv.12.1.4

Eksempel: \(x_3=4\) fordi den tredje observation er \(4\).

Øvelse 12.1.1

Med udgangspunkt i frekvenstabellen.

  • a) Aflæs \(x_4\)

Løsning 12.1.1

  • a) \(x_4=5\)

Hyppighed (\(h_i\))

Næste søjle viser hyppigheden, som er antallet af gange den enkelte observation optræder i observationssættet.

Eksempel: \(h_5=1\), da den femte observation er \(9\) og vi kun har ét nital blandt i vores oprindelige liste med observationer.

Øvelse 12.1.2

Med udgangspunkt i observationssættet.

  • a) Bestem \(h_3\)

Løsning 12.1.2

  • a) \(h_3=3\)

Frekvens (\(f_i\))

Frekvensens kaldes også den relative hyppighed fordi den viser, hvor stor en del den enkelte hyppighed udgør ud af det samlede antal observationer. Formlen for frekvens er

\[f_i=\frac {h_i}{n}\]

Eksempel \(f_4=\frac {2}{10}=0{,}2\):

At \(f_2=0{,}2\) betyder at \(20\%\) af de unge brugte \(3\) timer på sport (da \(x_2=3\)).

Øvelse 12.1.3

Med udgangspunkt i tabellen

  • a) Bestem \(f_5\)

  • b) Fortolk \(f_5\)

Løsning 12.1.3

  • a) \(f_5=0{,}1\)

  • b) At \(f_5=0{,}1\) betyder at \(10\%\) af de unge brugte 9 timer på sport .

Summeret frekvens (\(F_i\))

Summeret frekvens, også kaldet kumuleret frekvens, er frekvensen lagt sammen med af alle de foregående frekvenser. Altså

\begin{align*} F_1 = & f_1\\ F_2 = & f_1+f_2\\ F_3= &f_1+f_2+f_3\\ \vdots & \end{align*} Eksempel: \(F_3=f_1+f_2+f_3=0{,}2+0{,}2+0{,}3=0{,}7\)

At \(F_3=0{,}7\) betyder at \(70\%\) af de unge dyrkede sport højst \(4\) timer om ugen.

Øvelse 12.1.4

Med udgangspunkt i frekvenstabellen

  • a) Bestem \(F_2\)

  • b) Fortolk \(F_2\)

Løsning 12.1.4

  • a) \(f_2=0{,}4\)

  • b) At \(F_2=0{,}4\) betyder at \(40\%\) af de unge dyrkede sport højst \(2\) timer om ugen.

Outliers

En outlier er en observation, som afviger meget fra de andre. Er der en eller flere outliers, kan det skyldes fejl i datamaterialet, men sådan er det ikke altid. Hvis det er en fejl, bør man sortere outlieren fra, inden man laver statistik. Problemet er så bare, hvordan man afgør om der er tale om en fejl. Hvis man f.eks. undersøger, hvor langt elever har til deres skole, og der er en som svare 800 km, er det nok en fejl (måske mente eleven 800 m), og man bør sortere dette datapunkt fra. I andre tilfælde kan det være sværere at afgøre.

I vores observationssæt ligger observationen \(9\) lidt lang væk fra de andre og kan karakteriseres som en outlier, men der er ingen fast definition på hvad en outlier er og jeg ville personligt ikke karakterisere den som en outlier. Der findes flere mere eller mindre tekniske metoder til at afgøre om en observation er en outlier, men de giver forskellige resultater, da det i sidste ende er subjektivt. Outlier eller ej, så bør man ikke sortere observationen \(9\) fra, da vi ikke har grund til at tro, at der er noget galt med den observation.

Diagrammer for ugrupperede observationer

Ud fra frekvenstabellen kan vi nu tegne to diagrammer. Tabellen så altså således ud:

\(\begin {array}{|c|c|c|c|} \hline \text {Observation } (x_i) & \text {Hyppighed } (h_i) & \text {Frekvens } (f_i) & \text {Summeret frekvens } (F_i)\\ \hline 1 & 2 & 0{,}2 & 0{,}2\\ \hline 2 & 2 & 0{,}2 & 0{,}4\\ \hline 4 & 3 & 0{,}3 & 0{,}7 \\ \hline 5 & 2& 0{,}2 & 0{,}9\\ \hline 9 & 1 &0{,}1 & 1\\ \hline \end {array}\)

Pindediagram

Vi kan illustrere fordelingen af tidsforbrug med et pindediagram, som viser hyppighederne af hver observation.

(-tikz- diagram)

Ved at kigge på diagrammet kan vi få et hurtigt overblik over tidsforbruget. Pindediagrammer er ikke gode hvis man har for mange observationer, så der kommer uoverskuelig mange pinde. I det tilfælde er det bedre at gruppere observationerne (mere om det senere).

Trappediagram

Trappediagrammet bliver brugt til at illustrere den summerede frekvens. Vi afsætter et punkt for hver observation, hvor andenkoordinaten er den summerede frekvens. Derefter forbinder vi punkterne med linjer som vist her:

(-tikz- diagram)

Trappe diagrammet kan bruges til at bestemme en procentdel af observationerne. Går vi ud fra f.eks. \(30\%\) på andenaksen får vi observationen på \(2\). Denne observation kaldes \(30\%\)-fraktilen. At \(30\%\)-fraktilen er \(2\) betyder at mindst \(30\%\) af observationer er på \(2\) eller derunder. Vi betegner \(30\%\)-fraktilen med \(x_{0{,}3}\) (da \(30\%=0{,}3)\) og den er vist her:

(-tikz- diagram)

Vi vil vende tilbage til fraktilerne i næste afsnit.

Øvelse 12.1.5

Med udgangspunkt i observationssættet \(7,0,4,7\):

  • a) Lav en frekvenstabel.

  • b) Tegn et pindediagram med papir og blyant.

  • c) Tegn et trappediagram med papir og blyant.

  • d) Aflæs fraktilen \(x_{0{,}62}\) på trappediagrammet.

  • e) Forklar betydningen af \(x_{0{,}62}\) .

Løsning 12.1.5

  • a)
    \(\begin {array}{|c|c|c|c|} \hline \text {Observation } (x_i) & \text {Hyppighed } (h_i) & \text {Frekvens } (f_i) & \text {Summeret frekvens } (F_i)\\ \hline 0 & 1 & 0{,}25 & 0{,}25\\ \hline 4 & 1 & 0{,}25 & 0{,}5\\ \hline 7 & 2 & 0{,}5 & 1 \\ \hline \end {array}\)

  • b)

    (-tikz- diagram)

  • c)
    (-tikz- diagram)

  • d) \(x_{0{,}62}=7\)

  • e) Det byder at mindst \(62\%\) af observationerne var på \(7\) eller derunder.