CHAPTER 1 Overview and Descriptive Statistics
inference, we will present methods based on the sample mean for drawing conclu- sions about a population mean. For example, we might use the sample mean
computed in Example 1.14 as a point estimate (a single number that is our “best” guess) of crack length for all specimens treated as described.
The mean suffers from one deficiency that makes it an inappropriate measure of center under some circumstances: Its value can be greatly affected by the presence of even a single outlier (unusually large or small observation). In Example 1.14, the value is obviously an outlier. Without this observation,
; the outlier increases the mean by more than 1 If the 45.0 observation were replaced by the catastrophic value 295.0 a really extreme outlier, then , which is larger than all but one of the observations!
A sample of incomes often produces such outlying values (those lucky few who earn astronomical amounts), and the use of average income as a measure of location will often be misleading. Such examples suggest that we look for a meas- ure that is less sensitive to outlying values than , and we will momentarily pro- pose one. However, although does have this potential defect, it is still the most widely used measure, largely because there are many populations for which an extreme outlier in the sample would be highly unlikely. When sampling from such a population (a normal or bell-shaped population being the most important example), the sample mean will tend to be stable and quite representative of the sample.
The Median The word median is synonymous with “middle,” and the sample median is indeed the middle value once the observations are ordered from smallest to largest. When the observations are denoted by , we will use the symbol to represent the sample median.
x|x1, c, xn
x 5 694.8/21 5 33.09 mm,mm mm.x 5 399.8/20 5 19.99
x14 5 45.0
m 5 the true average x 5 21.18
The sample median is obtained by first ordering the n observations from smallest to largest (with any repeated values included so that every sample observation appears in the ordered list). Then,
The single middle value if n is odd The average of the two middle values if n is even
People not familiar with classical music might tend to believe that a composer’s instructions for playing a particular piece are so specific that the duration would not depend at all on the performer(s). However, there is typically plenty of room for interpretation, and orchestral conductors and musicians take full advantage of this. The author went to the Web site ArkivMusic.com and selected a sample of
5 an 1 1 2
b th ordered value
5 average of an 2 b thand an
2 1 1b th ordered values