91Ó°ÊÓ

Consider the data set (3,-5,7,4,8,2,8,-3,-6\\} (a) Find the average \(A\) of the data set. (b) Find the median \(M\) of the data set. (c) Consider the data set \\{3,-5,7,4,8,2,8,-3,-6,2\\} obtained by adding one more data point to the original data set. Find the average and median of this data set.

For Exercises 51 through \(54,\) you should use the following definition of an outlier: An outlier is any data value that is above the third quartile by more than 1.5 times the \(I Q R\) [Outlier \(>Q_{1}+1.5(I Q R) /\) or below the first quartile by more than 1.5 times the IQR [Outlier \(

Exercises 73 and 74 refer to histograms with unequal class intervals. When sketching such histograms, the columns must be drawn so that the frequencies or percentages are proportional to the area of the column. Figure 21 illustrates this. If the column. over class interval 1 represents \(10 \%\) of the population, then the column over class interval \(2,\) also representing \(10 \%\) of the population, must be one-third as high, because the class interval is three times as large (Fig. 21 ).. If the height of the column over the class interval \(20-30\) is one unit and the column represents \(25 \%\) of the population, then (a) how high should the column over the interval \(30-35\) be if \(50 \%\) of the population falls into this class interval? (b) how high should the column over the interval \(35-45\) be if \(10 \%\) of the population falls into this class interval? (c) how high should the column over the interval \(45-60\) be if \(15 \%\) of the population falls into this class interval?

Chebyshev's theorem. The Russian mathematician P. L. Chebyshev \((1821-1894)\) showed that for any data set and any constant \(k\) greater than \(1,\) at least \(1-\left(1 / k^{2}\right)\) of the data must lie within \(k\) standard deviations on either side of the mean \(A\). For example, when \(k=2\), this says that \(1-\frac{1}{4}=\frac{3}{4}(i, e ., 75 \%)\) of the data must lie within two standard deviations of \(A\) (i.e., somewhere between \(A-2 o\) and \(A+2 \sigma)\) (a) Using Chebyshev's theorem, what percentage of a data set must lie within three standard deviations of the mean? (b) How many standard deviations on each side of the mean must we take to be assured of including \(99 \%\) of the data? (c) Suppose that the average of a data set is \(A\). Explain why there is no number \(k\) of standard deviations for which we can be certain that \(100 \%\) of the data lies within \(k\) standard deviations on either side of the \(\operatorname{mean} A\)