,Click to Edit Master Title Style,Click to edit Master text styles,Second Level,Third Level,1,Slide,2011 Cengage Learning.All Rights Reserved.May not be scanned,copied,or duplicated,or posted to a publicly accessible website,in whole or in part.,Statistics for Businessand Economics,Anderson Sweeney Williams,Slides by,John Loucks,St.Edwards University,Statistics for Businessand Ec,Chapter 3,Part A Descriptive Statistics:Numerical Measures,Measures of Location,Measures of Variability,Chapter 3,Part A Descriptive,Measures of Location,If the measures are computed,for data from a sample,they are called,sample statistics,.,If the measures are computed,for data from a population,they are called,population parameters,.,A sample statistic is referred to,as the,point estimator,of the,corresponding population parameter.,Mean,Median,Mode,Percentiles,Quartiles,Measures of LocationIf the mea,Mean,The,mean,of a data set is the average of all the data values.,The sample mean is the point estimator of the population mean,m,.,Perhaps the most important measure of location is the,mean,.,The mean provides a measure of,central location,.,MeanThe mean of a data set is,Sample Mean,Number of,observations,in the sample,Sum of the values,of the,n,observations,Sample Mean Number ofSum of th,Population Mean,m,Number of,observations in,the population,Sum of the values,of the,N,observations,Population Mean mNumber ofSum,Seventy efficiency apartments were randomly,sampled in a small college town.The monthly rent,prices for these apartments are listed below.,Sample Mean,Example:Apartment Rents,Seventy efficiency apartm,Sample Mean,Example:Apartment Rents,Sample Mean Example:Apartm,Median,Whenever a data set has extreme values,the median,is the preferred measure of central location.,A few extremely large incomes or property values,can inflate the mean.,The median is the measure of location most often,reported for annual income and property value data.,The,median,of a data set is the value in the middle,when the data items are arranged in ascending order.,Median Whenever a data set h,Median,12,14,19,26,27,18,27,For an,odd number,of observations:,in ascending order,26,18,27,12,14,27,19,7 observations,the median is the middle value.,Median=19,Median12141926271827 For an,12,14,19,26,27,18,27,Median,For an,even number,of observations:,in ascending order,26,18,27,12,14,27,30,8 observations,the median is the average of the middle two values.,Median=(19+26)/2=22.5,19,30,12141926271827Median For an,Median,Averaging the 35th and 36th data values:,Median=(475+475)/2=475,Note:Data is in ascending order.,Example:Apartment Rents,MedianAveraging the 35th and 3,Trimmed Mean,It is obtained by deleting a percentage of the,smallest and largest values from a data set and then,computing the mean of the remaining values.,For example,the 5%trimmed mean is obtained by,removing the smallest 5%and the largest 5%of the,data values and then computing the mean of the,remaining values.,Another measure,sometimes used when extreme,values are present,is the,trimmed mean,.,Trimmed Mean It is obtained,Mode,The,mode,of a data set is the value that occurs with,greatest frequency.,The greatest frequency can occur at two or more,different values.,If the data have exactly two modes,the data are,bimodal,.,If the data have more than two modes,the data are,multimodal,.,Caution:If the data are bimodal or multimodal,Excels MODE function will incorrectly identify a,single mode.,Mode The mode of a data set,Mode,450 occurred most frequently(7 times),Mode=450,Note:Data is in ascending order.,Example:Apartment Rents,Mode450 occurred most frequent,Percentiles,A percentile provides information about how the,data are spread over the interval from the smallest,value to the largest value.,Admission test scores for colleges and universities,are frequently reported in terms of percentiles.,The,p,th percentile,of a data set is a value such that at least,p,percent of the items take on this value or less and at least(100-,p,)percent of the items take on this value or more.,Percentiles A percentile prov,Percentiles,Arrange the data in ascending order.,Compute index,i,the position of the,p,th percentile.,i,=(,p,/100),n,If,i,is not an integer,round up.The,p,th percentile,is the value in the,i,th position.,If,i,is an integer,the,p,th percentile is the average,of the values in positions,i,and,i,+1.,Percentiles Arrange the data i,80,th,Percentile,i,=(,p,/100),n,=(80/100)70=56,Averaging the 56,th,and 57,th,data values:,80th Percentile=(535+549)/2=542,Note:Data is in ascending order.,Example:Apartment Rents,80th Percentilei=(p/100)n=,80,th,Percentile,“At least 80%of the,items take on a,value of 542 or less.”,“At least 20%of the,items take on a,value of 542 or more.”,56/70=.8 or 80%,14/70=.2 or 20%,Example:Apartment Rents,80th Percentile“At least 80%o,Quartiles,Quartiles are specific percentiles.,First Quartile=25th Percentile,Second Quartile=50th Percentile=Median,Third Quartile=75th Percentile,Quartiles Q