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# Univariate Statistics of Dispersion - PowerPoint PPT Presentation

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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## PowerPoint Slideshow about ' Univariate Statistics of Dispersion' - teegan-chandler

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Presentation Transcript
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 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 ofDispersion

p 47

• The variance (S2X) is the square of the standard deviation:

• (3.7)

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

• 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

• 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

• S2Y = b2S2X and SY = |b|SX

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

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

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: Correlation Coefficient

p 54

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

• (3.18)