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Joint freq. Condit.% Charts Statistical dependence
Two-dimensional variables
Facultad de Comercio, Turismo y Ciencias Sociales Jovellanos
Two- dimensional variables
Joint freq. Condit.% Charts Statistical dependence
Introduction
Joint frequencies
Joint frequencies
Joint and marginal distributions
Conditional percentages
Scatter plot
Statistical dependence
Studying statistical dependence from double - entry tables
Studying statistical dependence from some coefficients
Two- dimensional variables
Intro Joint freq. Condit.% Charts Statistical dependence
Introduction: limitations of univariate tables and statistics
I Example: Consider a sample of n = 10 families. Let us study
two variables:
I (^) X =“household income (Eur per month)” I (^) Y =“household saving (Eur per month)”
The respective frequency tables are:
xi ni.
yj n.j
I You can calculate average income, average saving, variances,
percentiles, etc. (univariate statistics)
I But, the above frequency tables do not provide us any
information about the RELATION between both variables.
Two- dimensional variables
Intro Joint freq. Condit.% Charts Statistical dependence
Limitations of univariate tables and statistics (II)
I Frequency tables of X and Y do not provide us enough
information to answer the following questions:
I (^) Let us restrict ourselves to those families with an income higher than 2500 Eur. What percent of those families save more than 400 Eur? I (^) Is there a positive relation between both variables? Do higher incomes correspond to higher savings?
I But we can answer them if we look at the original dataset:
family 1 2 3 4 5 income 1500 2000 1500 2000 2000 saving 0 50 50 200 100
family 6 7 8 9 10 income 3000 2000 3000 4000 3000 saving 500 100 100 500 100
I We would need a different kind of table to summarize
bivariate data. Two- dimensional variables
Joint freq. Condit.% Charts Statistical dependence
Joint frequencies Joint and marginal distributions
Joint frequency tables
I Useful to summarize bivariate information when the sample
size is high and the number of different pairs of values is small.
I Example (just an illustration of the construction of joint
frequency tables. In this example, the sample size is small).
I (^) Row joint frequency table. income 1500 1500 2000 2000 2000 3000 3000 4000 saving 0 50 50 100 200 100 500 500 no. families 1 1 1 2 1 2 1 1 I (^) Double-entry table (each cell represents the number of individuals in each category) xi \yj 0 50 100 200 500 1500 1 1 0 0 0 2000 0 1 2 1 0 3000 0 0 1 2 1 4000 0 0 0 0 1 Two- dimensional variables
Joint freq. Condit.% Charts Statistical dependence
Joint frequencies Joint and marginal distributions
Joint and marginal distributions
I We can obtain the “marginal” frequency tables of X and Y
from the double-entry table:
xi \yj 0 50 100 200 500 ni. 1500 1 1 0 0 0 2 2000 0 1 2 1 0 4 3000 0 0 2 0 1 3 4000 0 0 0 0 1 1 n.j 1 2 4 1 2 n = 10
I The converse is not true: we cannot obtain the joint
frequencies from the marginal frequencies!!!
Two- dimensional variables
Intro Joint freq. Condit.% Charts Statistical dependence
Joint frequencies Joint and marginal distributions
Joint frequency table: general notation
I General nomenclature for joint frequency tables:
xi \yj y 1... yj... yl ni. x 1 n 11... n 1 j... n 1 l n 1.
......... n 23......... xi ni 1... nij... nil ni. ......... n 23......... xk nk 1... nkj... nkl nk. n.j n. 1... n.j... n.l n
I nij : number of individuals taking the pair of values (xi , yj ) in
the sample.
I ni.: number of individuals in the sample taking the value
X = xi.
I n.j : number of individuals in the sample taking the value
Y = yj.
Intro Joint freq. Condit.% Charts Statistical dependence
Joint frequencies Joint and marginal distributions
Joint and marginal distributions: general formulas
If, for instance, X takes k = 4, and Y takes l = 5 different values
in the sample, we use the following notation:
xi \yj y 1 y 2 y 3 y 4 y 5 ni. x 1 n 11 n 12 n 13 n 14 n 15 n 1. x 2 n 21 n 22 n 23 n 24 n 25 n 2. x 3 n 11 n 12 n 13 n 14 n 15 n 3. x 4 n 11 n 12 n 13 n 14 n 15 n 4. n.j n. 1 n. 2 n. 3 n. 4 n. 5 n
I Marginal frequencies of X :
I (^) n 1. = n 11 + n 12 + n 13 + n 14 + n 15 I (^) In general ni. = ni 1 +... + nil
I Marginal frequencies of Y :
I (^) n. 1 = n 11 + n 21 + n 31 + n 34 I (^) In general n.j = n 1 j +... + nkj
I Sample size: n = n 1. +... + nk. = n. 1 +... + n.l.
Joint freq. Condit.% Charts Statistical dependence
Double-entry tables Coefficients
Statistical dependence
INCREASING LEVEL OF DEPENDENCE
STATISTICAL DEPENDENCE
FUNCTIONAL DEPENDENCE
STATISTICAL INDEPENDENCE
Example: X: gender Y: eye color
Example: X: husband's age Y: wife's age
Example: X: number of working hours Y: total fee=300+50X
Two- dimensional variables
Joint freq. Condit.% Charts Statistical dependence
Double-entry tables Coefficients
Statistical independence
The value of X of them does not influence the value of Y and
viceversa. Examples:
I X =“age”, Y =“last figure in the street number”
I X =“score in Maths”, Y =“initial of grandmother’s first name
(numbered from 1 to 26)”
Two- dimensional variables
Intro Joint freq. Condit.% Charts Statistical dependence
Double-entry tables Coefficients
Statistical dependence
The value of X influences the value of Y and viceversa. Examples:
I (^) X =“height”, Y =“weight”, I (^) X =“husband’s age”, Y =“wife’s age” I (^) X =“elevation”, Y =“temperature” I (^) X =“income”, Y =“saving”
I High statistical dependence:
I Low statistical dependence:
Intro Joint freq. Condit.% Charts Statistical dependence
Double-entry tables Coefficients
Functional dependence
Y can be written as a mathematical function of X. Examples:
I X =“number of working hours per project”
Y =“total fee due for project” Y = 300 + 50X.
I X =“area of a bubble” Y =“radius of a bubble” Y =
X
Joint freq. Condit.% Charts Statistical dependence
Double-entry tables Coefficients
How to detect statistical independence from double - entry
tables
Example: eye color vs. gender:
gender \ eye color blue-gray hazel-green brown-black ni. male 8 24 16 48 female 9 27 18 54 n.j 17 51 34 n = 102
When X and Y are independent, all rows in the double-entry table
are proportional to each other.
Two- dimensional variables
Joint freq. Condit.% Charts Statistical dependence
Double-entry tables Coefficients
How to detect functional dependence from double - entry
tables
Example: Study of some defects in glass (bubbles). X =diameter
(in mm.) Y =area (in mm^2 ) :
diameter \ area 0.1 0.38 7.07 12.57 ni. 0.35 5 0 0 0 5 0.70 0 3 0 0 3 3 0 0 2 0 2 4 0 0 0 1 1 n.j 5 3 2 1 n = 11
I When Y is a function of X , there is only one figure different
from 0 in each column.
I When X is a function of Y , there is only one figure different
from 0 in each row.
Two- dimensional variables
Intro Joint freq. Condit.% Charts Statistical dependence
Double-entry tables Coefficients
Contingency coefficient
I Valid for categorical, ordinal and quantitative variables.
I Mathematical definition: c = χ
2
n[min{k,l}−1] ,^ where
I χ^2 is the χ^2 −coefficient,
χ^2 = n
∑^ k
i=
∑^ l
j=
n^2 ij
ni.n.j
I 0 ≤ c ≤ 1
I c = 0 : statistical independence
I c = 1 : functional dependence
Intro Joint freq. Condit.% Charts Statistical dependence
Double-entry tables Coefficients
Coefficient of determination
I Only valid for quantitative variables.
I Mathematical definition:
r 2 =
[
Cov (X , Y )
SD(X )SD(Y )
] 2
, where
I Cov (X , Y ) is the covariance of (X , Y )
I 0 ≤ r 2 ≤ 1
I r 2 = 0 represents linear independence
I r 2 = 1 represents linear dependence
