# GGR270H1 Lecture Notes - Covariance, Standard Score, Standard Deviation

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GGR270&Lecture& & October&3,&2012&

&

Measures&of&Dispersion&

• Coefficient&of&variation&(continued)&

o Would&calculate&for&each&individual&sample&and¬e&which&one&has&

the&highest°ree&of&variability.&

&

Practical&Significance&of&Standard&Deviation&

• Tchebysheff’s&Theorem&and&Empirical&Rule&

o We’re&relating&the&standard&deviation&to&the&normal&curve&

o On&a&normal&distribution&(bell&curve)&we&know&the&mean&is&in&the&

center.&

o &And&then&we&ask&how&do&we&interpret&the&standard&deviation?&

o We&always&say&that&the&standard&deviation&is&equal&to&plus/minus.&

o Typically&we&don’t&go&further&than&+/U&3&standard&deviations&on&a&

normal&curve.&&

o We&use&it&in&terms&of&understanding&distributions&of&marks&and&

predicting&the&possibility&of&particular&values&cropping&up,&we&can&also&

view&it&in&terms&of&probability.&

o Empirical&rule&only&works&with&normal&data.&

o See&charts&on¬es.&

• Z&scores&

o Standardizes&any&value&on&the&above&curve,&so&that&we&can&compare&

any&value&that&we&have&to&our&mean&and&standard&deviation.&

o Standard&scores&are&referenced&to&as&Z"Scores.&

o They&indicate&how&many&standard&deviations&separate&a&particular&

value&from&the&mean.&

o The&standard&deviation&is&+/U&1&value&from&the&mean&for&example,&1&is&

our&standardized&distance&–&but&this&approach&takes&any&value&for&

example&1.2&standard&deviations.&

o Z&scores&can&be&+&or&–&depending&on&if&they&are&>&or&<&than&the&mean.&&

o Z&score&of&the&mean&is&0&and&the&standard&deviation&is&+&or&–&1.&

o Table&of&Normal&Values&provides&probability&information&on&a&

standardized&scale.&

We&can&also&calculate&Z&scores.&

o Formula&involves&comparing&values&to&the&mean&value,&and&dividing&by&

the&standard&deviation.&

o Result&is&interpreted&as&the&‘number"of"standard"deviations"an"

observation"lies"above"or"below"the"mean’.&

o Z&=&X&U&&&&&&&&/&S&

&

Where:&S&–&Standard&deviation.&

&X&–&is&each&value&in&the&data&set.&

&&&&&&−&is&the&mean&of&all&values&in&the&data&set.&&

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## Document Summary

Where: s standard deviation. X is each value in the data set. Is the mean of all values in the data set: z scores. Consists of pairs of variables such as years of schooling or income. Positive: as x increases, y increases. Negative: as x increases, y decreases; as y increases, x decreases. Neutral: absolutely no association/relationship between variables; no direction: strength of the bivariate relationship. Perfect association: the variables move in the same direction; positive or negative; as x increases by 10% y increases by. 10%; absolutely perfect relationship between variables. Weak association: the weaker the relationship, the more scattered the plot. No association: blob of point; no direction; no strength; no relationship between the variables: correlation coefficients. More rigorous approach to observing and measuring strength and direction of a bivariate relationship. Most constructed have a maximum value of +1. 0 and can be positive or negative: +1. 0 perfect positive relationship. Most common measure is pearson"s product moment.