Chapter 1. Overview and Descriptive StatisticsSTAT 155Exercise 1.27The paper “Study on the Life Distribution of Microdrills” (J.of Engr. Manufacture, 2002: 301305) reported the following observations,listed in increasing order, on drill lifetime (number of holes that a drill ma-chines before it breaks) when holes were drilled in a certain brass alloy.11142023313639444750596165676871747678798184858991939699101104105105112118123136139141148158161168184206248263289322388513(a) Construct a relative frequency histogram based on the equal-widthclass intervals 0-<50, 50-<100, 100-<150,· · ·, and comment onfeatures of the histogram.(b) Construct a histogram of the natural logarithms of the lifetime obser-vations, and comment on interesting characteristics.(c) What proportion of the lifetime observations in this sample are lessthan 100? What proportion of the observations are at least 200?14
Chapter 1. Overview and Descriptive StatisticsSTAT 15515
Chapter 1. Overview and Descriptive StatisticsSTAT 1551.3 Measures of Location•mean•median•quartiles•trimmed meanMean– the arithmetic average of the data.Mean is calcu-lated as the sum of all observations divided by the number ofobservations.Sample meanxof observationsx1, x2, x3,· · ·, xnis:x=x1+x2+· · ·+xnn=∑ni=1xinFor reportingx, it is recommended to use a decimal accuracyof one digit more than the accuracy of thexi’s.The arithmetic mean is the most widely used measure of cen-tral location. However, it is oversensitive to extreme/outlyingvalues and must be used with caution.Median– the middle value.Sample medianexis the middle sorted observation. That is,we want a value such that half of the data is smaller than itand half is greater than it.Steps to find the sample median:1. Rank thenobservations from smallest to largest.16
Chapter 1. Overview and Descriptive StatisticsSTAT 1552. Ifnis odd, median equals to the middle value,ex= (n+12)thordered value;Ifnis even, there are two middle values whose averageequals the median,ex= average of(n2)thand(n+12)thordered values.Unlike the mean, the median is insensitive to extreme/outlyingvalues.In many samples, the relationship between the arithmetic meanand the sample median can be used to assess the shape of adistribution.