Chapter 3 Characterizing Data Visually. A picture is worth a thousand words.

Transcription

1 Chapter 3 Characterizing Data Visually A picture is worth a thousand words. Data are important because they allow ideas about the world to be proven wrong. In science, hypotheses are constructed and evaluated using data. While Thomas Henry Huxley actually wrote the apt words, "The great tragedy of Science--the slaying of a beautiful hypothesis by an ugly fact," he could have just as easily written that "the most elaborate of hypotheses can by brought down by the most modest of data." Equally important is the fact that new research questions and hypotheses often follow from a careful analysis of data. The acquisition of data begins with a question, and is followed by a series of observations on phenomena or behavior that provide us with the information necessary to address the question within our theoretical structure. For example, if we wish to know if education influences voting patterns, we must gather information on voting patterns and education. The same holds true if we wish to know what characteristics of the landscape influenced the kinds of tools people used, and as a consequence, the artifacts we encounter as archaeologists; we must gather information or data on characteristics of the landscape and the kinds of artifacts recovered. The immediate product of our problem driven observations is a simple list of numbers or qualities (i.e., variates) that are data. Often, especially with larger sets of data, it is impossible to fully identify trends or patterns from the raw data themselves. In order to accurately understand and interpret data, we must characterize them so that we can identify trends without having to remember every single variate. Characterizing data here is nothing more than the creation of an effective visual summary or description of them. Let us consider first a situation where our data consist of a variable that is continuously measured, providing us with a simple list of numbers. The summary or description of the list should take two forms that we see as sequential steps to the effective use of data in analysis. The first step is always the construction of a visual representation of the data. The second step is a numerical summary or description of the 3.1

2 data, which will be discussed in Chapter Four. We begin with a visual representation of the data because it provides information that is useful during the numerical summary, as well as providing a unique depiction of the shape or structure of the data. Consider the data presented in Table 3.1. The data are the depths of features observed on the Carrier Mills archaeological project (Jeffries and Butler 1982). Feature depths were recorded to assist in the determination of feature function. Do the data make much sense? Probably not. It would be a truly unique individual who could discern a pattern from this jumbled bunch of numbers. As fine an instrument as the human brain is, it has a difficult time organizing these numbers in any meaningful way without help. The difficulty of dealing with data sets is especially illustrated by the fact that there are only 91 variates here, many less than constitutes many data sets. Many individuals might be tempted to simply run head on into advanced statistical analysis of these data without a second thought. That would be a mistake. Each statistical method is developed for specific situations and requires its own assumptions about the characteristics of the data. While the specific assumptions associated with each statistical method will be discussed in future chapters, it is important to understand that the patterns in the data themselves in conjunction with the questions we ask determine what statistical procedures are acceptable. Statistics are like a fine piece of clothing that must be tailor fit for an individual. How can a suit or dress be altered if you don t know the shape of the individual who will wear it? Therefore, the best place to start any statistical analysis is with a simple visual description. Frequency Distributions Traditionally, analysts provide visual representations through the creation of a frequency distribution. Figure 3.1 is a frequency distribution of the data presented in Table 3.1. The continuous variable of feature depth has been divided into 22 classes or groups with a class interval (width) of 1cm. In Figure 3.1, Y represents the variable feature depth. Scanning down the column, we see that the values range from 3 to 24 cm. The implied limits represent our level of accuracy. For example, a variate with a value of 3 mm has a true depth that is 3.2

3 between The tally marks represent the number of times each value was observed and noted in Table 3.1. For example, the value 3 was observed 2 times in Table 3.1. The column under the symbol f represents the frequency with which each value was observed and is calculated by summing the tally marks. f contains the same information as the tally marks, but in different form. You have no doubt noticed the menacing presence of the figure f in the lower left corner of Figure 3.1. While the symbols look intimidating and complicated, is merely the capital Greek letter sigma, and represents the term for summation. This benign symbol will be used throughout the rest of the book. f simply represents the summation of all of our frequencies for the variable Y, or 91 ( ). The tally marks provide a visual perspective on how the data are distributed across the variable classes. This perspective is called a distribution, and tally marks are one way of creating a distribution. A visual inspection of the distribution created by the tally marks shows us that the observations (data) cluster toward the values 9 thruogh13 with fewer values toward the top and bottom of the table. Figure 3.2 shows these same data in a slightly different manner. Rather than employ a class interval of 1cm, a class interval of 2 cm was used, halving the number of classes. The result is a more clearly clustered set of tally marks. The tendency evident in Figure 3.1 for most observations to fall toward the center of the distribution and fewer toward the extremes is accentuated in Figure 3.2. Since two values from Figure 3.1 are collapsed for each class in Figure 3.2 (e.g., values 3 and 4 now constitute only one class in Figure 3.2, where they were two classes in Figure 3.1), it is no longer possible only to present these values under the column for the variable Y as in Figure 3.1. Instead, the class mark is used. The class mark represents the midpoint of the values combined into each of the eleven classes. The implied limits represent the new levels of accuracy, and the practical limits represent the limits of actual values that are encountered in Table 3.1. Once again, f represents the frequency of observations for each class, that is, a counting of the tally marks. f represents the sum of all frequencies in the figure. Figure 3.3 is similar to Figures 3.1 and 3.2 except that the class interval is 5 cm, and the class midpoints have been changed accordingly. 3.3

4 A consideration of the different number of classes and class intervals employed in Figures 3.1, 3.2, and 3.3 shows that the decision made regarding the number of classes and the size of the intervals affect the shape of our distribution. Excessive numbers of classes tend to spread out the distribution and can obscure patterns that interested us. Too few classes cause too many variates to be lumped together, which can also obscure meaningful patterns. Somewhere between these extremes is an adequate number of classes that will illuminate the characteristics of the distribution effectively. Unfortunately, there is no hard and fast rule to determine the ideal number of classes. A general rule of thumb is that 10 to 20 classes provide a good representation of most distributions, although some data sets may be so large and varied that many more than 20 classes will be needed. Researchers should try a number of class intervals to determine which interval best depicts the data. In our example, Figure 3.2 is the best. Figure 3.1 is quite spread out, and Figure 3.3 is so clustered that useful information is likely obscured. The number of classes and the class interval to be employed should be conscientiously considered by the investigator, but oftentimes the decision is more art than science. While many computer programs use algorithms that make good decisions regarding class intervals using a variety of techniques, we should not slavishly follow their direction. Most good programs allow the analyst the option of specifying the number of classes. In order to demonstrate how frequency distributions are constructed, Figure 3.4 presents the framework for the construction of a frequency distribution of heights of individuals in your class. Should your instructor choose to do so, you may construct your own frequency distribution. Histograms Another common way of visualizing data is through the construction of a histogram. Histograms graphically represent the frequency of data in various classes by listing the class midpoints on the horizontal axis and class frequencies on the vertical axis. A histogram looks similar to tally marks from a frequency table laid on their side. 3.4

5 Let us use an example of data gathered in the Gallina area of New Mexico to illustrate the histogram. Table 3.2 presents 100 measurements of minimum sherd thickness taken on sherds to examine differences in ceramic technology across settlements in the project area. These measurements are taken using size grades divided into.5 mm increments. Thus, the value 4.5 mm reflects a sherd with a minimum thickness between 4.5 and 4.9 mm. Again, patterns in the raw data are difficult to identify. To make sense of our data, we must first organize them. Figure 3.5 presents a frequency distribution of these data. Once a frequency distribution is constructed, it is quite easy to construct a histogram a visual that is slightly more aesthetically pleasing (and publishable) than the tallies, lines, and numbers in Figure 3.5. Figure 3.6 is a histogram of these data presented in Table 3.2. For this histogram, the metric scale of measurement is depicted on the horizontal axis of the graph. All variates are counted for each class, and the count of variates is depicted on the vertical axis represented by the vertical bars called bins. For example, there are 35 variates with a minimum sherd width of 5.0 mm. The bin associated with the class of 5.0 mm rises to the level associated with a frequency of 35 variates. This histogram provides an excellent visual presentation of the shape, or distribution, of the data. We can see that the distribution is roughly symmetrical, and that values toward the center are more common than extreme values. By grouping the observations into categories of a consistent interval we have been able to gain information about the distribution of our variates that could not be obtained through a simple visual inspection of the raw data. Visualize how an increase or decrease in the class interval would affect the shape of the distribution. A larger class interval would result in fewer bins, or bars, and they would have more variates in each class causing higher peaks. A smaller class interval would leave many empty bins because of the scale of measurement of the data. If our original measurements had been carried out to more decimal points, a smaller class interval would result in a lower peak, and a more widely spread distribution. In both cases, the distribution would remain roughly symmetrical. 3.5

6 Stem and Leaf Diagrams Another useful way to view data is through the construction of a stem and leaf diagram. Figure 3.7 is a stem and leaf diagram of the sherd thickness data presented in Table 3.2. While the diagram looks complex, it is easy to build and is very informative, once one knows how it is constructed. The stem is the vertical column of numbers to the left of the vertical lines. The leaf refers to those numbers to the right of the vertical line. Each leaf, along with the associated stem to its left, reflects an individual variate. In this particular example, the stem represents the number to the left of the decimal point in our data. The smallest value is 2.0, and the largest 7.5. Therefore, the limits of our stem are 2 (top) and 7 (bottom). The leaf reflects the value to the right of the decimal point in the data. In the Gallina data, the numbers to the right of the decimal points are all 0 or 5. We therefore know that all of our leafs will be 0 or 5. By looking at the stem and leaf diagram, we can reconstruct our raw data. For example, we can tell from the diagram the three variates are 2.0 mm, six are 3.0 mm, and two variates have a value of 3.5 mm. The stem and leaf can be constructed by reading across your data from any direction, which results in the leafs being unordered with respect to magnitude, as in Figure 3.7. Reordering the leafs as depicted in Figure 3.8 often makes the diagram easier to work with. Regardless of the reordering, the shape of the distributions are identical. In the same way that it is useful to change the numbers of classes in a frequency distribution, it is sometimes desirable to change the numbers of classes in a stem and lead diagram. Doing so is quite easy and is accomplished by dividing the leafs further. For example, Figure 3.9 is a stem and leaf diagram of the same data used in the construction of Figure 3.7, but it differs from Figure 3.7 in that each stem is repeated twice (there are two 2s, two 3s, two 4s, etc.). The first stem for 2 includes all stems between 2.0 and 2.4, whereas the second 2 will reflect all stems between 2.5 and 2.9. The same is true for all of the other number pairs. The leafs that are 0 are consequently grouped with the first in a pair of stems, whereas the leafs that are 5 are grouped with the second. Using these same principles, we can divide the stem and leaf diagram into as many or as few classes 3.6

7 as we wish. The only requirement is that each stem reflects the same increment of measurement. Knecht (1991) provides us with an additional set of data with which to construct a stem and leaf diagram (Table 3.3). These data are maximum thickness measurements of a sample of losange-shaped (leaf-shaped) Early Paleolithic projectile point bases. Use this data to finish placing the leafs on the two stems provided in Figure Ogives Another kind of frequency distribution that is very useful is the cumulative frequency distribution or ogive. Figure 3.11 is a cumulative frequency distribution for the Carrier Mills feature data (Table 3.1). It is similar to a normal frequency distribution except that the cumulative frequency corresponding to a given Y is created by summing the fs of all Y's of lower value with the f of the given Y. For example, for Y=4, f (cumulative) is equal to the sum of its frequency (2) with the frequency for the previous value of Y=3 (2). Therefore, f (cumulative) for Y=4 is equal to = 4. The f (cumulative) for the last Y, where Y=24, is the sum of all previous f's, or 91. The cumulative frequency distribution is often useful when we wish to compare the shapes of two distributions by contrasting their cumulative values visually, an aspect of the their use that will be discussed later in the book. Figure 3.12 provides a graphic presentation of Carrier Mills feature data (Table 3.1). Describing a Distribution After we build a distribution, we can characterize it in a number of ways. Often, though, we begin by identifying its degree of symmetry. Figure 3.13 presents a distribution that is perfectly symmetrical. This distribution may be more familiar to you as the normal distribution, which we will discuss in greater detail later. Nonsymmetrical distributions are described as skewed. Figure 3.14 presents a plot of a distribution that is skewed to the left and Figure 3.15 is a distribution skewed to the 3.7

8 right, a very common distribution in archaeological data. It is often difficult to remember the difference between distributions skewed left and distributions skewed right. Remember, a distribution is skewed in the direction with the longest tail. Remembering which direction is right and left is up to you. We are also concerned with the number of peaks that a distribution has. Figures 3.13 through 3.15 all have one peak and are therefore called unimodal distributions. Figure 3.16 presents a plot of a distribution that is bimodal, that is, it has two peaks. Distributions that have more than two peaks are often called multimodal. Note that we are using the word mode in a different way here then that with which you may be familiar. A mode can also refer to a statistical measure of location, as measured by the class in a distribution with the highest frequency. Here we are using the word to refer to a general assessment of the number of peaks a distribution has. This may be confusing at first, but it is important to know both usages. A characteristic that is often overlooked when describing distributions are their relative heights. Often a distribution will superficially look like another distribution, but will in fact be more spread out or narrow. Given similar numbers of variates, a narrow distribution will have a higher peak, whereas an overly broader distribution will have a lower peak. For example, the two distributions look similar, but the distribution illustrated in Figure 3.17 has a higher peak than the distribution in Figure Of course there are very big and important sounding words to describe these differences. The high peak of the distribution in Figure 3.17 is called leptokurtic, and the flat peak of the distribution in Figure 3.17 is platykurtic. The classic Bell-Shaped Curve depicted in Figure 3.19 is mesokurtic. If you use these words at parties, your friends will be impressed. Bar Chart One final distribution should be noted: a qualitative frequency distribution, often called a bar chart. Figure 3.20 provides a bar chart of the number of villages during the Early Monogahela period occupation of the Allegheny Plateau in several physiographic settings (Hasenstab and Johnson 2001:6). It is important to note that bar charts provide 3.8

9 abundance information on qualitative, not quantitative variables. Bar charts cannot be described in terms of symmetry, peakedness, or modality, because the order that the data are entered from left to right is arbitrary. For example, river terrace sites listed on Figure 3.20 could just as easily be placed after stream terrace sites as before them. It is important not to confuse bar charts with histograms, and one way to keep them distinct is to present gaps between the bars in the bar graph, and to close the bars together in the histogram. Bar charts are particularly useful when one wishes to compare differences within classes for multiple variables. For example, Figure 3.21 is a bar chart with the number of villages dating to the Early Monongahela, the Middle Monongahela, and the Late Monongahela/Protohistoric periods in different physiographic settings. By examining the relative heights of the bars, changes in settlement patterns can be identified. The statistical significance of such changes can then be evaluated using the various methods discussed later in this book. Now that you know how to characterize basic data visually, let us now learn procedures that allow us to make numerical comparison. That is the subject of Chapter 4. Figure 3.1 Original Frequency Distribution of Depths for Carrier Mills Features. Implied Limits Y Tally Marks f ll ll lll lllll l llll lllll lllll llll lllll lll lllll l llll 4 3.9

10 lllll ll lll lll lllll l lll ll llll lllll l llll lll ll 2 f 91 Figure 3.2. Frequency Distribution Resulting from Grouping Depths for Carrier Mills Features into 11 Classes: Interval=2cm. Practical Limits Implied Limits Class Mark Tally Mark f llll lllll llll lllll llll lllll lllll lllll l lllll lllll lllll lllll lllll llll

11 lllll lllll llll lllll lllll 5 f

12 Figure 3.3. Frequency Distribution Resulting from Grouping Depths for Carrier Mills Features into 5 Classes: Interval=5cm. Practical Limits Implied Limits Class Marks Tally Marks lllll lllll lllll ll lllll lllll lllll lllll lllll lllll l lllll lllll lllll lllll ll lllll lllll lllll lllll l 6 f 91 f 3.12

13 Figure 3.4. Frequency Distribution of Heights in Class. Implied Limits Y (inches*) Tally Marks f f * 1 inch = 2.54 cm 3.13

14 Figure 3.5. Frequency table of minimum sherd width of the Gallina ceramic data. Implied Limits Y Tally Marks f III IIIII I II IIIII IIIII IIIII I IIIII III IIIII IIIII IIIII IIIII IIIII IIIII IIIII IIIII IIIII III IIIII IIIII II I III I 1 f

15 Figure 3.6. Histogram of minimum sherd width of the Gallina ceramic data Frequency of Sherds Minimum Sherd Thickness (cm) Figure 3.7. Stem and leaf diagram of Gallina ceramic data

16 Figure 3.8. Ordered Stem and leaf diagram of Gallina ceramic data Figure 3.9. Stem and leaf diagram created using increments of.5 cm

17 Figure Stems that can be used to create a stem and leaf diagram using the data from Table

18 Figure Cumulative Frequency Distribution of Carrier Mills Feature Depths. Y f f (cumulative) f

19 Figure Plot of the Cumulative Frequency Distribution of Carrier Mills Feature Depths Cumulative Frequency Feature Depth (cm) Figure Plot of a Symmetrical, or Normal distribution. 3.19

20 Figure Plot of a Distribution that is skewed to the left. Figure Plot of a Distribution that is skewed to the right. Figure Plot of a Bimodal distribution. 3.20

21 Figure Plot of a Leptokurtic distribution. Figure Plot of a Platykurtic distribution. Figure Plot of a Mesokurtic distribution. 3.21

22 Figure Bar chart of the number of villages during the Early Monongahela period in the Lower Monongahela and Lower Youghiogheny river basins (Hasenstab and Johnson 2001:11) Frequency of Villages River Terrace Stream Terrace Upland Total Physiographic Provenience 3.22

23 Figure Bar chart of the number of villages during the Early Monongahela, Middle Monongahela, and Late Monongahela periods Number of Villages Early Middle Late 5 0 River Terrace Stream Terrace Upland Total Physiographic Provenience Table 3.1. Carrier Mills Feature Depths (cm) (from Jeffries and Butler 1982). Original Measurements

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25 Table 3.2. Minimum sherd thickness of Gallina, NM ceramics in mm Table 3.3. Measurements of the maximum thickness of Losange-shaped Early Paleolithic projectile points (Knetch 1991: )

26 3.26