MÓDULO ESTADÍSTICA DESCRIPTIVA 100-105
Milton Fernando Orte!n Pa"a Dire#tor In$ Fran#i%#o Ca&rera A#reditador
U'IVERSIDAD U'IVERSIDAD 'ACIO'AL A(IERTA ) A DISTA'CIA * U'AD * FACULTAD DE CIE'CIAS (+SICAS E I',E'IERÍA U'IDAD DE CIE'CIAS (+SICAS I&a.$ /00
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UNIVERSIDAD NACIONAL ABIERTA Y A DISTANCIA – UNAD
ESCUELA DE CIENCIAS BÁSICAS, TECNOLOGÍA E INGENIERÍA CONTENIDO DIDÁCTICO DEL CURSO: 100105 – Estadística Desci!ti"a
COMIT DIRECTIVO Jaime Alberto Leal Afanador
Re#tor
Gloria C. Herrera Sánchez Vi#erre#tor A#ad.2i#o 3 de in"e%tia#i!n Roberto Salazar Ramos Vi#erre#tor de Medio% 3 Media#ione% Maribel Córdoba Guerrero Se#retaria ,eneral
MÓDULO CURSO ESTADÍSTICA DESCRIPTIVA © Co!ri"th #ni$ersidad %acional Abierta ! a &istancia
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UNIVERSIDAD NACIONAL ABIERTA Y A DISTANCIA – UNAD
ESCUELA DE CIENCIAS BÁSICAS, TECNOLOGÍA E INGENIERÍA CONTENIDO DIDÁCTICO DEL CURSO: 100105 – Estadística Desci!ti"a
CO'TE'IDO P4$ I'TRODUCCIÓ'
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'.'. CO'CEPTUALI6ACIÓ' CO'CEPTUALI6ACIÓ' DE TRMI'OS ESTADÍSTICOS '.'.'. 78#9 0S LA 0S+A&3S+*CA: '.'.;. C,%C02+,S / Reresentati$idad de la muestra '.'.;.; Muestra aleatoria simle '.'.;.) Muestra Aleatoria sistemática '.'.;.? Muestra Aleatoria 0stratificada '.'.;. Muestra Aleatoria de con"lomerados '.'.;.5. &eter. 2oblación@ muestra ! tio de muestreo a utilizar
1$/
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'.'.? CARAC+0R*AC*-% &0 #%A 1AR*A/L0 C#AL*+A+*1A '.'.?.' tablas de frecuenciasB '.'.?.; reresentaciones "raficasB '.'.?.;.' dia"ramas de barras '.'.?.;.; dia"rama circularB '.'.?.) moda
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UNIVERSIDAD NACIONAL ABIERTA Y A DISTANCIA – UNAD
ESCUELA DE CIENCIAS BÁSICAS, TECNOLOGÍA E INGENIERÍA CONTENIDO DIDÁCTICO DEL CURSO: 100105 – Estadística Desci!ti"a
'.;.;. R0C,L0CC*-% '.;.;.' Se"Dn la cobertura '.;.;.; Se"Dn la forma de obser$ación
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'.;.). ,RGA%*AC*-% &0 LA *%4,RMAC*-% '.;.).' Combinación o arre"lo ordenado '.;.).; Arre"lo de tallo ! hoas '.;.).) +abulación de la información
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MEDIDAS DE TE'DE'CIA CE'TRAL O DE POSICIO' 0S+A&*GRA4,S M0&*A AR*+M9+*CA M0&*A%A M,&A ,+RAS M0&*&AS &0 +0%&0%C*A C0%+RAL
5( 5( 5' 5? 5= 6'
ACTIVIDADES DE AUTOEVALUACIÓ' AUTOEVALUACIÓ' DE LA U'IDAD 1 LA(ORATORIO E' E8CELL U'IDAD 1-DIA,RA 1 -DIA,RAMAS MAS ESTADISTICOS
=' F6
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U'IDAD DIDACTICA / MEDIDAS DE DISPERSIÓ' ) ESTADÍSTICAS ESTADÍSTICAS (IVARIA'TES (IVARIA'TES *%+R,CC*,% A LA #%*&A& ,/J0+*1,S 0S20C*4*C,S /$ MEDIDAS DE DISPERSIÓ' ) ESTADÍSTICAS (IVARIA'TES /$1 MEDIDAS DE DISPERSIÓ' ) ASIMETRIA
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UNIVERSIDAD NACIONAL ABIERTA Y A DISTANCIA – UNAD
ESCUELA DE CIENCIAS BÁSICAS, TECNOLOGÍA E INGENIERÍA CONTENIDO DIDÁCTICO DEL CURSO: 100105 – Estadística Desci!ti"a
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ACTIVIDADES ACTIVIDADES DE AUTOEVALUACIO' U'IDAD /
LA(ORATORIO E' E8CELL U'IDAD /-RE,RESIÓ' ) CORRELACIÓ'
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UNIVERSIDAD NACIONAL ABIERTA Y A DISTANCIA – UNAD
ESCUELA DE CIENCIAS BÁSICAS, TECNOLOGÍA E INGENIERÍA CONTENIDO DIDÁCTICO DEL CURSO: 100105 – Estadística Desci!ti"a
LISTA DE TA(LAS P4$
U'IDAD DID+CTICA 1 Con#eto% Preli2inare% ) Medida% De Tenden#ia Central +ablas de contin"encia Ta&la 1$1 +abla de Contin"encia de 4recuencia Relati$as Ta&la 1$/ +abla de Contin"encia de orcentaes Ta&la 1$7 %Dmero de e"resados de la #%A& en el erEodo 'FF?I Ta&la /$1 ;((? Clasificación de estudiantes or C0A& en la ona Ta&la /$/$ ,ccidente durante el rimer semestre de ;(( Clasificación de emleados or car"o Ta&la /$7$ Clasificación de la estatura de los estudiantes de un "ruo Ta&la /$:$ de uinto "rado &istribución de frecuencias simle de latidos cardiacos de Ta&la /$5$ )( ersonas %Dmero de inter$alos de clases su"erido en función del Ta&la /$;$ tamao de la muestra &istribución de frecuencias a"ruadas de la $elocidad de Ta&la /$<$ ulsaciones &istribución de frecuencias absolutas@ relati$as ! Ta&la /$=$ acumuladas ascendentes de la $elocidad de ulsaciones &istribución de frecuencias simle de $isita al odontólo"o Ta&la /$$ de nios entre los 5 ! '; aos Ta&la /$10$ 0"resados de la #%A& en el erEodo ;(((I;((? Ta&la /$11$ 1entas or deartamento al contado ! a crKdito en marzo de ;(( &istribución de frecuencias a"ruadas Ta&la 7$1$ &istribución de frecuencias a"ruadas Ta&la 7$/$ &istribución de frecuencias de la asistencia a cine Ta&la 7$7$ &istribución de frecuencias a"ruadas de la asistencia a Ta&la 7$:$ cine Comaración de la media@ mediana ! moda Ta&la 7$5$ &istribución de frecuencias a"ruadas Ta&la 7$;$ &istribución de frecuencias a"ruadas Ta&la 7$<$
;F )( )( )= )= )F )F ?' ?) ? ? ?F ? 5 5) 55 5F 5F 6' 6; 65
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ESCUELA DE CIENCIAS BÁSICAS, TECNOLOGÍA E INGENIERÍA CONTENIDO DIDÁCTICO DEL CURSO: 100105 – Estadística Desci!ti"a
Ta&la :$1$ Ta&la :$/$
&istribución de frecuencias de las calificaciones de estudiantes de 0stadEstica &istribución de frecuencias de las calificaciones de estudiantes de 0stadEstica
&istribución de frecuencias de las calificaciones de rimer semestre en 1alleduar Cálculo de ara la distribución de frecuencias de las Ta&la :$:$ calificaciones de Cometencias Comunicati$as Cálculo de ara la distribución de frecuencias de las Ta&la :$5$ calificaciones de 0stadEstica &escriti$a Relación de $entas de un roducto ! la emisión del Ta&la 5$1$ comercial en tele$isión Grado de correlación lineal Ta&la 5$/$ Gastos indirectos de roducción Ta&la 5$7$ 2recios ! cantidades $endidas en una farmacia en ;(() ! Ta&la 5$:$ ;((?
Ta&la :$7$
''? ''6 ';5 ';= ')( ') ')F '?; '?F
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UNIVERSIDAD NACIONAL ABIERTA Y A DISTANCIA – UNAD
ESCUELA DE CIENCIAS BÁSICAS, TECNOLOGÍA E INGENIERÍA CONTENIDO DIDÁCTICO DEL CURSO: 100105 – Estadística Desci!ti"a
LISTA DE FI,URAS P4$
U'IDAD DID+CTICA 1 Con#eto% Preli2inare% ) Medida% De Tenden#ia Central Fira 1$1$ &ia"rama de /arras Medios de *nformación Fira 1$/ &ia"rama Circular medios Comunicación Fira /$1$ &ia"rama de tallo ! hoas ara los datos de ulsaciones del eemlo =.'. Fira /$/$ &ia"rama de doble tallo ! hoas ara los datos de ulsaciones del eemlo =.'. Fira /$7$ &ia"rama de frecuencias absolutas de $isita al odontólo"o de nios entre los 5 ! '; aos Fira /$:$ &ia"rama de frecuencias absolutas acumuladas de $isita al odontólo"o de nios entre los 5 ! '; aos Fira /$5$ Histo"rama de frecuencias absolutas de la $elocidad de ulsaciones Fira /$;$ 2olE"ono de frecuencias absolutas de la $elocidad de ulsaciones Fira /$<$ ,i$a ascendente de la $elocidad de ulsaciones Fira /$=$ ,i$a descendente de la $elocidad de ulsaciones Fira /$$ ,i$a ascendente ! descendente de la $elocidad de ulsaciones Fira /$10$ &ia"rama de lEneas. 0"resados de la #%A& en el erEodo ;(((I;((?
;6 ;= )5 )6 ?F ( ' ; ) ) ?
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UNIVERSIDAD NACIONAL ABIERTA Y A DISTANCIA – UNAD
ESCUELA DE CIENCIAS BÁSICAS, TECNOLOGÍA E INGENIERÍA CONTENIDO DIDÁCTICO DEL CURSO: 100105 – Estadística Desci!ti"a
Fira 7$/$ ,i$a orcentual ascendente Fira 7$7$ &ia"rama de fluo ara el PIKsimo ercentil Fira :$1$ Cur$a normal o camana de Gauss Fira :$/$ Cur$a asimKtrica ositi$a. 2olE"ono de frecuencias de calificaciones de Ló"ica Matemática Fira :$7$ Cur$a simKtrica laticDrtica. 2olE"ono de frecuencias de calificaciones de Cometencias Comunicati$as Fira :$:$ Cur$a asimKtrica ne"ati$a. 2olE"ono de frecuencias de calificaciones de Cultura 2olEtica Fira :$5$ Cur$a simKtrica letocDrtica. 2olE"ono de frecuencias de calificaciones de 0stadEstica &escriti$a Fira 5$1$ Gráficas de disersión. aN linealO bN cur$ilEneaO cN sin relación Fira 5$/$ &ia"rama de disersión de $entas de un roducto ! la emisión del comercial en tele$isión Fira 5$7$ Gráficas de disersión lineal. aN ositi$aO bN ne"ati$a
66 6= ';' ';6 ';= ';F ')' ')) ') ')6
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UNIVERSIDAD NACIONAL ABIERTA Y A DISTANCIA – UNAD
ESCUELA DE CIENCIAS BÁSICAS, TECNOLOGÍA E INGENIERÍA CONTENIDO DIDÁCTICO DEL CURSO: 100105 – Estadística Desci!ti"a
I'TRODUCCIÓ' La 0stadEstica es una discilina ue se alica en muchos camos de la acti$ acti$ida idad d del del ser huma humano no.. 0s mu! frecu frecuent ente e encont encontrar rarse se en las dife difere rent ntes es disc disci ililina nass del del saber saber con con incer incertitidu dumb mbres res como como el rono ronost stic icar ar el creci crecimi mien ento to oblacional de un aEs@ el crecimiento económico de una emresa o el crecimiento de roducción ! $enta de un roducto esecEfico@ el conocer la efecti$idad de dif diferen erente tess abon abonos os en el cam camo o a"ra a"rari rio@ o@ el dete determ rmiinar nar la tend tenden enci cia a de contaminación del a"ua o el aire@ la clasificación de ersonal en una emresa ara efectos de una buena ! sana olEtica laboral@ etc. Habi Habitu tual alme ment nte@ e@ el roó roósi sito to de la 0sta 0stadE dEst stic ica a Ali Alica cada da es el de sacar sacar conclusiones de una oblación en estudio@ eQaminando solamente una arte de ella ella denomi denominada nada muestra. muestra. 0ste 0ste roceso roceso@@ llamad llamado o Inferen Inferencia cia Estadí Estadísti stica, ca, suele $eni $enirr rec reced edid ido o de otro otroBB la Estadística Estadística Descriptiva Descriptiva@ en el ue los datos son ordenados@ ordenados@ resumidos resumidos ! clasificados clasificados con obeto de tener una $isión $isión más recisa recisa ! conunta de las obser$aciones@ intentando descubrir de esta manera osibles relaciones entre los datos@ $iendo cuáles toman $alores arecidos@ cuáles difieren "randemente del resto@ destacando hechos de osible interKs@ entre otros. 0n todos los camos de la in$esti"ación se reuiere a menudo el uso racional racional de los MKtodos 0stadEsticos. 0stadEsticos. Los rocesos de laneación@ laneación@ control ! toma de decisiones económicas@ administrati$as ! financieras se basan en resultados obtenidos mediante el análisis estadEstico de los fenómenos en ellos in$olucrados.
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UNIVERSIDAD NACIONAL ABIERTA Y A DISTANCIA – UNAD
ESCUELA DE CIENCIAS BÁSICAS, TECNOLOGÍA E INGENIERÍA CONTENIDO DIDÁCTICO DEL CURSO: 100105 – Estadística Desci!ti"a
en un conunto de datos. > es recisamente allE donde resalta la imortancia de la Estadística Descriptiva como rimer aso en la determinación de decisiones e inferencias ue ueden concluirse de la $ariada información ue nos lle"a en forma de datos numKricos. Con el resente módulo@ se busca ue el estudiante se encuentre en caacidad de interretar@ discriminar ! relacionar los fundamentos básicos de la Estadística Descriptiva @ a tra$Ks del análisis de datos tomados de un fenómeno roio de su discilina ! ue describa@ eQamine ! sintetice adecuadamente la información mediante mKtodos estadEsticos sencillos. 0l curso acadKmico de Estadística Descriptiva hace arte de la formación básica discilinar de los ro"ramas ue oferta la #ni$ersidad %acional Abierta ! a &istancia #%A&. Consta de dos ;N crKditos acadKmicos@ el sistema adotado or la #%A& como estándar curricular en la formación uni$ersitaria@ ! es de tio teór teóric ico@ o@ en tant tanto o ue ue busc busca a la iden identitififica caci ción ón ! el reco recono noci cimi mien ento to de las las roblemáticas@ ersecti$as teóricas@ concetos@ cate"orEas@ mKtodos ! tKcnicas indisensables ara la formación rofesional. 0ste teQto contiene dos unidades didácticas '@ correlacionadas directamente con el nDmero de crKditos acadKmicos acadKmicos asi"nados al curso acadKmico. La rimera de ellas@ considera los Concetos /ásicos necesarios ara el cumlimiento de los roósitos ! obeti$os del curso. 0n esta unidad se identifican al"unos concetos estadEsticos como oblación@ muestra@ $ariable@ dato@ etc.O ! se reconocen cada uno de los asos a se"uir ara una correcta ! acertada in$esti"ación estadEstica como son la laneación@ la recolección de la información@ su or"anización ! su resentación "ráfica. 0n la se"unda unidad didáctica se reconocen al"unas de las medidas estadEsticas más comunes@ tanto uni$ariantes como /i$ariantes. 0ntre las rim rimera erass se cont contem eml lan an las las medi medidas das de tend tendenc encia ia cent centra ral@l@ las las medi medida dass de
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UNIVERSIDAD NACIONAL ABIERTA Y A DISTANCIA – UNAD
ESCUELA DE CIENCIAS BÁSICAS, TECNOLOGÍA E INGENIERÍA CONTENIDO DIDÁCTICO DEL CURSO: 100105 – Estadística Desci!ti"a
estudiantes ara ue lo"ren@ mediante análisis cuantitati$os@ la interretación de diferentes fenómenos roios de su discilina de formación ! del entorno social@ económico ! olEtico. Aunta al maneo estadEstico de datos@ dar las autas en la recolección laneada de los mismos ! roorcionar un conunto de tKcnicas a artir de las cuales se lo"ra resentar@ resumir e interretar datos ue ueden corresonder a una muestra o a un "ruo total. 0l módulo no retende reemlazar las diferentes referencias biblio"ráficas clás clásic icas as de la 0sta 0stadE dEst stic ica@ a@ busca busca entr entre"a e"arr los los conc concet etos os de un modo modo más más didáctico@ enfocado en el autoarendizae ! en relación directa con la GuEa de Acti$idades referenciada en el rotocolo del resente curso. Al final de cada unidad@ el estudiante encontrará las referencias biblio"ráficas básicas@ ero no Dnic Dnicas@ as@ ara ara ue ue con con ella ellass refu refuer erce ce en conce concet tos os ! defi defini nici cione ones. s. Además Además@@ encont encontrará rará una serie serie de á"ina á"inass eb recomenda recomendadas das ue amlEan amlEan los temas trat tratad ados os.. Se trat trata a ues ues de un mate materi rial al didá didáct ctic ico o de ao! ao!o o ara ara el curs curso o de Estadística Estadística Descriptiva Descriptiva de la #%A&@ como arte de las diferentes ! di$ersas herramientas didácticas en las ue se ao!a el arendizae autónomo.
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UNIVERSIDAD NACIONAL ABIERTA Y A DISTANCIA – UNAD
ESCUELA DE CIENCIAS BÁSICAS, TECNOLOGÍA E INGENIERÍA CONTENIDO DIDÁCTICO DEL CURSO: 100105 – Estadística Desci!ti"a
Unidad Did4#ti#a Uno
CO'CEPTOS PRELIMI'ARES ) MEDIDAS DE TE'DE'CIA CE'TRAL
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UNIVERSIDAD NACIONAL ABIERTA Y A DISTANCIA – UNAD
ESCUELA DE CIENCIAS BÁSICAS, TECNOLOGÍA E INGENIERÍA CONTENIDO DIDÁCTICO DEL CURSO: 100105 – Estadística Desci!ti"a
U'IDAD 1$ CO'CEPTOS PRELIMI'ARES ) MEDIDAS DE TE'DE'CIA CE'TRAL
Lecc Lecció ión n'
8ue es la est estadE adEstic stica a
Lección ;
Conc oncetos /ásicos
Lecc Lecció ión n)
1ari 1ariab ablles est estad adEs Estticas icas
Lecc Lecció ión n?
Carac aractteri erizaci zación ón de una $ariable cualitati$a
Lecc Lecció ión n
+abl +ablas as de cont contiin"en n"enci cia a
CAPITULO 1. GENERALIDADES GENERALIDADES Y VARIABLES VARIABLES ESTADÍSTICAS ESTADÍSTICAS
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UNIVERSIDAD NACIONAL ABIERTA Y A DISTANCIA – UNAD
ESCUELA DE CIENCIAS BÁSICAS, TECNOLOGÍA E INGENIERÍA CONTENIDO DIDÁCTICO DEL CURSO: 100105 – Estadística Desci!ti"a
Lecc Lecció ión n '' '' 0sta 0stadE dE"r "raf afos os.. Lecc Lecció ión n '; Medi Media a Arit AritmK mKtitica ca CAPITULO 3.
Lecc Lecció ión n ') Media ediana na..
MEDIDAS DE TENDENCIA CENTRAL O DE POSICIÓN
Lección '? '? Moda Lecc Lecció ión n ' ,tra ,trass medi medida dass de de tendencia central.
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UNIVERSIDAD NACIONAL ABIERTA Y A DISTANCIA – UNAD
ESCUELA DE CIENCIAS BÁSICAS, TECNOLOGÍA E INGENIERÍA CONTENIDO DIDÁCTICO DEL CURSO: 100105 – Estadística Desci!ti"a
0n el caEtulo uno se amliará al"unas definiciones de tKrminos básicos de la estad estadEs Estitica ca como como obl oblaci ación@ ón@ mues muestr tra@ a@ $ari $ariab able le@@ dato dato@@ etc. etc.@@ busca buscand ndo o ue ue el estudiante los identifiue en eemlos sencillos de la $ida diaria. 0n el caEtulo dos se reconocerán cada uno de los asos a se"uir ara una correcta ! acertada in$esti"ación estadEstica como son la laneación@ la recolección de la información@ su or"anización ! su resentación "ráfica ! en el caitulo *** se mencionaran las medidas de tendencia central ! de osición ara lo cual se hace indisensable recordar al"unas nociones aritmKticas ! al"ebraicas básicas en estadEstica@ es or esto ue se recomienda al lector iniciar el caEtulo reasando la sumatoria como roiedad roiedad aritmKtica aritmKtica fundamental fundamental ara entender entender las medidas estadEstic estadEsticas as de una oblación o muestra. +odo cuanto tiene ue $er con sumatoria ! roductoria uede ser reasado ! consultado en el aneQo A@ ue se encuentra al final del teQto.
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UNIVERSIDAD NACIONAL ABIERTA Y A DISTANCIA – UNAD
ESCUELA DE CIENCIAS BÁSICAS, TECNOLOGÍA E INGENIERÍA CONTENIDO DIDÁCTICO DEL CURSO: 100105 – Estadística Desci!ti"a
in$esti"ación estadEstica. • Ad$ertir la imortancia de las distribuciones de frecuencias ara la
descrición de datos. • Alicar los concetos de frecuencia@ marca de clase ! distribución de
frecuencias a un conunto de datos estadEsticos. • Construir diferentes tios de distribuciones de frecuencias ara conuntos
de datos. • Reconocer al"unas caracterEsticas ue debe tener una "ráfica ara ue
reresente meor una situación.
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UNIVERSIDAD NACIONAL ABIERTA Y A DISTANCIA – UNAD
ESCUELA DE CIENCIAS BÁSICAS, TECNOLOGÍA E INGENIERÍA CONTENIDO DIDÁCTICO DEL CURSO: 100105 – Estadística Desci!ti"a
artiendo de una in$esti"ación realizada en un "ruo menor@ denominado muestra cu!os elementos se seleccionan aleatoriamente o al azar.
1$1$1$ LECCIO' LECCIO' 1$ @. @. e% la E%tad> E%tad>%ti#aB %ti#aB Antes de dar a conocer los concetos de los tKrminos estadEsticos ue lle$en a entablar el len"uae comDn ue se utilizará en adelante@ es necesario saber uK es la 0stadEstica ! en uK consiste la 0stadEstica &escriti$a. 0mEricamente se sabe ue la 0stadEstica tiene ue $er con datos ! la manera en ue estos son a"ruados. 0sto se reconoce en muchos casos de la $ida cotidiana ue in$olucran información numKrica ! el conteQto en ue esta
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UNIVERSIDAD NACIONAL ABIERTA Y A DISTANCIA – UNAD
ESCUELA DE CIENCIAS BÁSICAS, TECNOLOGÍA E INGENIERÍA CONTENIDO DIDÁCTICO DEL CURSO: 100105 – Estadística Desci!ti"a
Si esto es asE@ 7uK le corresonde entonces a la Estadística Descriptiva : 0sta tiene or fin ele$ar los asectos caracterEsticos del "ruo de datos ero sin intentar obtener más conocimiento del ue ueda aduirirse or sE mismos. 0s or ello ue la 0stadEstica &escriti$a es el unto de artida del análisis de un "ruo de datos ue in$olucran una cierta comleidad@ o bien uede ser el todo de un análisis básico ! limitado del "ruo de datos.
1$1$/$ LECCIÓ' LECCIÓ' /$ CO'CEP CO'CEPTOS TOS (+SICOS (+SICOS Población es el conunto de medidas@ indi$iduos u obetos ue comarten
una caracterEstica en comDn. La oblación se basa en cuatro caracterEsticasB contenido@ tio de unidades ! elementos@ ubicación esacial ! ubicación temoral. &e la oblación es eQtraEda la muestra..
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UNIVERSIDAD NACIONAL ABIERTA Y A DISTANCIA – UNAD
ESCUELA DE CIENCIAS BÁSICAS, TECNOLOGÍA E INGENIERÍA CONTENIDO DIDÁCTICO DEL CURSO: 100105 – Estadística Desci!ti"a
Parámetro son auello auelloss $alores $alores ue caracte caracteriz rizan an numKric numKricame amente nte a la
oblación como tal. 0l arámetro oblacional de interKs es Dnico media@ $arianza@ etc.N@ ero una oblación uede tener muchas caracterEsticas o arámetros de interKs. 2or el contrario@ un estadístico es una ma"nitud corresondiente a una muestra aleatoria eQtraEda de la oblación@ or lo ue cambiando la muestra cambiará entonces el estadEstico media muestral@ $arianza muestral@ etc.N. 0n ocas alabras alabras se uede decir ue arámetro arámetro es a oblación como estadEstico estadEstico es a muestra. 0s comDn desi"nar los arámetros con letras minDsculas del alfabeto "rie"o ! los estadEsticos con letras de nuestro alfabeto. 0n la #nidad &idáctica &os@ se amliará más estos dos concetos.
E?EMPLO /$1$
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! aleatoria orue debe ser esco"ida al azar@ de tal manera ue uien realiza el estudio no ueda influir en la elección de los indi$iduos or encuestar. Cada Cada elem elemen ento to de la obl oblac ació ión n debe debe tene tenerr la mism misma a oor oortu tuni nida dad d de ser ser seleccionado. 0l muestre aleatorio uede serB
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UNIVERSIDAD NACIONAL ABIERTA Y A DISTANCIA – UNAD
ESCUELA DE CIENCIAS BÁSICAS, TECNOLOGÍA E INGENIERÍA CONTENIDO DIDÁCTICO DEL CURSO: 100105 – Estadística Desci!ti"a
0n el caitulo anterior nos dedicamos a estudiar los mKtodos ue deben ser alicados en el roceso de a"ruar@ or"anizar ! resentar los datos en cualuier tio de in$esti"ación estadEstica. Ahora trataremos de resentar otros mKtodos ara estudiar o medir el comortamiento de los elementos ue constitu!en una oblación. Si bien es cierto ue ue los los cuad cuadro ross ! "raf "rafic icas as desc descri ribe ben n el fenó fenóme meno no@@ no lo hace hacen n en form forma a satisfactoria@ ! or tanto ha! necesidad de acudir a ciertas medidas denominadas parámetros de la obla oblaci ción ón@@ cuan cuando do se hace hacen n sobr sobre e el total otal de Kst Ksta ! estadígrafos( cuando corresonden a una arte de la oblación o muestra.. Los estadE"rafos ermiten hallar un $alor numKrico@ el mismo ue reresenta a la muestra. +eniendo en cuenta la anterior la clasificación@ los sEmbolos ue se refieren a la obl oblac ació ión n se rere reresen senta tan n "ener "eneral alme ment nte e or or letras letras "rie "rie"as "as o or las las letr letras as ma!Dsculas de nuestro alfabeto ! los ue se refieren a la muestra@ or letras minDsculas. 2ara 2ara el anál anális isis is de una una $ari $ariab able le o de una una dist distri ribu buci ción ón unid unidim imen ensi sion onal al se consideran las si"uientes clases de medidasB • Medidas de tendencia central • Medidas de osición • Medidas de disersión • Medidas de asimetrEa ! auntamiento
1$7$/ LECCIÓ' 1/ MEDIA ARITMTICA 0s la medida más conocida ! la más fácil de calcular. Se define como la suma de los $alores de una cantidad dada de nDmeros di$idido entre la cantidad de nDmeros. n
∑ X
i
x
=
i =1
n
dondeB n Z cantidad de elementos X i i Z $alor de cada elemento x Z media aritmKtica@ o simlemente s imlemente media
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UNIVERSIDAD NACIONAL ABIERTA Y A DISTANCIA – UNAD
ESCUELA DE CIENCIAS BÁSICAS, TECNOLOGÍA E INGENIERÍA CONTENIDO DIDÁCTICO DEL CURSO: 100105 – Estadística Desci!ti"a
0l recio de la bolsa de un litro de leche en diferentes suermercados fueB '.)((@ '.) '.)( (@@ '.; '.;(@ (@ '.? '.?(( (( ! '.); '.);. . 0l $alo $alorr rom romedi edio o o medi media a arit aritmKt mKtic ica a es entoncesB
x =
1.300 300 +1.350 +1.250 +1.400 400 +1.325 5
= $1.325
La media aritmKtica tiene la roiedad de asi"nar a cada elemento de la suma el mismo $alor@ o sea el $alor romedio. Si se cono conoce ce el $alor alor de la media edia ! el nDme nDmero ro n de elem elemen enttos u obser$aciones@ obser$aciones@ se uede conocer el $alor de la suma total multilican multilicando do la media or el nDmero de elementos. 0sto esB n
∑ X
i
= n ⋅ x
i =1
E?EMPLO 1/$/ 110$110$1 Las $entas romedio de un almacKn durante el rimer semestre del ao fueron )?;;.(((O hallar el total de $entas de este erEodo de tiemo. 1enta total rimer semestre Z 5 Q )?;;.(((N Z ;();.((( +ambiKn uede suceder ue los elementos ue se analizan se encuentren a"ruados@ en este caso ara encontrar el $alor de la media aritmKtica se debe realizar la onderación de estos elementos a"ruados@ es decir@ encontrar el eso ue le corresonde a cada $alor. 0sto da lu"ar a la media aritm%tica ponderada .
E?EMPLO 1/$7 110$110$1 #n a"ricultor $ende la cosecha de aas de la si"uiente formaB )( sacos a ;5.(((@ '= sacos a ;5?.((( ! ; sacos a ;5'.((. 7Cuál es el recio romedio del saco de aa $endida or el a"ricultor:
2recio romedio saco de aa Z Z#5)*(5&
30(256.000) + 18( 264.000) + 25( 261.500) 30 + 18 + 25
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UNIVERSIDAD NACIONAL ABIERTA Y A DISTANCIA – UNAD
ESCUELA DE CIENCIAS BÁSICAS, TECNOLOGÍA E INGENIERÍA CONTENIDO DIDÁCTICO DEL CURSO: 100105 – Estadística Desci!ti"a n
x
=
m1 X 1 . + m2 X 2 . + + mn X n . m1 + m2
+ + mn
∑ m X = ∑m i
i
i =1 n
i
i =1
Sien Siendo do X 1 X 2,… onderaciones.
X n@
las cantidades onderadas !
m1, m 2,,…, ,,…, mn los
esos o
#n caso similar al anterior consiste en la media de una distribución de frecuencias frecuencias agrupadas@ donde los esos o onderaciones onderaciones corresonderEan corresonderEan a las frecuencias de los $alores de las marcas de clase@ recordando ue la marca de clase es el $alor romedio de un inter$alo de clase. 0sta similitud entre la media de una distribución distribución de frecuencias frecuencias a"ruadas ! la media aritmKtica aritmKtica onderada se muestra en el si"uiente eemlo.
E?EMPLO 1/$: 110$110$1 &ada &ada la si"u si"uiient ente dist distri ribu buci ción ón de corresondiente media aritmKticaB
frecu recuen enci cias as a"ru a"rua ada das@ s@ calc calcul ule e
Ta&la 7$1$ &istribución de frecuencias a"ruadas Inter"alo '5I;( ;'I; ;5I)( )'I) )5I?( Total
Mar#a de #la%e
Fre#en#ia
X
f
'= ;) ;= )) )=
? 5 6 ) /5
x =
∑ f ⋅ X . = 685 = 27.4 ∑ f 25
f .X
6; ')= 'F5 '5 ''? ;=5
su
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UNIVERSIDAD NACIONAL ABIERTA Y A DISTANCIA – UNAD
ESCUELA DE CIENCIAS BÁSICAS, TECNOLOGÍA E INGENIERÍA CONTENIDO DIDÁCTICO DEL CURSO: 100105 – Estadística Desci!ti"a n
∑ f X ∑ f ⋅ X x = = n ∑ f i
i
i =1 n
i
i =1
&ada la imortancia ue tiene el cálculo de la media aritmKtica ! su frecuente uso@ se hace necesario considerar al"unas de sus roiedadesB • "a suma de las desviaciones respecto a la media aritm#tica es igual a cero.
#na des$iación es la diferencia ue se resenta entre los $alores ue toma la $ari $ariab able le ! un $alo $alorr const constat ate@ e@ en este este caso caso es la medi media a arit aritmK mKtitica ca.. 0sta 0sta roiedad@ al i"ual ue las demás@ es $álida ara datos a"ruados o no a"ruados. > en tKrminos aritmKticos ella lanteaB ∑( X − x ) = 0 +en"a en cuenta ue cuando los datos están a"ruados en una tabla de frecuencias@ las des$iaciones con resecto a la media deben onderarse. Si la distribución es simKtrica no ha! necesidad de onderar. • "a suma de los cuadrados de las desviaciones desviaciones respecto respecto a la media es siempre menor menor $ue la suma suma de los los cuadr cuadrado ados s de las las desvi desviac acion iones es con con respec respecto to a cual$uier otro valor.
0sto uiere decir ue sólo la media aritmKtica hace mEnima la suma de los cuadrados de las des$iaciones en torno a ella . 0sta imortante roiedad se retomará más adelante cuando se estudie re"resión lineal ! el mKtodo de los mEnimos cuadrados ara auste de cur$as. 0n sEntesis@ la media o romedio aritmKtico es la medida de tendencia central más comDnmente usada@ además de ser la Dnica de las medidas de tendencia cent centra rall ue ue erm ermitite e un trat tratam amien iento to al"e al"ebr brai aico. co. Sin Sin emba embar"o r"o no siem siemre re es recomendable usarla como un romedio@ !a ue es mu! sensible a los $alores eQtremos del conunto de datos. 2or otra arte@ la media es li"eramente más difEcil de calcular a mano ue las otras medidas ue se $erán en se"uida@ uesto ue reuiere sumar todo el conunto de datos@ ue bien odrEan ser bastantes@ ! di$idir entre el nDmero de elementos del conunto.
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UNIVERSIDAD NACIONAL ABIERTA Y A DISTANCIA – UNAD
ESCUELA DE CIENCIAS BÁSICAS, TECNOLOGÍA E INGENIERÍA CONTENIDO DIDÁCTICO DEL CURSO: 100105 – Estadística Desci!ti"a
La medi median ana a se simb simbol oliz iza a como como Me. 0s meno menoss usad usada a ue ue la medi media a aritmKtica. 2ara su cálculo es necesario ue los datos estKn ordenados. Cuando la cantidad de datos es imar@ fácilmente se identifica la medianaO ero cuando el nDmero de datos es ar@ la mediana se calcula hallando el $alor medio entre los dos $alores centrales ! no coincidirá con nin"uno de los $alores del conunto de datos.
E?EMPLO 17$1 110$110$1 &ados los $aloresB 'F@ '@ ;)@ ;=@ '?@ ;5@ '=@ ;(@ )(@ determinar su media. a$ Lo rimero ue debe hacerse es ordenar los datosB '?
'
'=
'F
/0
;)
;5
;=
)(
Como el nDmero de datos es F@ el $alor del medio de estos datos es la mediana@ uesto ue dea cuatro $alores or debao ! cuatro $alores or encima. 0ste $alor es /0.
&$
Hallar la media del si"uiente conunto de datos ordenadosB
'?
'
'=
'F
/0
/7
;5
;=
)(
);
,bser$e ue son '( datos@ un nDmero ar de datos. 0n este caso se toman los dos $alores del medio ! se romedianB Me =
20 + 23 2
= 21.5
Cuando los datos se encuentran a"ruados@ a"ruados@ se calcula calcula el $alor de
n 2
! con Kl se
busca@ en las frecuencias acumuladas@ el inter$alo de clase en donde este se encue encuent ntra ra o se aroQ aroQim ime e meo meor. r. 0sta 0sta clase clase recibe recibe el nomb nombre re de clase de la mediana. *dentificada la clase de la mediana@ se considera ue los $alores en esa clase se distribu!en uniformemente de modo ue se ueda calcular la mediana or el mKtodo de la interolación lineal. 0n el si"uiente eemlo se describe aso a aso el cálculo de esta medida de tendencia central.
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UNIVERSIDAD NACIONAL ABIERTA Y A DISTANCIA – UNAD
ESCUELA DE CIENCIAS BÁSICAS, TECNOLOGÍA E INGENIERÍA CONTENIDO DIDÁCTICO DEL CURSO: 100105 – Estadística Desci!ti"a
2rimero se identifica la clase de la mediana la clase ue contiene a la medianaN. n
2
=
25 2
= 12.5
La clase de la mediana es ;5I)(N@ ues el nDmero de frecuencias acumuladas es el $alor más cercano a ';..
Ta&la 7$/$ &istribución de frecuencias a"ruadas Inter"alo '5I;( ;'I; ;5I)( )'I) )5I?( Total
Fre#en#ia F
? 5 6 ) /5
Fre#en#ia a#2lada ? 10 '6 ;; ;
Clase de la mediana
Ha! '( obser$aciones or debao del lEmite inferior de la clase de la mediana. 12.5 −10 = 2.5
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UNIVERSIDAD NACIONAL ABIERTA Y A DISTANCIA – UNAD
ESCUELA DE CIENCIAS BÁSICAS, TECNOLOGÍA E INGENIERÍA CONTENIDO DIDÁCTICO DEL CURSO: 100105 – Estadística Desci!ti"a
0n muchas muchas referen referencia ciass biblio biblio"ráf "ráfica icass se eQone eQone una ecuaci ecuación ón ara el cálculo de la mediana cuando los datos se encuentran a"ruados. 0lla se deri$a del análisis hecho en el eemlo anterior ! se describe de la si"uiente maneraB n Me = 2
− F k −1 f k
× Ak + Lk
&ondeB n es el tamao de la muestra o la suma de todas las frecuencias. F k-1 k-1 es la frecuencia absoluta acumulada de la clase anterior de la clase de la mediana. f k k es la frecuencia absoluta de la clase de la mediana. Ak es la amlitud de la clase de la mediana. Lk es el lEmite real inferior de la clase de la mediana.
E?EMPLO 17$7 10$110$1 del eemlo &etermine la mediana de la distribución de frecuencias1a"ruadas ').;.@ haciendo uso de la ecuación ara su cálculo. 2rimero@ se identifica cada $alorB n Z ; F k-1 k-1 Z '( f k k Z 6 Ak Z ? Lk Z ;5 n Me =
2
− F k −1 f k
× Ak + Lk
⇒
25 − 10 2 Me = × 4 + 26 = 1.4 + 26 = 27.4 7
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UNIVERSIDAD NACIONAL ABIERTA Y A DISTANCIA – UNAD
ESCUELA DE CIENCIAS BÁSICAS, TECNOLOGÍA E INGENIERÍA CONTENIDO DIDÁCTICO DEL CURSO: 100105 – Estadística Desci!ti"a
Se conclu!e entonces ue la mediana no está afectada or los $alores eQtremos del conunto de datos@ sean estos "randes o eueos. %o influencian en lo absoluto como sE lo hacen en el cálculo de la media. Cuando la distribución de los datos es mu! simKtrica@ no ha! casi diferencia entre la media ! la mediana. 0l cálculo de la mediana es simle@ ero siemre reuiere ue los datos se encue encuent ntre ren n orde ordenad nados@ os@ condi condici ción ón ue ue no reu reuie iere re el cálcu cálculo lo de la medi media. a. 4inalmente@ se odrEa decir ue la mediana no es una medida mu! confiable ara describir el conunto de datos@ ues en su cálculo sólo inter$ienen los $alores más centrales sin tener en cuenta los demás ! su comortamiento "eneral.
1$7$ 1$7$<< LECC LECCIÓ IÓ' ' 1: 1: MOD MODA A Se trata del $alor más frecuente en un conunto de datos. Se considera como el $alor más reresentati$o reresentati$o o tEico de una serie de $alores. 0s simbolizada simbolizada como Mo. Si dos $alores tienen la misma frecuencia se dice ue el conunto es bimodal . Cuando más de dos $alores ocurren con la misma frecuencia ! Ksta es la más alta@ todos los $alores son modas@ or lo ue el conunto de datos recibe el nombre de multimodal o o polimodal . Cuando los datos se encuentran a"ruados la moda es la marca de clase del inter$alo de clase ue contiene la ma!or frecuencia. 0s usual tambiKn hacer uso de la si"uiente ecuación ara su cálculoB Mo =
&ondeB
f k −1 f k −1 + f k +1
× Ak + Lk
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UNIVERSIDAD NACIONAL ABIERTA Y A DISTANCIA – UNAD
ESCUELA DE CIENCIAS BÁSICAS, TECNOLOGÍA E INGENIERÍA CONTENIDO DIDÁCTICO DEL CURSO: 100105 – Estadística Desci!ti"a
es decir untos ue corresonden a una ma!or densidad de frecuencias. 0sto sucede cuando se trabaa con "ruos de datos hetero"Kneos.
E?EMPLO 1:$1 110$110$1 Las si"uientes si"uientes tablas de frecuencias indican indican el nDmero de ersonas de acuerdo a su edad ue asistieron al estreno de una elEcula. 0n la tabl tabla a ).). ).).@@ dond donde e los los dato datoss está están n sin sin a"ru a"rua ar@ r@ la moda moda es //@ $alo $alorr corresondiente a la ma!or frecuencia ue es . 0n la tabla ).?.@ los datos se encuentran a"ruados@ la moda se encuentra en el inte inter$ r$al alo o de clase clase 'F. 'F. ;;. ;;. ! corre correso sond nde e a la marca marca de clas clase e ue ue es /1. Además@ usando la ecuación ara el cálculo de la moda@ se tieneB Mo =
9 9 +9
× 3 + 19.5 = 21
,bsK ,bsKr$ r$es ese e ue ue aun aunue ue sean sean el mism mismo o con conun unto to de dato datos@ s@ la moda moda $arE $arEa a deendiendo de su tratamiento@ es decir@ de cómo estos se a"ruan.
Ta&la 7$7$ &istribución de de frecuencias de la asistencia a cine
Ta&la 7$:$ &istribución de de frfrecuencias a" a"ruadas de la asistencia a cine Inter"alo% de #la%e
Mar#a de #la%e
Fre#$
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UNIVERSIDAD NACIONAL ABIERTA Y A DISTANCIA – UNAD
ESCUELA DE CIENCIAS BÁSICAS, TECNOLOGÍA E INGENIERÍA CONTENIDO DIDÁCTICO DEL CURSO: 100105 – Estadística Desci!ti"a X
f
X
F
'? ' '5 '6 '= 'F ;( ;' //
' ( ' ; ) ? ? ? Total
;) ;? ; ;5 ;6 ;= ;F )( )'
? ) ; ? ) ; ( ( ' :7
La moda no es tan usada como la media o la mediana. mediana. 2ara encontrarla encontrarla se reuiere reuiere ue los datos estKn ordenados. ordenados. Su cálculo cálculo es oco reciso debido debido a ue no se uede eQresar en tKrminos al"ebraicos. Se han han $ist $isto o hast hasta a ahora ahora tres tres medi medida dass de tend tendenc encia ia cent central ralBB medi media@ a@ mediana ! moda. &eterminar cuál de ellas usar en un tratamiento estadEstico deende mucho de la información información ue se ten"a ! del obeti$o ue se ersi"ue. La media@ a diferencia de la mediana ! la moda@ resenta una li"era estabilidad en el muestreo@ es or eso ue su uso es más frecuente. Si la distribución es casi simKtrica@ cualuiera de ellas uede usarse ! resultarán aroQimadamente i"uales.
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UNIVERSIDAD NACIONAL ABIERTA Y A DISTANCIA – UNAD
ESCUELA DE CIENCIAS BÁSICAS, TECNOLOGÍA E INGENIERÍA CONTENIDO DIDÁCTICO DEL CURSO: 100105 – Estadística Desci!ti"a
medi median ana a se encu encuen entr tran an a la izu izuie ierd rda a de la moda moda.. Mient Mientras ras ue ue los los dato datoss ses"ados a la derecha sesgo positivo N oseen una cola derecha más lar"a ! su mediana ! media están a la derecha de la moda $er fi"ura ).'.N Fira 7$1$
&istribuciones ses"adas aN Ses"ada a la derechaO bN Ses"ada a la izuierdaO cN SimKtrica
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UNIVERSIDAD NACIONAL ABIERTA Y A DISTANCIA – UNAD
ESCUELA DE CIENCIAS BÁSICAS, TECNOLOGÍA E INGENIERÍA CONTENIDO DIDÁCTICO DEL CURSO: 100105 – Estadística Desci!ti"a
Medida de tenden#i a #entral
Media
Mediana
Moda
@To2a @Se "e @Reiere @. tan en ae#tada e lo% @Ei%te #o2Hn #enta or lo% dato% e%t.n %ie2reB e%B #ada "alore% ordenado%B "alorB etre2o%B 0s la más comDn
Si
Si
Si
%o
&e uso comDn
Si
%o
%o
Si
%o
%o
Si
2odrEa no eQistir o #sada en haber ocasiones más de
VentaGa% 3 de%"entaGa% 2resenta una li"era estabilidad frente al muestreo. %o es mu! confiable ara describir el conunto de datos@ ues en su cálculo sólo inter$ienen los datos más centrales. 0s más recisa cuando los datos no están a"ruados.
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UNIVERSIDAD NACIONAL ABIERTA Y A DISTANCIA – UNAD
ESCUELA DE CIENCIAS BÁSICAS, TECNOLOGÍA E INGENIERÍA CONTENIDO DIDÁCTICO DEL CURSO: 100105 – Estadística Desci!ti"a
Mg = n
n
∏ X
ni
i
= n X 1n ⋅ X 2 n ⋅ ...⋅ X n n 1
2
i
i =1
a$
E?EMPLO 15$1 110$110$1
Hallar la media "eomKtrica de ;@ ?@ 5@ F@ ';@ ' Mg = 6
6
∏ X = i
6
2 ⋅ 4 ⋅ 6 ⋅ 9 ⋅ 12 ⋅ 15 = 6 77.760 = 6.53
i =1
b. Hall Hallar ar la medi media a "eom "eomKt Ktri rica ca de la si"u si"uie ient nte e dist distri ribu buci ción ón de frec frecue uenc ncia iass a"ruadas. Ta&la 7$;$ &istribución de frecuencias a"ruadas
Inter"alo% #la%e (. '. '. ;. ;. ). ) ?
de Ma Mar# r#a% a% de #la%e #la%e Fre# Fre#e en# n#ia ia%% ' ; ) ?
; =
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UNIVERSIDAD NACIONAL ABIERTA Y A DISTANCIA – UNAD
ESCUELA DE CIENCIAS BÁSICAS, TECNOLOGÍA E INGENIERÍA CONTENIDO DIDÁCTICO DEL CURSO: 100105 – Estadística Desci!ti"a
1 Mh
=
1 50
+
2
1 40
=
0.045 1 = 0.0225 ⇒ Mh = = 44.44 2 0.0225
??.?? minutos es el tiemo medio reuerido. Los cuartile cuartiles( s( deciles deciles ! percentiles son medidas ue se utilizan ara dete determ rmin inar ar los los inte inter$ r$al alos os dent dentro ro de los los cual cuales es ued uedan an roor roorci cion onal alme ment nte e reartidos los tKrminos de la distribución. 2ara calcular los cuartiles se di$ide la distribución en cuatro artes i"uales@ de manera ue cada una tendrá el ;W de las obser$aciones. obser$aciones. Los tres untos de searación de los $alores son los cuartiles. 0l cuartil inferior Q1N es auel $alor de la $ariable ue reresenta el ;W de las obser$aciones ! a la $ez@ es suerado or el 6W restante. 0l se"undo cuartil Q 2N corresonderá a la mediana de la distribución. 0l tercer cuartil Q3N es auel $alor ue reresenta el 6W ! es suerado or el ;W restante de las obser$aciones. 2ara calcular estos tres romedios se rocede de manera semeante al cálculo de la media aritmKtica.
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UNIVERSIDAD NACIONAL ABIERTA Y A DISTANCIA – UNAD
ESCUELA DE CIENCIAS BÁSICAS, TECNOLOGÍA E INGENIERÍA CONTENIDO DIDÁCTICO DEL CURSO: 100105 – Estadística Desci!ti"a
Clase del Q1 Clase del Q3
Ha! ? obser$aciones or debao del lEmite inferior de la clase del rimer cuartil. 6.25 − 4 = 2.25
0l $alor de ;.; se interola en la amlitud de la clase del rimer cuartil ue es ?. 4recuencia absoluta 5 ;.; X =
Ancho de clase ? 8 2.25 × 4 6
= 1.5
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UNIVERSIDAD NACIONAL ABIERTA Y A DISTANCIA – UNAD
ESCUELA DE CIENCIAS BÁSICAS, TECNOLOGÍA E INGENIERÍA CONTENIDO DIDÁCTICO DEL CURSO: 100105 – Estadística Desci!ti"a
4recuencia absoluta '.6 X =
Ancho de clase ? 8 1.75 × 4 5
= 1.4
0l tercer cuartil se encontrará '.? unidades más del lEmite inferior de su claseB Q3 = 31 +1.4 = 32.4
Lo ue uiere decir ue el ;W de los $alores está or debao de ;;.O el (W
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ESCUELA DE CIENCIAS BÁSICAS, TECNOLOGÍA E INGENIERÍA CONTENIDO DIDÁCTICO DEL CURSO: 100105 – Estadística Desci!ti"a
&istribución de frecuencias a"ruadas Inter"alo
Fre#en#ia a&%olta
Fre#en#ia a#2lada
Fre#en#ia relati"a
'5I;( ;'I; ;5I)( )'I) )5I?( Total
? 5 6 ) /5
? '( '6 ;; ;
'5 W ;? W ;= W ;( W '; W 100
Fre#en#ia relati"a a#2lada '5 W ?(W 5=W ==W '((W
Con Con las las frecu frecuen enci cias as relat relati$ i$as as acumu acumula ladas das se const constru! ru!e e la oi$ oi$a a orc orcent entua uall
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ESCUELA DE CIENCIAS BÁSICAS, TECNOLOGÍA E INGENIERÍA CONTENIDO DIDÁCTICO DEL CURSO: 100105 – Estadística Desci!ti"a
&ia"rama de fluo ara el cálculo del IKsimo ercentil Hacer arre"lo ordenado ascendenteN
Calcular &ondeB n Z nDmero de $alores Z ercentil en cuestión
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ESCUELA DE CIENCIAS BÁSICAS, TECNOLOGÍA E INGENIERÍA CONTENIDO DIDÁCTICO DEL CURSO: 100105 – Estadística Desci!ti"a
no es un nDmero entero@ este $alor debe ser redondeado al si"uiente entero más "rande ! el $alor de 2 corresonderá a la osición LIKsima.
E?EMPLO 15$5 10$110$1#no sobre la +ome el arre"lo ordenado del eemlo =.'.@ de la #nidad1&idáctica e$aluación de los latidos cardEacos de un "ruo de )( ersonas desuKs de cierta acti$idad fEsica ! calcule los si"uientes ercentiles. = 5( ;/ ;:
6( 6? <5 65
=( =; =/ =;
= = = =6
== F' F' F;
F? F F F
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0l $alor del ercentil ='@ P81 #$ Se calcula el $alor de LB 81
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). ULa estadEstica estadEstica estudia estudia el comortamiento comortamiento de fenómenos fenómenos colecti$os colecti$os ! nunca de una obser$ación indi$idualT 78uK le dice este rinciio:
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AUTOEVALUACIÓ' AUTOEVALUACIÓ' LECCIO' 7 '. &eterminar &eterminar si cada una de las si"uientes si"uientes $ariables es cualitati cualitati$a $a o cuantitati$ cuantitati$a. a. 0n caso de ser cuantitati$a@ clasificarla en discreta o continua.
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UNIVERSIDAD NACIONAL ABIERTA Y A DISTANCIA – UNAD
ESCUELA DE CIENCIAS BÁSICAS, TECNOLOGÍA E INGENIERÍA CONTENIDO DIDÁCTICO DEL CURSO: 100105 – Estadística Desci!ti"a
2ara efectuar comaraciones entre series de obser$aciones distintas@ en estadE estadEsti stica ca se usa el coeficie coeficiente nte de variaci variación ón ! asE se uede determinar cuál serie tiene ma!or o menor $ariabilidad relati$a.
CV =
s x
100% × 100
Cuando el coeficiente de $ariación es mu! alto se dice ue la media aritmKtica no es lo suficientemente reresentati$a en la distribución.
/$1 /$1$7$1
De%"ia# a#ii!n 2ed 2ediia
Se define como la media aritmKtica de las des$iaciones resecto a la media@ tomadas en $alor absoluto . 0s una de las medidas más fáciles de calcular ! or ello@ mu! usada. 0lla toma todos los $alores de la $ariable ! es menos afectada ue la des$iación estándar or los $alores eQtremos. Su $alor siemre será menor ue la des$iación estándar. 2ara datos no a"ruadosB DM =
∑ X − x n
2ara datos a"ruadosB DM =
∑ f ⋅ X − x n
Cuanto ma!or sea el $alor de la des$iación media@ ma!or será la disersión de los datosO sin embar"o este $alor no roorciona una relación matemática recisa con la osición de un dato dentro de la distribución !@ uesto ue se toman los $alores absolutos@ mide la des$iación de una obser$ación sin determinar si está or encima o or debao de la media aritmKtica. &e la misma manera ue la des$iación estándar@ a la des$iación media uede calculársele el coeficiente de desviación media B
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ESCUELA DE CIENCIAS BÁSICAS, TECNOLOGÍA E INGENIERÍA CONTENIDO DIDÁCTICO DEL CURSO: 100105 – Estadística Desci!ti"a CVM CVM =
DM x
× 100%
E?EMPLO 1=$1 10$110$1 en miles de Los si"uientes datos corresonden a los salarios de '( 1emleados esosN de dos emresas de alimentos. alimentos. Calcular Calcular los coeficiente coeficientess de $ariación $ariación ! de des$iación media.
E2re%a A ?;( 5=( 5F( 6;( 6;( 6;( 6)( 6?( 6?( 65( E2re%a ( ?' ?=( '( 5( 6(( 6(( 6)( 6) 6( 65( E2re%a A Media aritmKticaB x = 692 1arianzaB s 2 = 8716 &es$iación estándarB s 93,36 &es$iación mediaB DM 57,2 =
=
Coeficiente de $ariaciónB
CV =
93,36 692
Coeficiente de des$iación mediaB
× 100% = 13,49%
CVM =
57,2 692
100% = 8,27% × 100
E2re%a ( Media aritmKticaB x = 643 1arianzaB s 2 = 14396 &es$iación estándarB s 119,98 &es$iación mediaB DM 104,86 =
=
Coeficiente de $ariaciónB
CV =
119,98 643
Coeficiente de des$iación mediaB
0l CVM es menor ue el des$iación estándar.
×100% = 18,66%
CVM =
CV debido debido
104,86 643
× 100% = 16,31%
a ue la des$iación media es menor ue la
0stos resultados lle$an a las si"uientes conclusionesB 0l salario romedio de los '( emleados de la emresa A es de 5F;.(((@
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UNIVERSIDAD NACIONAL ABIERTA Y A DISTANCIA – UNAD
ESCUELA DE CIENCIAS BÁSICAS, TECNOLOGÍA E INGENIERÍA CONTENIDO DIDÁCTICO DEL CURSO: 100105 – Estadística Desci!ti"a
• 0l coeficiente de $ariación ! el coeficiente de $ariación media de la emresa /
son menores a los coeficientes calculados ara la emresa A@ esto indica la $ariación relati$a de los salarios en ambas emresas.
/$1$: /$1 $: LECCIÓ LECCIÓ' ' 1 PU'TA? PU'TA?E E TÍPICO TÍPICO O ESTA'DA ESTA'DARI6A RI6ADO DO Cuando se tiene una distribución simKtrica@ su olE"ono de frecuencias re$elará una forma de camana mu! comDn en estadEstica. 0sta cur$a es llamada curva normal @ de error @ de probabilidad o o campana de auss. 0n ella la media aritmKtica aritmKtica se localiza en la mitad de la distribución. distribución. 0n el ee horizontal horizontal se ubican los $alores ue toma la $ariable ! en el $ertical la frecuencia absoluta o relati$a. 0l área bao la cur$a tendrá un $alor del '((W
Fira :$1$ Cur$a normal o camana de Gauss
0l punta/e típico o estandariado o variable normaliada @ es una medida de disersión mu! utilizada como $ariable estadEstica estadEstica en este tio de distribución@ distribución@ denominada distribución normal . 0l untae estandarizado mide la des$iación de una obser$ación con resecto a la media aritmKtica@ en unidades de des$iación estándar@ determinándose asE la osición relati$a de una obser$ación dentro del conunto de datos. 2or lo "eneral se simboliza or Z @ ero cuando el tamao de la muestra es menor de )(@ se simboliza or t .
Z =
X − x s
2or ser adimensional adimensional@@ el untae untae Z es es Dtil ara comarar datos indi$iduales indi$iduales de distribuciones ue tienen distintas unidades de medida@ asE como diferentes medias ! des$iaciones estándar. &entro de sus roiedades@ las más imortantes
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UNIVERSIDAD NACIONAL ABIERTA Y A DISTANCIA – UNAD
ESCUELA DE CIENCIAS BÁSICAS, TECNOLOGÍA E INGENIERÍA CONTENIDO DIDÁCTICO DEL CURSO: 100105 – Estadística Desci!ti"a
Al terminar el semestre@ un "ruo de '( estudiantes de rimer semestre de Re"encia de 4armacia del C0A& de MedellEn obtu$ieron los si"uientes resultados en el untae final de los cursos Ló"ica Matemática ! 0stadEstica &escriti$aB • Ló"ica MatemáticaB untuación media de ).F ! $arianza ).;. • 0stadEstica &escriti$aB untuación media de ).6 ! des$iación estándar '.6.
a$ &$
70n cuál curso hubo ma!or dispersión absoluta: 70n cuál hubo ma!or dispersión relativa : Si un estudiante obtu$o como nota final en Ló"ica Matemática ).= ! en 0stadEstica &escriti$a ).. 70n cuál curso fue su untuación relati$a suerior:
2ara determinar la disersión absoluta@ basta con hacer una comaración a$ entre sus des$iaciones estándar. ,bser$e ue en los datos suministrados@ !a se tien tiene e el $alo $alorr de la des$ des$ia iaci ción ón está estánd ndar ar de las las cali calififica caci cione oness de 0sta 0stadE dEst stic ica a &esc &escri rititi$a $a en camb cambio io@@ se tien tiene e la $ari $arianz anza a de las las cali califificac cacio ione ness de Ló"i Ló"ica ca Matemática. Recuerde ue la des$iación estándar es la raEz cuadrada de la $arianza. → s = 3.2 s = 3.2 = 1.79 2ara Ló"ica MatemáticaB Se tiene entonces ue en Ló"ica Matemática hubo una ma!or disersión absoluta@ ues '.6F'.6@ aunue no es mucha la diferencia. 2
2ara saber en cuál hubo ma!or disersión relati$a@ se recurre al coeficiente de $ariaciónB 2ara Ló"ica MatemáticaB
CV =
2ara 0stadEstica &escriti$aB
CV =
1.79 3.9 1.7 3.7
× 100 = 45.9%
× 100 = 46%
0n 0stadEstica &escriti$a hubo una ma!or disersión relati$a ?5W?.FW 2ara 2ara el cálc cálcul ulo o de la unt untua uaci ción ón rela relatiti$a $a@@ se hace hace uso uso del del unt unta ae e &$ estandarizado. 0s decir@ se reuiere estandarizar las calificaciones con$irtiKndolas en untuaciones Z . Ló"ica MatemáticaB
Z =
x − x
=
3.8 − 3.9
= −0 06
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UNIVERSIDAD NACIONAL ABIERTA Y A DISTANCIA – UNAD
ESCUELA DE CIENCIAS BÁSICAS, TECNOLOGÍA E INGENIERÍA CONTENIDO DIDÁCTICO DEL CURSO: 100105 – Estadística Desci!ti"a
0stos 0stos $alores $alores de untuac untuación ión Z ne"ati$os ne"ati$os indican ue ambas calificaciones se encuentran or debao de la media. 0ste es un rinciio del untae estandarizadoB Siempre $ue un valor sea menor $ue la media, su puntuación Z correspondiente correspondiente ser( negativa.
0stos resultados afirman entonces ue el estudiante con calificaciones de ).= en Ló"ica Matemática Matemática ! ). en 0stadEstica 0stadEstica &escriti$a@ &escriti$a@ está or debao del romedio del "ruo en ambos cursos. &ado ue I(.(5 se encuentra más cera a ( la media de la $ariable estandarizadaN@ se dice dice ue ue la unt untua uaci ción ón rela relatiti$a $a del del estu estudi dian ante te fue fue sue sueri rior or en Ló"i Ló"ica ca Matemática.
/$1$5 LECCIÓ' /0 MEDIDAS DE ASIMETRÍA ) APU'TAMIE'TO &esuKs de conocer cómo $arEa un "ruo de datos resecto a su media e identificar otras medidas de $ariación@ se trabaará a continuación unas nociones básicas sobre cur$as asimKtricas. asimKtricas. 0n cursos más a$anzados@ a$anzados@ como 2robabilidad@ 2robabilidad@ este tema se rofundiza rofundiza más@ ero ara los obeti$os obeti$os ue se trazan en este curso basta con las nociones ue se desarrollan a continuación.
/$1$5$1
A%i2etr>a
>a se ha mencionado al"o sobre los efectos de la asimetrEa resecto a la media@ mediana mediana ! moda $er sección '.'.?.). '.'.?.). de la #nidad &idáctica &idáctica unoN. 0n una distribución sim%trica se tiene ueB x
= Me = Mo
0n las las dist distri ribu buci cion ones es asim%tricas la media se corre en el sentido del alar"amiento alar"amiento o sesgo or efecto de las frecuencias ! de los $alores eQtremos de la $ariableO la mediana tambiKn se corre ero menos ue la media !a ue en ella sólo influ!en las frecuenciasO en tanto ue la moda no es influenciada ni or las frecuencias ni or los $alores eQtremos $er fi"ura ).'. de la resente #nidad &idácticaN. La distribu ribucción es asim%tr asim%trica ica positiva positiva cuan cuando do res resen enta ta un alar"amiento o ses"o a la derecha !B
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UNIVERSIDAD NACIONAL ABIERTA Y A DISTANCIA – UNAD
ESCUELA DE CIENCIAS BÁSICAS, TECNOLOGÍA E INGENIERÍA CONTENIDO DIDÁCTICO DEL CURSO: 100105 – Estadística Desci!ti"a
x
< Me < Mo
Las asimetrEas ositi$as son las más frecuentes ue las ses"adas hacia la izuierda@ orue con frecuencia es más fácil obtener $alores eQcecionalmente "ran "rande dess ue ue $alo $alore ress eQce eQcec cio iona nalm lmen ente te eu eue eos os.. 0em 0eml lo o de ello ello es la distribución de $alores en los consumos de ser$icios Dblicos@ las calificaciones en ruebas@ los sueldos@ etc. Se reconocen@ entre otras@ las si"uientes medidas ara calcular el "rado de la asimetrEaB • "oeficiente de Pearson . AsimetrEa en función de la media ! la moda. 1arEa
entre j) ! es ( en la distribución normal. As =
x − Mo s
⇔
As =
3 ⋅ ( x − Me) s
• Media cuartil de asimetría o media de 0o1le2 . 1arEa entre j' ! es ( en la
distribución normal. As =
Q1 + Q3 − 2Q2 Q3 − Q1
Si As = 0 la distribución es sim#trica. Si As > 0 la distribución es asim#trica positiva. Si As < 0 la distribución es asim#trica negativa.
/15/
Anta2iento o #rto%i%
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UNIVERSIDAD NACIONAL ABIERTA Y A DISTANCIA – UNAD
ESCUELA DE CIENCIAS BÁSICAS, TECNOLOGÍA E INGENIERÍA CONTENIDO DIDÁCTICO DEL CURSO: 100105 – Estadística Desci!ti"a
∑ Z Ap =
4
i
n ⋅ s
Si Si Si
⋅ f i 4
Ap = 3 la
distribución es normal o o mesoc+rtica. Ap > 3 la distribución es apuntada o leptoc+rtica. Ap < 3 la distribución es achatada o platic+rtica.
,tra medida de curtosis ue se emlea está basada en el ran"o semiintercuartElico ! los ercentiles '( ! FB Ap =
Q D 2 P 90 − P 10
=
Q3 − Q1 2( P 90 − P 10 )
0n el si"uiente eemlo se eQlicarán meor las medidas de asimetrEa ! auntamiento.
E?EMPLO /0$1 110$110$1 0l coordinador coordinador acadKmico acadKmico del C0A& de 1alleduar desea conocer el rendimiento rendimiento acadKmico de los estudiantes de rimer semestre en el ;((@ en los cursos de Ló"ica Matemática@ Matemática@ Cometencias Cometencias Comunicati$ Comunicati$as@ as@ Cultura Cultura 2olEtica@ 2olEtica@ 0stadEstica 0stadEstica &escriti$a &escriti$a ! Herramientas Herramientas *nformáticas. *nformáticas. 2ara esto selecciona selecciona una muestra de estudiantes de los distintos ro"ramas ue se ofrecen en el C0A&. La si"uiente tabla@ arroa los resultados de la in$esti"ación realizada or el funcionario.
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UNIVERSIDAD NACIONAL ABIERTA Y A DISTANCIA – UNAD
ESCUELA DE CIENCIAS BÁSICAS, TECNOLOGÍA E INGENIERÍA CONTENIDO DIDÁCTICO DEL CURSO: 100105 – Estadística Desci!ti"a
00 05 10 15 /0 /5 70 75 :0 :5 50 Total
' ? 6 F F = 5 ? ) ; ; 55
) ) 5 6 6 6 5 ) ) 55
; ; ) ? 5 = F F 6 ? ' 55
' ' ; ? '' '? '; 5 ) ' ( 55
' ; ) 6 F '' F 6 ) ; ' 55
0n la tabla si"uiente se reorta un resumen de las medidas estadEsticas or cada uno de los cursos kcomruKbeloNB
Medida x )e )o
s 2
s Q1
Q2 Q3
L!i#a Mate24ti#a ;.; ;. ( '. ! ;.( '.? '.;( '. ;. ( ). (
Co2eten#ia% Co2ni#ati"a% ;. ;. ;.(@ ;. ! ).( '.=? '.)5 '. ;. ).
Cltra Pol>ti#a ;.6 ).( ).( ! ). '.? '.;( ;.( ).( ).
/$1$5$7 L!i#a Mate24ti#a N A%i2.tri#a Po%iti"a
E%tad>%ti#a De%#riti"a ;.) ;. ;. (.65 (.=6 ;.( ;. ).(
Jerra2ienta% Inor24ti#a% ;. ;. ;. '.'; '.(5 ;.( ;. ).?
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UNIVERSIDAD NACIONAL ABIERTA Y A DISTANCIA – UNAD
ESCUELA DE CIENCIAS BÁSICAS, TECNOLOGÍA E INGENIERÍA CONTENIDO DIDÁCTICO DEL CURSO: 100105 – Estadística Desci!ti"a
2olE"ono de frecuencias de calificaciones de Ló"ica Matemática '( F = a i c n e u c e r 4
6 5 ? ) ; ' (@(
(@
'@(
'@
;@(
;@
)@(
)@
?@(
?@
@(
Calificación
La cur$a lle$a a concluir concluir ue la ma!orEa ma!orEa de los estudiantes estudiantes están or debao de la media en el curso de Ló"ica Matemática ! son ocos los estudiantes ue la sueran.
/$1$5$: Co2eten#ia% Co2ni#ati"a% NSi2.tri#a Se obser$a obser$a ue Mo = Me = x @ lo ue indica ue la distribución es simKtrica. 2ara confirmarlo se hace uso del coeficiente de /ole!@ ues la distribución tiene tres modas ! no ermite un resultado se"uro con el coeficiente de 2earson.
As
=
Q1
+ Q3 − 2Q2 1.5 + 3.5 − 2(2.5) = =0 Q3 − Q1 3.5 − 1.5
0l olE"ono de frecuencias frecuencias de las calificaciones calificaciones de Cometencias Cometencias Comunicati$as Comunicati$as
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UNIVERSIDAD NACIONAL ABIERTA Y A DISTANCIA – UNAD
ESCUELA DE CIENCIAS BÁSICAS, TECNOLOGÍA E INGENIERÍA CONTENIDO DIDÁCTICO DEL CURSO: 100105 – Estadística Desci!ti"a
2ara determinar el "rado de auntamiento o curtosis@ se debe determinar el untae tEico o estandarizado de cada clase ! lue"o alicar la fórmula ue lo calcula. 0n la si"uiente tabla se indican estos $alores.
Ta&la :$:$ Cálculo de 4 ara ara la distribución de frecuencias de las calificaciones de Cometencias Comunicati$as Calii#a#i!n
6
(@( (@ '@( '@ ;@( ;@ )@( )@ ?@(
) ) 5 6 6 6 5
I'@=)=;);F? I'@?6(==;) I'@'(;F?''65 I(@6);F?''= I(@)565?6(F ( (@)565?6(F (@6);F?''= '@'(;F?''65
4
Z i f i )?@;');= '?@()(F(;? 6@)FF'(=5F '@6)=5;= (@';6===) ( (@';6===) '@6)=5;= 6@)FF'(=5F
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UNIVERSIDAD NACIONAL ABIERTA Y A DISTANCIA – UNAD
ESCUELA DE CIENCIAS BÁSICAS, TECNOLOGÍA E INGENIERÍA CONTENIDO DIDÁCTICO DEL CURSO: 100105 – Estadística Desci!ti"a
/$1$5$5 Cltra Pol>ti#a NA%i2.tri#a 'eati"a Se obser bser$a $a ue ue Mo > Me > x @ lo ue indica ue la distribución es asimKtrica ne"ati$a. 2ara confirmarlo se hace uso de la media de /ole!@ ues la distribución tiene dos modas ! no ermite un resultado se"uro con el coeficiente de 2earson.
As =
Q1 + Q3 − 2Q2 Q3 − Q1
=
2.0 + 3.5 − 2(3.0) 3.5 − 2.0
= −0.33 < 0
0l olE"ono de frecuencias de las calificaciones de Cultura 2olEtica confirma los resultados.
Fira :$:$ Cur$a asimKtrica ne"ati$a 2olE"ono de frecuencias de calificaciones de Cultura 2olEtica '( F =
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UNIVERSIDAD NACIONAL ABIERTA Y A DISTANCIA – UNAD
ESCUELA DE CIENCIAS BÁSICAS, TECNOLOGÍA E INGENIERÍA CONTENIDO DIDÁCTICO DEL CURSO: 100105 – Estadística Desci!ti"a As =
Q1
+ Q3 − 2Q2 2.0 + 3.0 − 2(2.5) = =0 Q3 − Q1 3.0 − 2.0
2ara determinar el "rado de auntamiento o curtosis@ se debe determinar el untae tEico o estandarizado de cada clase ! lue"o alicar la fórmula ue lo calcula. 0n la tabla si"uiente tabla se indican estos $alores.
Ta&la :$5 Cálculo de 4 ara ara la distribución de frecuencia de las calificaciones de 0stadEstica &escriti$a Calii#a#i!n
6
(@( (@ '@( '@ ;@( ;@ )@( )@ ?(
' ' ; ? '' '? '; 5 )
I;@F(=(?F66 I;@))))))))) I'@6=5;(5F I'@'=)F(=(?5 I(@5(F'F?(; I(@()??=;6F (@?(;;F== '@''?F?;;F ' 5=F5'6;
4
Z i f i 6'@'5)(5 ;F@5?'F6) 'F@')('5?6 6@==)F;5 '@'(;;6 '@F6F?0I( '@(;;'()5 F@;6'6)=5 ;? ?'F?6
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UNIVERSIDAD NACIONAL ABIERTA Y A DISTANCIA – UNAD
ESCUELA DE CIENCIAS BÁSICAS, TECNOLOGÍA E INGENIERÍA CONTENIDO DIDÁCTICO DEL CURSO: 100105 – Estadística Desci!ti"a
/$1$5$< Jerra2ienta% Inor24ti#a% Se obser$a obser$a ue Mo = Me = x @ lo ue indica ue la distribución es simKtrica. 2ara confirmarlo se hace uso del coeficiente de 2earsonB As =
x − Mo s
=
2.5 − 2.5 1.06
=0
0l olE"ono de frecuencias de las calificaciones de Herramientas *nformáticas confirma confirma los resultados. resultados. La cur$a es simKtrica mesocDrtica mesocDrtica o normal. normal. 1erifEuelo 1erifEuelo ! constru!a la "ráfica.
/$/ CAPITULO 5 MEDIDAS ESTADÍSTICAS (IVARIA'TES Hasta ahora se ha estudiado el análisis de una sola $ariable@ calculando los estadEsticos de muestras ue ermiten describir e interretar la distribución de esa
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UNIVERSIDAD NACIONAL ABIERTA Y A DISTANCIA – UNAD
ESCUELA DE CIENCIAS BÁSICAS, TECNOLOGÍA E INGENIERÍA CONTENIDO DIDÁCTICO DEL CURSO: 100105 – Estadística Desci!ti"a
La alabra re"resión la utilizamos ara si"nificar la estimación de una $ariable en función de otro $alor conocido@ corresondiente a la otra $ariable. &ecimos ue la meor lEnea ue se austa a un conunto de untos es auella en donde la suma de los cuadrados de las diferencias entre los $alores reales ! los estimados es mEnima. 2ara determinar el "rado de correlación entre las $ariables@ no basta con calcul calcular ar la $arianz $arianza a eQlic eQlicada@ ada@ ues eQiste eQiste el coefici coeficient ente e de determ determina inació ción n o coeficiente coeficiente de correlación correlación al cuadradoO cuadradoO sin embar"o@ embar"o@ frecuentemente frecuentemente se utiliza utiliza un coeficiente de correlación rectilEneo@ r siendo siendo este un $alor entre I' ! '. Si el coeficiente coeficiente de correlación r es i"ual o menor ue uno@ nos indica ue tanto la co$arianza@ como los coeficientes coeficientes an"ulares@ son ne"ati$os ne"ati$os ! or tanto la recta será descendente@ or ser la endiente ne"ati$a. Además si es i"ual a I'@ nos indica ue eQiste una erfecta correlación correlación en otras alabras@ alabras@ cada $alor de la $ariable deberá ser eQactamente i"ual al estimado@ ! or tanto la $arianza residual es i"ual a cero@ ! la $arianza eQlicada i"ual a la $arianza total.
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UNIVERSIDAD NACIONAL ABIERTA Y A DISTANCIA – UNAD
ESCUELA DE CIENCIAS BÁSICAS, TECNOLOGÍA E INGENIERÍA CONTENIDO DIDÁCTICO DEL CURSO: 100105 – Estadística Desci!ti"a
Fira 5$1$ Gráficas de disersión aN linealO bN cur$ilEneaO cN sin relación )
)
Na
/$/$7 $
8
)
N&
8
N#
LECCIÓ' /7 RE,RESIÓ' LI'EAL SIMPLE
La re"resión eQamina la relación entre dos $ariables restrin"iendo una de ellas resecto a la otra@ con el obeto de estudiar las $ariaciones de la rimera cuando la otra ermanece constante. La re"resión es un mKtodo ue se emlea ara ronosticar o redecir el $alor de una $ariable en función de los $alores dados de la otra o de las otras@ cuando se trabaa más de dos $ariablesN.
8
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UNIVERSIDAD NACIONAL ABIERTA Y A DISTANCIA – UNAD
ESCUELA DE CIENCIAS BÁSICAS, TECNOLOGÍA E INGENIERÍA CONTENIDO DIDÁCTICO DEL CURSO: 100105 – Estadística Desci!ti"a
0n esta esta ecua ecuaci ción ón ha! ha! dos dos $alo $alore ress desc descon onoc ocid idas asBB a ! b@ ue ue debe deben n determinarse alicando el criterio de los mEnimos cuadrados@ buscando asE la meor recta ue se auste a los datos. Se tiene entoncesB
=
∑ X! − ∑ X ∑ ! n ∑ X − ( ∑ X )
n
2
2
a
=
∑ ! − ∑ X
&ondeB 2endiente de de la la recta B a B *nterce rceto de de la la $a $ariabl able Y
n
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UNIVERSIDAD NACIONAL ABIERTA Y A DISTANCIA – UNAD
ESCUELA DE CIENCIAS BÁSICAS, TECNOLOGÍA E INGENIERÍA CONTENIDO DIDÁCTICO DEL CURSO: 100105 – Estadística Desci!ti"a
@; 6@' '( ';@F ';@' '?@?
5 = '' '; ') '?
2ara conocer el tio de relación ue uede eQistir entre estas dos $ariables@ el rimer aso es determinar es si el dia"rama de disersión efecti$amente insinDa una tendencia lineal.
Fira 5$/$ &ia"rama de disersión de $entas de un roducto ! la emisión del comercial en tele$isión
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UNIVERSIDAD NACIONAL ABIERTA Y A DISTANCIA – UNAD
ESCUELA DE CIENCIAS BÁSICAS, TECNOLOGÍA E INGENIERÍA CONTENIDO DIDÁCTICO DEL CURSO: 100105 – Estadística Desci!ti"a
';@F ';@' '?@? < 0 1
=
'; ') '? <7
n∑ X! − ∑ X ∑! n∑ X − ( ∑ X ) 2
a=
2
=
'?@= '6@) ;('@5 <= < 7
'55@?' '?5@?' ;(6@)5 < ; = 1
7 × 787 787,3 − (70,1)(73) 7 × 768,19 − (70,1)
2
=
393 393,8 463,32
= 0,85
415 ∑! − ∑ X = 73 − (0.85 × 70,1) = 13,415 = 1,92 n
7
7
&e modo ue la ecuación de la recta austada está dada orB
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UNIVERSIDAD NACIONAL ABIERTA Y A DISTANCIA – UNAD
ESCUELA DE CIENCIAS BÁSICAS, TECNOLOGÍA E INGENIERÍA CONTENIDO DIDÁCTICO DEL CURSO: 100105 – Estadística Desci!ti"a
/$/$ /$ /$:: LECC LECCIÓ IÓ' ' /: CORR CORRELA ELACI CIÓ' Ó' La correlación entre dos $ariables busca determinar el "rado de relación ue eQiste entre ellas dos. 0lla se calcula con los coeficientes de correlación . Los coeficientes de correlación son nDmeros ue $arEan entre ]' ! I'. Su ma"nitud indica el "rado de asociación entre las $ariables@ si es ( indica ue no eQiste relación al"una ! los $alores eQtremos ]' ! I' indican una correlación erfecta ositi$a o ne"ati$a resecti$amente.
Fira 5$7$
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UNIVERSIDAD NACIONAL ABIERTA Y A DISTANCIA – UNAD
ESCUELA DE CIENCIAS BÁSICAS, TECNOLOGÍA E INGENIERÍA CONTENIDO DIDÁCTICO DEL CURSO: 100105 – Estadística Desci!ti"a
&ondeB #e B 0rro 0rrorr est están ánda darr del del esti estima mado do ! B 1alo 1alore ress de la $ari $ariab able le dee deend ndie ient nte e 1alo 1alore ress esti estima mado doss de la ecua ecuaci ción ón ! B nB +amao de la muestra 0sta ecuación imlica demasiadas demasiadas oeraciones@ oeraciones@ or lo ue suele utilizarse utilizarse un mKtodo más bre$eB #e =
∑!
2
− a∑! − ∑ X!
n −2
#na $ez obtenido el error estándar del estimado@ es necesario medir uK orcentae de la información es reco"ida o eQlicada or el modelo de re"resión
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Correlación eQcelente Corre orrela lacción ión ace acet tab ablle Correlación re"ular Correlación mE mEnima %o ha! correlación
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ESCUELA DE CIENCIAS BÁSICAS, TECNOLOGÍA E INGENIERÍA CONTENIDO DIDÁCTICO DEL CURSO: 100105 – Estadística Desci!ti"a
0l dear dearta tame ment nto o de ubl ublic icid idad ad reu reuer erirá irá de 'F comer comerci cial ales es de tele tele$i $isi sión ón diariamente ara ue el nDmero de $entas ascienda a ;((( unidades mensuales.
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ESCUELA DE CIENCIAS BÁSICAS, TECNOLOGÍA E INGENIERÍA CONTENIDO DIDÁCTICO DEL CURSO: 100105 – Estadística Desci!ti"a
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2252.5 = 85a +13754.171 +1204.172 − 2260 = −85a − 37561 −12112 − 7.5 = −1.831 − 6.832
(4)
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Los n3mero n3meros s índic índice e son son cifr cifras as rela relatiti$a $ass eQr eQres esad adas as en tKrm tKrmin inos os orcentuales@ ue sir$en ara indicar las $ariaciones ue sufre una serie de $alores resecto a una de ellas@ tomada como unto de referencia ! a la cual se le denomina base.
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