**Continuous data** are data without natural categories. These are usually
measurements such as height, weight, age, temperature, or cholesterol.
For weight one might think that 200 is a natural category, but in kilograms
200 pounds is 90.8 KG which is not even an integer. Because we can not
measure infinitely precise, measurements are approximations.

Example: Here is a sample of head sizes (maximum measurement across the top of the skull in mm) of 25 Etruscans. This data was taken form the data set Etruscan-Italian head sizes data set given in Appendix A.

141 148 132 138 154 142 150 146 155 158 150 140 147 148 144 150 149 145 149 158 143 141 144 144 126So what do we need? A picture, of course. The above picture for the discrete data is a nice visual summary. So we need a sampling distribution of these numbers. Since continuous data have no natural categories we have to create some categories. This results in

12 6 13 28 14 182607849593144 15 4058008Do you like the picture? No, neither do I. The numbers are too bunched up. We need more categories (stems). But this is easy to do with stem-leaf plots. Lets split each stem into two. In this case the leaves 0 through 4 go on the lower stem while the leaves 5 through 9 go on the upper stem. The picture is

12 6 13 2 13 8 14 12043144 14 8678959 15 4000 15 588This picture is better than the first. I wouldn't split the stems again. Although we only have 25 numbers here, certainly the picture is much more informative than looking at the above string of numbers. We can see immediately that in this sample most Etruscans have head sizes between 140-150 mm and there are a few with smaller head sizes.

Note that a stem-leaf plot is also a

- 1.
- Consider again Carrie's baseball data given in Appendix A. Glance through the weights (second column) the baseball players. What does a typical baseball player weigh? Do more baseball players weigh over 200 pounds than under 170?
- 2.
- Obtain a stem-leaf plot of the weights of the baseball players. Now answer the questions in the last problem. For your stem-leaf plot, should the stems be split or grouped together?
- 3.
- The typical American male weighs about 170-175 pounds. Based on your stem-leaf plot, how would you compare the weights of baseball players to typical American males?
- 4.
- The typical American male height is 70 inches. What about the heights of baseball players? Base your answer on a stem-leaf plot of the baseball players' heights.
- 5.
- Obtain a stem-leaf plot of the following using the summary module.
Data 14 117 77 81 205 21 22 157 134 69 193 8 162 0 156 194 17 100 50 53 235 29 191 81 167 29 158 105 171 2 8 89 82 11 247 149 106 61 18 172

Try the same example data (given below). Choose stemleaf in the **summary** module after entering the data.

12 18 25 15 9 14 21 25 28 125

We need a little on **shapes of distributions** so that we can discuss
them. We will classify distributions as **symmetric** or **asymmetric** .
Symmetric distributions are (approximately if it's a sample distribution)
symmetric about a point on the data axis. An example of a symmetric sample
distribution is given by:

Low: 49 6 : 4 6 : 78 7 : 14 7 : 556788 8 : 0122334 8 : 67799 9 : 01122223334 9 : 5555666677788889999 10 : 000000001122223444 10 : 568889 11 : 000001134 11 : 599 12 : 123 12 : 89 13 : 2 13 : 56 14 : 0 High: 161To avoid many empty stems on the ends of stem-leaf plots sometimes, as in the above plot, the low and the high points are just indicated, as 49 and 161 are here. The point of symmetry in this plot is close to 95.

The above plot is unimodal , a single

-1 : 2 -0 : 0 : 2 1 : 5 2 : 5669 3 : 1125677 4 : 223445556699 5 : 0111233344444566667788888999 6 : 001122244444445555555567788899 7 : 014445566778899 8 : 0122334455677799 9 : 011122223334455556666777788889999 10 : 000000001122223444568889 11 : 0000011123334599 12 : 0122389 13 : 256 14 : 0 15 : 16 : 1A distribution is a

0 : 1223444444 0 : 556666667777777888999 1 : 0011111123333344 1 : 5555566788889999 2 : 011222333334 2 : 56666789999 3 : 0114 3 : 668 4 : 02 4 : 58 5 : 02 High: 617Another plot for continuous data that we will frequently use is the

. . . . . . : .. : .. . :: : .. : -+---------+---------+---------+---------+---------+----- 126.0 132.0 138.0 144.0 150.0 156.0

An interesting application of dotplots concerns **comparison dotplots**
of several data sets. Suppose we have several data sets that we want to
compare. Simply draw one number line. Then for each sample put a row of
dots corresponding to the measurements. For example here are skull
measurements of 20 modern Italians taken from the data set Etruscan-Italian
head sizes data set.

133 128 136 140 127 136 131 131 128 132 125 133 134 136 134 129 132 139 143 138

Here is the comparison dotplot between the Italian skull sizes and the above Etruscan skull sizes.

. . Etruscan . . . . : .. : .. . :: : .. : -+---------+---------+---------+---------+---------+----- . Italian . .: . : : :: : . .. . -+---------+---------+---------+---------+---------+----- 126.0 132.0 138.0 144.0 150.0 156.0Any conclusions about the Etruscan and Italian skull sizes? It appears that the Etruscans have larger heads than the Italians. As the exercise below shows, this difference occurs when all the data are used. We will discuss inference based on this data set in Chapter 8.

- 1.
- Obtain a dot plot of the weights of the twenty students discussed above and listed again as :
Weights 122 146 65 162 148 155 136 151 151 153 201 156 235 157 160 171 178 197 142 131

- 2.
- For Carrie's baseball data, obtain comparison dotplots of the batting averages (6th column of the data for the hitters only, (signified by a 1 in the 5th column)) by the side of the plate they hit from R, L or Switch, signified by a 1, 2 or 3 in column 3.
- 3.
- Obtain comparison dotplots of the the Etruscan and Italian data given in Appendix A. Note that the Etruscans formed an ancient civilization in Truscany, Northern Italy, that predated the Romans. There is some question as to where the Etruscans came from. Were they native to Italy or not? Draw conclusions about this mystery based on the comparison dotplots.
- 4.
- Obtain stem-leaf plots and comparison dotplots for the following 3 samples. Comment on the shape of each.
Sample 1 76 183 125 24 8 59 25 179 29 101 55 108 68 128 5 12 35 25 122 39 59 91 90 81 66 20 178 111 186 26 5 123 124 45 13 79 158 20 92 23 Sample 2 66 9 62 21 11 39 21 24 21 19 67 71 67 0 4 82 32 91 152 124 20 108 5 63 1 10 23 125 59 25 Sample 3 59 54 19 79 22 81 18 67 61 53 71 14 10 87 76 49 21 16 35 11 7 77 90 6 79 55 83 28 11 60 55 43 9 65 25