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MooreIntro-3620056ipsSeptember 24, 20107:541.2 Describing Distributions with Numbers43LINEAR TRANSFORMATIONSAlinear transformationchanges the original variablexinto the newvariablexnewgiven by an equation of the formxnew=a+bxAdding the constantashifts all values ofxupward or downward by thesame amount. In particular, such a shift changes the origin (zero point)of the variable. Multiplying by the positive constantbchanges the size ofthe unit of measurement.E X A M P L E1.30Change the units(a) If a distancexis measured in kilometers, the same distance in miles isxnew=0.62xFor example, a 10-kilometer race covers 6.2 miles. This transformationchanges the units without changing the origin—a distance of 0kilometers is the same as a distance of 0 miles.(b) A temperaturexmeasured in degrees Fahrenheit must be reexpressedin degrees Celsius to be easily understood by the rest of the world. Thetransformation isxnew=59(x32)= −1609+59xThus, the high of 95F on a hot American summer day translates into35C. In this casea= −1609andb=59This linear transformation changes both the unit size and the origin ofthe measurements. The origin in the Celsius scale (0C, thetemperature at which water freezes) is 32in the Fahrenheit scale.Linear transformations do not change the shape of a distribution.If measure-

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