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3.1. Tus primeros datos

 

Glosario

Desvirtuar

Dos monigotes enfrentados, uno con un círculo rojo y el otro con un círculo transparente y un símbolo de interrogación sobre él

Definición:

Alterar la verdadera naturaleza de algo. Quitarle su valor, haciendo que los datos sean engañosos.

Ejemplo:

Estudiando los sueldos de una empresa, al añadir el del dueño subió muchísimo la media y desvirtuó el propósito de este estudio.

Datos desde 0 hasta 100

Tus primeros datos

Sala de partos con personal médico y uno de ellos con un bebé en el brazo

¿Sabías que, nada más nacer, se recogieron los primeros datos sobre ti?

El personal médico te pesó y te midió y, luego, anotó los datos en una gráfica estandarizada.

Estas gráficas aparecen en las tablas de crecimiento.

Son curvas de percentiles, o lo que es lo mismo, medidas que dividen los datos de la población estudiada en 100 partes (centiles).

Forman parte de las medidas de posición, igual que la media, la mediana y la moda.

Si sientes curiosidad, en la red hay calculadoras que hallan los percentiles al nacer.

Accede en este enlace a una de ellas: https://www.ihan.es/calculadora-de-percentiles/.

¡De Q!

Monigote con medida de su alturaRecordarás que la mediana es un valor que marca la mitad de una lista de datos cuando están ordenados.

El cuartil (Q) tiene el mismo objetivo: marcar la mitad de la mitad inferior y de la superior.

  • Q1: el primer cuartil indica el valor que deja la cuarta parte de los datos (el 25 %) por debajo.
  • Q2: el segundo cuartil indica el valor que deja la mitad de los datos (el 50 %) por debajo. Es la mediana.
  • Q3: el tercer cuartil indica el valor que deja las tres cuartas partes de los datos (el 75 %) por debajo.

También puede interesar el valor más bajo (mínimo) y el más alto (máximo) que, con los cuartiles, dan una imagen de cómo se organizan los datos.

Boxplot

Se te plantea un nuevo estudio: analizar cómo están distribuidas las edades en un pueblo remoto de la montaña lucense.

Para ello, recoges los datos de las personas mayores de 80 años y obtienes este conjunto: {85, 86, 88, 88, 89, 90, 91, 92, 94, 100, 102}.

Lo ideal es hacer un diagrama de caja (en inglés boxplot o box plot).

Diagrama de caja de datos de longevidad

Observa que el 50 % de la población se concentra en la zona de la caja, entre Q1 = 88 y Q3 = 94.

A esta medida, Q3 - Q1; se le llama recorrido intercuartílico.

Caja y bigote

Este gráfico está formado por unas líneas laterales (los bigotes) y una caja entre ellas.

  • La primera línea va desde el valor mínimo hasta Q1. En el ejemplo, va desde 85 a 88; 3 años.
  • La caja va desde Q1 hasta Q3. En el ejemplo, va desde 88 a 94; 6 años.
  • La línea final va desde Q3 hasta el valor máximo. En el ejemplo, va desde 95 a 102; 8 años.
  • La línea central es la mediana. En el ejemplo, es la edad a = 90 años.

La altura o grosor de la caja no tiene significado estadístico; su propósito es solo visual, para facilitar la lectura del gráfico.

Outliers

Personas en la tercera edad corriendo

Si aparece algún valor muy separado del resto de datos, a veces, se elimina de la lista. Son los llamados valores atípicos o outliers.

Una opción para detectarlos es multiplicar la base de la caja (Q3-Q1) por 1,5 y, si un dato se separa más de ese valor, es un outlier.

En el ejemplo 6 · 1,5 = 9, ningún dato se separa de la caja más de 9 años. Por tanto, no hay valores atípicos.

GeoGebra encaja

Usa la Suite

Ejemplos de diagrama de caja

GeoGebra permite hacer un diagrama de caja de forma rápida a partir de una lista de datos usando la Suite Calculadora.

El comando: DiagramaCaja la dibuja automáticamente.

Los números que van entre paréntesis, en este comando, son:

  • El primero: punto vertical en donde se dibuja el centro de la caja.
  • El segundo: ancho de la caja en vertical.
  • Lista entre llaves: conjunto de datos.

Si cambias un número de la lista, el diagrama de caja se cambiará automáticamente.

Y el botón derecho

Haciendo clic con el botón derecho del ratón, en los tres puntos que hay al lado de la lista, puedes obtener los valores estadísticos que se ven en el gráfico de caja.

Captura de pantalla del menú del botón derecho de GeoGebra

También puedes duplicar la entrada y hacer modificaciones a partir de la lista original.

En el pueblo de al lado...

Supón que tienes los datos de otro pueblo cercano. Son muy similares, aunque no hay centenarios: {85, 86, 88, 88, 89, 90, 91, 92, 94}.

Ahora no es sencillo localizar el primer cuartil, ya que hay dos posibles candidatos, el 86 y el 88.

En estos casos se hace la media aritmética de ambos, obteniendo el valor 87.

Lo mismo para el tercer cuartil, se hace la media de 91 y 92, obteniendo 91,5.

En esta lista: Min = 85; Q1 = 87; Q2 = 89; Q3 = 91,5 y Máx = 94.

Así quedarían ambas gráficas con GeoGebra:

Ejemplos de comparativa de diagramas de caja

En la primera, el 50 % de los datos están entre 87 y 91,5; y están poco dispersos (bigotes cortos).

En la segunda, el 50 % de los datos están entre 88 y 94; y están más dispersos (el bigote derecho es largo).

Centrando las ideas

Visto en 1.º

En 1.º de ESO estudiaste algunas medidas estadísticas que sirven para resumir un grupo de datos de variables cuantitativas.

Esos valores se clasifican en dos categorías: de posición y de dispersión.

Dentro de las de posición están las centrales, muy importantes porque se usan como resumen.

A continuación, puedes repasar los vídeos que explican cómo calcular cada uno de ellos con una hoja de cálculo.

1. Media

Valor calculado a partir de la suma de todos los datos dividida entre el total:

\( \bar{x} = \dfrac{\sum x_i}{N} \)

2. Mediana

Valor que divide al conjunto de datos ordenados en dos partes iguales:

\( Me\) = Valor de la posición \(\dfrac{N}{2} \)

3. Moda

Valor con mayor frecuencia (que más se repite):

\( Mo\) = Valor con la \( f_{max} \)

El centro de la esperanza

En la siguiente actividad vas a estudiar la edad de las personas mayores de 64 años en un ayuntamiento.

Completa la tabla de frecuencias y después calcula las medidas indicadas.

En caso de haber cifras decimales, recuerda escribir siempre dos, aunque alguna sea 0.

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Tabla de frecuencias: edad de las personas > 64.
Intervalos de edad Marca de clase \(x_i\)

Número de personas \(f_i\)

Frecuencia absoluta acumulada \(F_i\)
[65,70) 67,5 855 @@855@@
[70,75) 72,5 179 @@1034@@
[75,80) 77,5 163 @@1197@@
[80,85) 82,5 92 @@1289@@ = N
N=\(\displaystyle \sum_{i=1}^{4}f_{i} \)=@@1289@@

     Histograma con los datos de las edades y el polígono de frecuencias

  • La media de edad de personas con más de 64 años del ayuntamiento es:

\(\bar{x}\) = @@70,53@@ años.

  • El intervalo que contiene mayor número de personas (moda) es:

Mo = [@@65@@,@@70@@).

  • El intervalo que contiene el centro de los datos ordenados (mediana) de edades es:

Me = [@@65@@,@@70@@).

  1. Ahora fíjate en el histograma proporcionado, ¿aporta alguna información?
  2. ¿La información que has calculado, encaja con la que aporta el histograma?
  3. Reflexiona si podrías haber representado los datos a través de un diagrama diferente.
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De la niñez a la ancianidad

En esta actividad analizarás el número de pediatras en la provincia de A Coruña en 2023 usando una hoja de cálculo o la calculadora.

La secuencia para llegar a los datos en la web del IGE es:

Datos ➤ Sanidad ➤ Recursos humanos y materiales ➤ Recursos humanos en atención primaria

Datos de las comarcas de A Coruña

En la imagen puedes ver cómo elegir todas las comarcas de Galicia. Una vez descargados los datos selecciona sólo las que pertenecen a la provincia de A Coruña.

De esta tabla, elimina las tres grandes ciudades:

1501 Arzúa 2
1502 Barbanza 8
1503 A Barcala 2
1504 Bergantiños 8
1505 Betanzos 4
1506 A Coruña [+] 48
1507 Eume 4
1508 Ferrol [+] 21
1509 Fisterra 0
1510 Muros 2
1511 Noia 4
1512 Ordes 6
1513 Ortegal 1
1514 Santiago [+] 24
1515 O Sar 2
1516 Terra de Melide 1
1517 Terra de Soneira 2
1518 Xallas 1

El selector de la derecha permite marcar por bloques:

Captura de pantalla de la web del IGE con el selector 115 con las provincias y comarcas seleccionadas.

Tabla de frecuencias

A partir de la tabla con las 15 comarcas calcula las medidas que se indican a continuación.

Te puede resultar útil pegar esta tabla en una hoja de cálculo y ordenar la columna de número de pediatras de menor a mayor.

También puedes usar tu calculadora.

Al escribir las respuestas con resultados decimales recuerda redondear y poner las centésimas, aunque sean 0.

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  • La media de pediatras en A Coruña, por comarcas, excluyendo A Coruña, Ferrol y Santiago es:

\(\bar{x}\) = @@3,13@@ pediatras.

  • ¿Cuál es el valor con frecuencia absoluta más alta? En este caso, la moda es:

Mo = @@2@@ pediatras.

  • ¿Hay un único valor en el medio de los datos? En este caso, la mediana es:

Me = @@2@@ pediatras.

  1. A raíz de los datos centrales obtenidos, reflexiona sobre cuál es el dato central de pediatras en las comarcas rurales de A Coruña.
  2. ¿Tiene sentido que la media no sea un número entero? ¿Por qué?
  3. ¿Crees que las comarcas rurales están suficientemente atendidas?
  4. Aun despreciando los datos de las tres grandes ciudades, estás comparando datos de comarcas con unas poblaciones muy distintas. ¿Cómo podrías mejorar este estudio?
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El motor de la vida

En esta actividad vas a analizar el número de centros educativos con ESO y/o enseñanzas "posobligatorias" en los ayuntamientos de las comarcas de Ortegal, Eume y A Mariña occidental en 2025.

En este caso, los datos los obtendrás de la gráfica que está a continuación.

Si te interesan otros lugares puedes llegar a los datos en la web del IGE con la secuencia:

Datos ➤ Educación ➤ Enseñanza universitaria, no universitaria y formación no reglada ➤ Centros de régimen general en funcionamento según las enseñanzas que imparten

Al escribir las respuestas con resultados decimales recuerda redondear y poner hasta las centésimas, aunque sean 0.

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Diagrama de barras con datos y una línea que une los puntos más altos de las barras.Foto con el IES de Ortigueira al fondo y una señal de zona escolar.

Tabla de frecuencias: n.º de institutos por ayuntamientos
N.º de institutos \(x_i\) N.º de ayuntamientos \(f_i\)

Frec. absoluta acumulada \(F_i\)

Frec. relativa \(h_i\)

 \( x_i \cdot f_i \)
0 @@7@@ @@7@@ @@0,50@@ @@0@@
1 @@4@@ @@11@@ @@0,29@@ @@4@@
2 @@2@@ @@13@@ @@0,14@@ @@4@@
3 @@1@@ N= @@14@@ @@0,07@@ @@3@@
N=\(\displaystyle \sum_{i=1}^{4}f_{i} \)=@@14@@ \(\displaystyle \sum_{i=1}^{4}h_{i} \)=1 \(\displaystyle \sum_{i=1}^{4}x_{i}\cdot f_i \)=@@11@@
  • ¿Solo observando el diagrama de barras, podrías decir cuál es la moda?

Mo = @@0@@ institutos.

  • ¿Y cuál es la mediana? Ten cuidado, son datos pares.

Me = @@0,50@@ institutos.

  • ¿Cuál es la media?

\(\bar{x}\) = @@11@@/@@14@@ = @@0,79@@ institutos. 

  1. ¿Cuál es tu opinión sobre la cantidad de ayuntamientos sin institutos? ¿Lo sabías?
  2. Has obtenido tres medidas centrales sobre el número de institutos en una zona rural de Galicia. ¿Crees que son suficientes para hacer un análisis de estos datos? ¿Por qué?
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