This document discusses central tendency measures such as the mean, median, and mode. It defines each measure and explains how to calculate them from both grouped and ungrouped data. The mean is the average value of all data points. The median is the middle value when data is arranged in order. The mode is the value that occurs most frequently. The document provides formulas for calculating the mean, median, and mode in different data situations. It also discusses situations where there may be more than one mode.
The lesson begins with students engaging in a review of some measures of central tendency by considering a numerical example. Students are also asked to examine both strengths and limitations of these measures. Assessments will be given to students on their ability to calculate these measures, and also to get an overall sense of whether they recognize how these measures respond to changes in data values.
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La Medida de Tendencia Central: Es un número ubicado hacia el centro de la distribución de los valores de una serie, en la que se encuentra ubicado el conjunto de los datos.
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2. Medidas de Tendencia Central
Toda la información del conjunto en una sola medida que se
considera un número representativo.
Concentran
Las destacadas:
Media Mediana Moda
la media de una variable se
define como la suma
ponderada de los valores
de la variable por sus
frecuencias relativas
Es el valor central de la variable,
es decir, supuesta la muestra
ordenada en orden creciente o
decreciente, el valor que divide
en dos partes la muestra.
Es el valor de la variable
que tenga mayor
frecuencia absoluta, la
que más se repite.
3. Media
Aritmética
La medida
mas
utilizada en
la
estadística
𝑋 =
𝑖=1
𝑛
𝑋𝑖 ∗ 𝑓
𝑛
Siempre y
cuando la
distribución se
organice en una
tabla de
frecuencias. Se
calcula mediante
la expresión:
Xi = Variable
Si los datos están
organizados en una tabla
con datos agrupados, la Xi
representa la marca de
clase.
f = frecuencia absoluta
La media se interpreta
como una esperanza,
como la medida justa que
le corresponde a cada uno
de los elementos de la
muestra.
Las
conclusiones
se toman con
respecto a la
población.
Datos
Agrupados
Cuando el número de
datos que constituyen
la base de datos son
muy numerosos y
vienen de una
variable continua. Los
datos se “agrupan”
Los datos son presentados
en pequeños paquetes que
abarcan todos los datos
contenidos entre dos valores
determinados de la variable
4. Procedimiento
para agrupar datos
4. Redefina el rango: siempre y cuando el valor de la
longitud del intervalo no sea exacta.
D = Rnuevo – Rango
Esta diferencia determina el valor mínimo y el valor
máximo de la tabla
3. Calcule la longitud de intervalo
𝐶 =
𝑅𝑎𝑛𝑔𝑜
𝑚
2. Calcule el numero de intervalos
m= 1 + 3,3*log(n), donde n es la muestra,
m es un valor entero
1. Calcule el rango
R = Xmax – Xmin
5. Mediana
Si N es Par, hay dos
términos centrales, la
mediana será la media
de esos dos valores.
Para calcular la mediana debemos tener en cuenta
Discreta
si la variable es
Continua
¿cómo calcularlo?
Teniendo en cuenta el
tamaño de la muestra:
Si N es Impar, hay un
término central, el
término que será el
valor de la mediana.
6. Si n es Par, hay dos términos
𝑛
2
, 𝑜
𝑛+1
2
centrales, la
mediana será la media de esos dos valores. Los valores
encontrados en estos cocientes representan una posición
Mediana en
datos no
agrupados
Si n es Impar, hay un término
𝑛+1
2
central, el
término que será el valor de la mediana.
7. N par N impar
1,4,6,7,8,9,12,16,20,
24,25,27 N=12
1,4,6,7,8,9,12,16,20,
24,25,27,30 N=13
Términos Centrales el 6º
y 7º 9 y 12
Término Central el 7º ,
12
8. Mediana en datos agrupados
En datos agrupados, la mediana se calcula de la siguiente manera
𝑀𝑒 = 𝐿𝑖 +
𝑛
2
−𝐹𝑎
𝑓𝑜
*C
Donde
Li = Limite inferior real de la clase mediana
n = muestra
Fa = Frecuencia absoluta anterior a la observada en la clase mediana
fo = Frecuencia absoluta observada en la clase mediana
C = Longitud del intervalo
Numero de intervalo impar
Numero de intervalo par
9. MODA
Es la única medida de centralización que tiene
sentido estudiar en una variable cualitativa, pues no
precisa la realización de ningún cálculo.
Por su propia definición, la moda no es única, pues puede haber
dos o más valores de la variable que tengan la misma frecuencia
siendo esta máxima. En cuyo caso tendremos una distribución
bimodal o polimodal según el caso.
10. Moda en datos no agrupados
En una distribución de frecuencias de datos
discretos sin agrupar, la moda equivale al
valor que mas se repite
Pueden existir varias modas o
uno o puede no existir moda
en una distribución. Esta es la
única medida de tendencia
central que admite esta
cualidad.
Si existe una moda: Unimodal
Si existen dos modas: Bimodal
Si existen mas de dos modas: Polimodal
11. Moda en datos agrupados
La moda en datos agrupados se calcula mediante la formula
𝑴𝒐 = 𝑳𝒊 +
∆𝟏
∆𝟏+∆𝟐
*C
𝑳𝒊= Limite inferior de la clase modal, representada por
el intervalo con mayor frecuencia absoluta.
∆𝟏 = fo - fa
∆𝟐 = fo - fs
fo = la mayor frecuencia absoluta observada
fa = Frecuencia absoluta anterior fo
fs = Frecuencia absoluta siguiente a fo
13. 1- Discapacidad en tiempos del covid (DANE)
2- Pulso Social (DANE)
3- Boxplot (Medidas de posición)
4- Medidas de tendencia central y dispersión
5- Graficos estadisticos en el covid
6- Correlacion lineal
7- Correlacion no lineal