9.10: Collaborative Activity
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)This collaborative activity will consider computing the numerical summary measures studied in this chapter on some data that will be simulated by rolling some dice. You will be provided with a six-sided, a ten-sided, and a twenty-sided die. You will need to designate one person in your group to oversee rolling the dice, while another person will collect the data. This activity will allow you to investigate how these measures work for three different types of known models for generating data which correspond to the three dice.
For the first part of the activity, you will use the six-sided die. The process for obtaining the data is quite simple. The six-sided die will be rolled ten times, and the outcomes will be listed on the data sheet provided in Table 9.13. This process will then be used for the ten-sided die and the twenty sided-die.
Table 9.13. The data table for the collaborative activity.
|
Dice |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
10 |
|
Six-Sided |
||||||||||
|
Ten-Sided |
||||||||||
|
Twenty-Sided |
For example, suppose that when the six-sided die is rolled, you observe the rolls 6, 4, 1, 3, 1, 4, 4, 3, 6, and 5. The first roll of 6 is entered into the table in the row corresponding to the six-sided die under the column for the first roll. See Table 9.14. The remaining rolls are then listed under the appropriate columns in this roll as shown in the table. Next, suppose that when the ten-sided die is rolled you observe the rolls 7, 10, 8, 8, 2, 5, 1, 6, 5, and 7. As with the six-sided die, these rolls are listed under the appropriate column and row. Finally, the process is repeated for the twenty-sided die. The example rolls are shown on the table.
Table 9.14. Example data table for the collaborative activity.
|
Dice |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
10 |
|
Six-Sided |
6 |
4 |
1 |
3 |
1 |
4 |
4 |
3 |
6 |
5 |
|
Ten-Sided |
7 |
10 |
8 |
8 |
2 |
5 |
1 |
6 |
5 |
7 |
|
Twenty-Sided |
7 |
2 |
18 |
13 |
12 |
6 |
18 |
5 |
15 |
17 |
Once the rolls have been completed, the group can work on calculating the mean, median, range, and standard deviation of the observed data. Begin by computing the mean of the rolls for each of the dice. This is done as demonstrated in this chapter: Add up all the observations and then divide by the number of observations. For the example data shown in Table 9.14, the mean roll for the six-sided dies is computed as
\[(6+4+1+3+1+4+4+3+6+5)\div 10=37\div 10=3.7. \]
Similarly, for the ten-sided die the mean roll is
\[(7+10+8+8+2+5+1+6+5+7)\div 10=59\div 10=5.9. \]
Finally, for the twenty-sided die the mean roll is
\[(7+2+18+13+12+6+18+5+15+17)\div 10=113\div 10=11.3. \]
The means can be reported on a table like the one shown in Table 9.15.
Table 9.15. The data table for the mean, median, range, and standard deviation for the collaborative activity.
|
Dice |
Mean |
Median |
Range |
Standard Deviation |
|
Six-Sided |
||||
|
Ten-Sided |
||||
|
Twenty-Sided |
To compute the median roll for each dice, you will first need to order the values from smallest to largest. Once they have been ordered, you will need to find a value that separates the smallest half of the data from the largest half of the data. Because there are ten values, the median will be somewhere between the fifth largest roll and the sixth largest roll. Following the suggestion from the chapter, we will average these two values together.
For the example data, the rolls for the six-sided die sorted from smallest to largest are 1, 1, 3, 3, 4, 4, 4, 5, 6, and 6. The fifth largest value is 4 and the sixth largest value is also 4, so when we take the mean of the two values we get \((4+4)\div 2=4\). Therefore, the median roll for the six-sided die is 4. Continuing with the rolls for the ten-sided die, the sorted values are 1, 2, 5, 5, 6, 7, 7, 8, 8, and 10. The fifth largest value is 6 and the sixth largest value is also 7, so when we take the mean of the two values we get \((6+7)div 2=6.5\). Therefore, the median roll for the six-sided die is 6.5. Finally, for the rolls for the twenty-sided die, the sorted values are 2, 5, 6, 7, 12, 13, 15, 17, 18, and 18. The fifth largest value is 12 and the sixth largest value is 13, so when we take the mean of the two values we get \((12+13)\div 2=12.5\). Therefore, the median roll for the six-sided die is 12.5.
To compute the range, we will use the ordered data values that we used when computing the median. The range is computed by subtracting the smallest value from the largest value. For the example data, the rolls for the six-sided die sorted from smallest to largest are 1, 1, 3, 3, 4, 4, 4, 5, 6, and 6. The largest value is 6 and the smallest value is 1; therefore the range is \(6-1=5\). Continuing with the rolls for the ten-sided die, the sorted values are 1, 2, 5, 5, 6, 7, 7, 8, 8, and 10. The largest value is 10 and the smallest value is 1, so the range is \(10-1=9\). Finally, for the rolls for the twenty-sided die, the sorted values are 2, 5, 6, 7, 12, 13, 15, 17, 18, and 18. The largest value is 18 and the smallest value is 2. The range is \(18-2=16\).
The final calculations required for this activity are to compute the standard deviation of the observed rolls for each die. Because the standard deviation is somewhat more complicated to compute than the other measures, it is helpful to use a table like the one shown in Table 9.6. We will consider computing the standard deviation for the example data for the six-sided die. The corresponding table is shown in Table 9.16. From this table we can observe that the variance of the example data for the six-sided die is 3.12. To compute the standard deviation, we take the square root of this value to get \(\sqrt{3.12}\approx 1.77\). The corresponding tables for the ten-sided and twenty-sided die are shown in Tables 9.17 and 9.18. From these tables we can observe that the variance for the example data for the ten-sided die is 7.65, which corresponds to a standard deviation of \(\sqrt{7.65}\approx 2.76\), and the variance for the example data for the twenty-sided die is 34.67, which corresponds to a standard deviation of \(\sqrt{34.67}\approx 5.89\).
The values of the summary statistics for the example data are shown in Table 9.19.
Table 9.16 The calculations used in computing the standard deviation of the of the data for the example data from the collaborative activity for the six-sided die. The data values are given in the first column, while for convenience the mean is listed in the second column. The squared absolute deviations are given in the fourth column.
|
Value |
Mean |
Absolute Deviation |
Squared Absolute Deviation |
|
6 |
3.7 |
2.3 |
5.29 |
|
4 |
3.7 |
0.3 |
0.09 |
|
1 |
3.7 |
2.7 |
7.29 |
|
3 |
3.7 |
0.7 |
0.49 |
|
1 |
3.7 |
2.7 |
7.29 |
|
4 |
3.7 |
0.3 |
0.09 |
|
4 |
3.7 |
0.3 |
0.09 |
|
3 |
3.7 |
0.7 |
0.49 |
|
6 |
3.7 |
2.3 |
5.29 |
|
5 |
3.7 |
1.3 |
1.69 |
|
Sum |
28.1 |
||
|
Variance |
3.12 |
||
Table 9.18 The calculations used in computing the standard deviation of the of the data for the example data from the collaborative activity for the ten-sided die. The data values are given in the first column, while for convenience the mean is listed in the second column. The squared absolute deviations are given in the fourth column.
|
Value |
Mean |
Absolute Deviation |
Squared Absolute Deviation |
|
7 |
5.9 |
1.1 |
1.21 |
|
10 |
5.9 |
4.1 |
16.81 |
|
8 |
5.9 |
2.1 |
4.41 |
|
8 |
5.9 |
2.1 |
4.41 |
|
2 |
5.9 |
3.9 |
15.21 |
|
5 |
5.9 |
0.9 |
0.81 |
|
1 |
5.9 |
4.9 |
24.01 |
|
6 |
5.9 |
0.1 |
0.01 |
|
5 |
5.9 |
0.9 |
0.81 |
|
7 |
5.9 |
1.1 |
1.21 |
|
Sum |
68.9 |
||
|
Variance |
7.65 |
||
Table 9.19 The calculations used in computing the standard deviation of the of the data for the example data from the collaborative activity for the twenty-sided die. The data values are given in the first column, while for convenience the mean is listed in the second column. The squared absolute deviations are given in the fourth column.
|
Value |
Mean |
Absolute Deviation |
Squared Absolute Deviation |
|
7 |
11.3 |
4.3 |
18.49 |
|
2 |
11.3 |
9.3 |
86.49 |
|
18 |
11.3 |
6.7 |
44.89 |
|
13 |
11.3 |
1.7 |
2.89 |
|
12 |
11.3 |
0.7 |
0.49 |
|
6 |
11.3 |
5.3 |
28.09 |
|
18 |
11.3 |
6.7 |
44.89 |
|
5 |
11.3 |
6.3 |
39.69 |
|
15 |
11.3 |
3.7 |
13.69 |
|
17 |
11.3 |
5.7 |
32.49 |
|
Sum |
312.10 |
||
|
Variance |
34.68 |
||
Table 9.19. Example data table for the mean, median, range, and standard deviation for the collaborative activity.
|
Dice |
Mean |
Median |
Range |
Standard Deviation |
|
Six-Sided |
3.7 |
4.0 |
5 |
1.76 |
|
Ten-Sided |
5.9 |
6.5 |
9 |
2.76 |
|
Twenty-Sided |
11.3 |
12.5 |
16 |
5.89 |
Questions
- Compare the means of the ten rolls of the three different dice. Given what you know about the dice, namely the number of sides of each and the fact that each face has an equal probability, do the relative values of the means provide good information about the relative values of a typical roll from each die? If you rolled each die a very large number of times, what value do you think the mean roll for each die would be equal to?
- Compare the medians of the ten rolls of the three different dice. Given what you know about the dice, namely the number of sides of each and the fact that each face has an equal probability, do the relative values of the medians provide good information about the relative values of a typical roll from each die? How do the medians compare to the means? If you rolled each die a very large number of times, what value do you think the mean roll for each die would be equal to?
- Compare the ranges of the ten rolls of the three different dice. Given what you know about the dice, namely the number of sides of each and the fact that each face has an equal probability, do the relative values of the ranges provide good information about the relative variability of the rolls from each die? If you rolled each die a very large number of times, what value do you think the range of the rolls for each die would be equal to?
- Compare the standard deviations of the ten rolls of the three different dice. Given what you know about the dice, namely the number of sides of each and the fact that each face has an equal probability, do the relative values of the ranges provide good information about the relative variability of the rolls from each die?