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Univariate Statistics of Dispersion. p 47. Very useful properties of S X occurs when the data are normally distributed (i.e. symmetrically distributed and not extremely concentrated or dispersed about the mean), and there is a large number of observations available:

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univariate statistics of dispersion
Univariate Statistics ofDispersion

p 47

  • Very useful properties of SX occurs when the data are normally distributed (i.e. symmetrically distributed and not extremely concentrated or dispersed about the mean), and there is a large number of observations available:
    • Approximately 68% of the observations should have values that fall within 1 standard deviations from the mean (i.e. within the interval - SX to ( + SX)
univariate statistics of dispersion1
Univariate Statistics ofDispersion

p 47

  • Approximately 95% of the observations should have values that fall within 2 standard deviations from the mean (i.e. within the interval - 2SX to ( + 2SX)
univariate statistics of dispersion2
Univariate Statistics ofDispersion

p 47

  • The variance (S2X) is the square of the standard deviation:
    • (3.7)
univariate statistics of dispersion3
Univariate Statistics ofDispersion

p 47

  • It provides the same information about the variable of interest contained in the standard deviation, but it is often used as the main measure of dispersion in statistics
  • The numerator in the variance is considered a measure of the total variation in
linear transformations
Linear Transformations
  • In applied statistics, sometimes is convenient to define and create a new variable as a transformation of an existing one, i.e.:
    • Yi = f(Xi) for all i
linear transformations1
Linear Transformations
  • If we know and SX, and the transformation is linear, there is a simple way to calculate and SY directly from and SX; for instance if:
    • Yi = a + bXi for all i, then
    • = a + b
linear transformations2
Linear Transformations
  • In addition:
    • S2Y = b2S2X and SY = |b|SX
bivariate statistics
Bivariate Statistics

p 53

  • The ultimate objective of regression analysis is to determine if and how certain (independent) variables influence another (dependent) variable
  • Bivariate statistics can be used to examine the degree in which two variables are related, without implying that one causes the other
bivariate statistics1
Bivariate Statistics

p 54

  • In Figure 3.3 (a) Y and X are positively but weakly correlated while in 3.3 (b) they are negatively and strongly correlated
bivariate statistics covariance
Bivariate Statistics: Covariance

p 53

  • The covariance is one measure of how closely the values taken by two variables Y and X vary together:
    • (3.17)
    • A disadvantage of the covariance statistic is that its magnitude can not be easily interpreted, since it depends on the units in which we measure Y and X
bivariate statistics c orrelation coefficient
Bivariate Statistics: Correlation Coefficient

p 54

  • The related and more used correlation coefficient remedies this disadvantage by standardizing the deviations from the mean:
    • (3.18)
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