Moments, ctd

skewness =

= measure of asymmetry of a distribution

· skewness = 0: distribution is symmetric

· skewness > (<) 0: distribution has long right (left) tail

kurtosis =

= measure of mass in tails

= measure of probability of large values

· kurtosis = 3: normal distribution

· skewness > 3: heavy tails (“ leptokurtotic ”)

2 random variables: joint distributions and covariance

· Random variables X and Z have a joint distribution

· The covariance between X and Z is

cov(X, Z) = E [(X – m B _{X B})(Z – m B _{Z B})] = s B _{XZ B}

· The covariance is a measure of the linear association between X and Z; its units are units of X ´ units of Z

· cov(X, Z) > 0 means a positive relation between X and Z

· If X and Z are independently distributed, then cov(X, Z) = 0 (but not vice versa!!)

· The covariance of a r.v. with itself is its variance:

cov(X, X) = E [(X – m B _{X B})(X – m B _{X B})] = E [(X – m B _{X B})P^2P] =

The covariance between Test Score and STR is negative:

so is the correlation …
The correlation coefficient is defined in terms of the covariance:

corr(X, Z) = = r B _{XZ B}

· –1 £ corr(X, Z) £ 1

· corr(X, Z) = 1 mean perfect positive linear association

· corr(X, Z) = –1 means perfect negative linear association

· corr(X, Z) = 0 means no linear association

The correlation coefficient measures linear association

(c) Conditional distributions and conditional means

Дата добавления: 2014-01-07