Timetostart24 timetostart25 use your knowledge 159

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TIMETOSTART24TIMETOSTART25USE YOUR KNOWLEDGE1.59Effect of an outlier on theIQR.Find theIQRfor the time to start abusiness with and without Suriname. What do you conclude about thesensitivity of this measure of spread to the inclusion of an outlier?Choosing measures of center and spreadHow do we choose between the five-number summary andxandsto describethe center and spread of a distribution? Because the two sides of a stronglyskewed distribution have different spreads, no single number such assde-scribes the spread well. The five-number summary, with its two quartiles andtwo extremes, does a better job.CHOOSING A SUMMARYThe five-number summary is usually better than the mean and standarddeviation for describing a skewed distribution or a distribution withstrong outliers. Usexandsonly for reasonably symmetric distributionsthat are free of outliers.Remember that a graph gives the best overall picture of a distribution. Nu-merical measures of center and spread report specific facts about a distribution,but they do not describe its entire shape.Numerical summaries do not disclosethe presence of multiple modes or gaps, for example.Always plot your data.Changing the unit of measurementThe same variable can be recorded in different units of measurement. Ameri-cans commonly record distances in miles and temperatures in degreesFahrenheit, while the rest of the world measures distances in kilometers andtemperatures in degrees Celsius. Fortunately, it is easy to convert numericaldescriptions of a distribution from one unit of measurement to another. Thisis true because a change in the measurement unit is alinear transformationofthe measurements.
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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Term
Spring
Professor
Yukheev
Tags
Statistics, Normal Distribution, Bar chart, Histogram

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