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UW-L Journal of Undergraduate Research X (2007)
Analysis of Archaeological Sampling Methods Using the Complete
Surface Data from the Pirque Alto Site in Cochabamba, Bolivia
Elizabeth Green
Faculty Sponsor: Timothy McAndrews, Department of Sociology/Archaeology
ABSTRACT
Sampling is an extremely important aspect of archaeological fieldwork, and it can significantly
influence interpretation; therefore, the effects of these processes must be understood and
controlled. Various experimental sampling methods are tested against the surface recovery data
collected from the Pirque Alto site in Cochabamba, Bolivia. The project addresses the following
questions: do results from the sampling methods reflect the total surface recovery; what method
gives the most accurate results for this site and why; and at what percentage of quadrat (unit)
samples taken per method do the results show validity. Finally, in what way are the methods
biased and how can these biases be addressed? The project has yielded information about how
various methods would or would not have affected the interpretation of the Pirque Alto site, and
the results were then applied to a broader study and understanding of sampling methods.
INTRODUCTION
Sampling, a less visible but just as essential tool in archaeology as the trowel, has been used since the start of
the discipline. “Archaeologists have always sampled… archaeologists have been choosing, among alternatives,
where to look for archaeological sites, which sites to excavate, and where on selected sites to dig” (Hole 1980:217).
However, despite the fact that archaeologists have always sampled, there seems to be a disproportionately small
effort put forth to truly understand and test the methods so heavily relied upon. Sampling exists and is used for many
reasons, including: financial, labor and time restraints and the logistical necessity of controlling the sheer volume of
physical materials and non-physical data generated by the process of archaeology.
The processes of sampling have far-reaching consequences and influences. For instance, a sampling of units on
the surface of a site can directly affect an archaeologist’s perception of the site (its function, chronology, scope, etc),
as well as directly affecting the direction of the research questions posed in the future. It is important to know that
these critical procedures and methodologies are the most conclusive possible so that their influence on the
excavation and interpretation of sites is as constructive as possible. “Given that sampling is essential, we need to
ensure that it is carried out in a way that enables us to make best use of the data that it provides” (Orton 2000:8).
Due to this, it is important that experiments with comparable results be conducted in order to gain a better
understanding of the act of sampling and its effects. The purpose of this specific project is to do just that — to act as
a testing mechanism for common sampling procedures. The data used in this endeavor comes from the work done at
the site of Pirque Alto, located in the Department of Cochabamba, Bolivia. Details of this site will be discussed in
the background section to follow; however, it is important to note that the work conducted at this site included a
surface collection of ceramics and lithics of the entire site. Ceramics bigger than a nickel and all of those featuring
paint were collected.
The collected ceramics data act as a control group in testing various sampling methods with the goal of
answering the following questions:
1. Do results from the sampling methods equal or come close to the spatial (ex. identified ceramic clusters)
and numeric results indicated from surface collection?
2. What method gives the most spatially and numerically accurate results for this site and why?
3. At what percentage of unit samples taken per method do the results show the greatest spatial and numerical
accuracy, (10%, 30%, or 50%)?
4. In what way are the methods of sampling biased, and how can these biases be addressed or corrected?
The ultimate goal of this project is to determine the validity and precision of common archaeological sampling
methods and identify the most accurate methods. Because the data being used as the control group comes from a real
site, looking at how the sampling methods would have affected its interpretation help to present the gravity of
influence from sampling.
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UW-L Journal of Undergraduate Research X (2007)
BACKGROUND
The Site of Pirque Alto
During the 2005 field season, archaeological fieldwork was conducted by several students in the UW-L
Archaeological Studies program under the supervision of Dr. Timothy McAndrews at Pirque Alto, in the
Department of Cochabamba, Bolivia. One goal of this project was to investigate the influence of the Tiwanaku
culture in the area through the cultural remains present at the site, such as ceramics, stone artifacts, and features.
Tiwanaku was a powerful polity operating out of the southern part of the Lake Titicaca Basin that influenced a broad
region throughout the central Andes from A.D. 500 to A.D. 1150 (Janusek 2004:xvii). Now viewed as “the first true
state in the south-central Andes,” the Tiwanaku culture is typified by stratified societal levels, monumental
architecture, and sophisticated material culture (Young-Sanchez 2004:17-18). Their material culture is unique,
especially the ceramics which are “extremely distinctive and rather easy to recognize both in terms of slip and paint,
and in diameter of base and rim,” which allows for easier identification (Burkholder 1997: 130). The nature of the
spread of the Tiwanaku is a topic currently under fierce debate, and as such this site could help shed light on the
presence of the Tiwanaku in the Cochabamba Valley area.
Through the surface collection of 406 5 x 5 meter units, a total of over 13,200 diagnostic ceramics (rims, bases,
handles, painted pieces), and over 52,000 non-diagnostic (those pieces bigger than a nickel not belonging to the
diagnostic category) were collected and subsequently analyzed. Early conclusions regarding this site indicated that it
was a multi-component site occupied from the Formative Period though Inca times. This site is located on a bluff
Figure 1. Site of Pirque Alto as viewed from opposite valley wall
(see Figure 1) overlooking the Rio Tapacari and exists in a natural travel corridor leading to the Altiplano and
Titicaca Basin. Since this site, in terms of location and artifact assemblage, can offer important addition to the
debate concerning Tiwanaku presence in the Cochabamba Valley, it serves as an excellent case of how sampling
would and could affect site interpretation and how in turn that can affect the archaeology of an entire area
(McAndrews 2006).
History of Sampling
The history of sampling in archaeology is an interesting one, and the current debates within the subject
showcase the breadth of thought concerning its importance and its influence, both on the process of archaeology and
on archaeologists themselves. One of the few comprehensive histories of sampling in archaeology is contained in
Sampling in Archaeology, by Clive Orton (2000). In this book, Orton discusses the changes in the actual sampling
themes throughout time but also talks about the changes in archaeologists’ attitudes towards sampling.
At the turn of the twentieth century, there was more attention directed towards sampling; however, without the
later developments that statistics would bring, “it would not be reasonable to expect more than an informal
approach” (Orton 2000:112). Preliminary attempts at sampling were carried out in a rather intuitive way. For
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instance, if there appeared to be a large amount of sherds, you took what you believed to be “representative.” It was
not until the developments of the 1940s that sampling procedures became more concrete and statistically
representative. The developments in statistical theory came about as a result of the need for quality control in
response to industrial growth but eventually became incorporated into the sampling of shell middens. This helped to
cut down on the tedium of going through mountains of low grade data. At first glance, the beginning of sampling in
archaeology would seem to suggest a rather haphazard approach designed to make archaeological evaluation easier
and less tedious.
It was during the 1950s and 1960s that more formal sampling methods, which Orton describes as “samples
selected from well-defined populations according to rigorous statistical procedures,” became more commonplace
(Orton 2000:1-2). Vescelius was the first to discuss the use of probabilistic sampling, and he directed his attention
towards finding a way to estimate artifact proportions compared to all others within a site (Vescelius 1960:457-470).
However, “no other single article has altered more radically the recent course of archaeology than Lewis Binford’s
1964 consideration of research design” (Hole 1980:218).
Lewis Binford contributed greatly to the field of archaeology, especially in relation to field methodology and
research design, which included sampling. Although Binford’s regard to sampling was directed regionally rather
than at a site level, he instituted tremendous changes in the way archaeology was carried out. Orton describes there
being three major changes advocated by Binford: “the promotion of the region to the position of primary unit of
archaeological research, the explicit design of archaeological research programs, and the explicit use of sampling
theory as the way of linking the two” (Orton 2000:4).
Though Binford’s contributions to the field of archaeology cannot be underestimated, such an enforcement and
indoctrination of sampling methods caused a severe backlash that emerged in the 1970s. Sampling appeared as an
evil necessity that one would never escape and as a compromise with the constraints of real life. This feeling
increased with the development of sieving methods at the feature level, which created even more data which
required sampling from (Orton 2000:4). Part of this backlash was to criticize sampling itself rather than how it was
being carried out, as though sampling was something impressed upon archaeology by fields that had nothing to do
with archaeology.
Though the 1970s are characterized by a backlash against sampling, they were also a time of reflection on the
progress of sampling due to the publication of the San Francisco Symposium (which was the first formal gathering
to discuss sampling) and Kent Flannery’s Early Mesoamerican Village (Flannery 1970). James Mueller, one of the
big names in sampling theory, describes this reactionary period as extremely varied. “The responses and reactions
have been varied – categorical rejection, blind acceptance, and skeptical positivism,” the last which seems to
dominate in the 1970s (Mueller 1975: xi).
This evaluation of the progress of sampling had the curious result of bringing viewpoints regarding it back full
circle to a more intuitive process. “By the 1980s in the USA and the 1990s in the UK, sampling had become firmly
entrenched in the methodology of the ‘contract’ wing of archaeology” and had become something almost instinctive
among archaeologists (Orton 2000:6). Though sampling has again become something rather intrinsic to archaeology,
“[it] is still on the agenda even if it is not discussed as openly or explicitly as it was, say, in the 1970s and 1980s”
(Orton 2000: 207).
Debates within Sampling
Due to the importance of sampling, there exists a great deal of controversy throughout the literature. The
debates range from the general importance of sampling (Rootenberg 1964) to the proper methods while conducting
sampling (Schiffer, Sullivan and Klinger 1978) to the correct math and proportions (Plog and Hegmon 1993, Nance
1981). According to Bonnie Laird Hole, “Despite the widespread availability of understandable presentations of the
theory of sampling and the powerful potential of its appropriate application in archaeological research, its
contributions to archaeological knowledge have been disappointing” (Hole 1980:217).
Articles run the gamut from math-entrenched to highly theoretical and behavioral. It is interesting to note that
within most of the literature there exists a great deal of disagreement expressed through journal publications. At
times some archaeologists would rather attack one another through publication than to correspond directly. Though
opinions are varied, it is important to mind the theoretical milieu from which the disagreements arise.
A great debate going on in sampling right now is the degree to which post-depositional factors contribute to the
bias of a sample. Many in the Processual archaeology group bypassed looking at this question by examining things
which were the least affected. The Cultural History school of thought also tended to ignore the issue. (Orton
2000:43). This debate continues today, especially in regards to issues like the value of the number of identified
specimens over the minimum number of individuals.
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A debate more relevant to this project is the degree to which surface remains reflect the subsurface, and “is a
key issue for the practice of archaeological survey, at both the regional and the site level. Opinions, both as
expressed and as implicit as fieldwork practices, seem to vary widely” (Orton 2000:57). For the site of Pirque Alto,
the clusters of certain identified phases are so distinct and identifiable that they are assumed to represent a reliable
indication of the subsurface.
Obviously, these are hardly the only ongoing debates within sampling, and more research is being conducted in
order to present specific debates concerning individual sampling methods; i.e. random vs. judgmental, etc. Some of
these disagreements will be explored later in the paper as they relate to the results of the project.
Descriptions of the Sampling Methods used in this Project
Below are basic descriptions of the types of sampling methods relied upon in this project. Information
concerning the experimental procedures will be discussed in the methodology section of this paper. Lastly, more
specific pros and cons for each method will be discussed in the results section as they correlate directly.
Archaeological sampling methods can be roughly divided into two categories: probabilistic and non-probabilistic.
Archaeology employs probabilistic sampling methods as an “attempt to improve the probability that generalizations
from the sample will be correct” (Renfrew and Bahn 2000: 76). Charles Redman describes probabilistic sampling
methods as “good for estimated total population values for common artifacts or features… probability sampling is
poor for locating rare feature or artifacts, dealing with clustered distributions, or illuminating contiguous spatial
patterns” (Redman 1987: 251). Non-probabilistic methods, described subsequently, are generally looked upon less
favorably.
The most basic probabilistic sampling method is a simple random sample, in which numbers are assigned to the
quadrats and those to be sampled are chosen through a random number chart till a predetermined number or
percentage of units is sampled. “It implies that an equal probability of selection is assigned to each unit of the frame
at the time of sample selection” (Binford 1964:141).
Systematic random sampling involves the sectioning out of a site grid into groups of a predetermined number of
quadrats, and then randomly selecting quadrats within each section (Drennan 1996: 246). The general advantage of
this method is that it ensures there will be broader coverage across the entire site; “Systematic sampling ensures an
equal dispersion of sample units” (Binford 1964:153).
Next there is stratified random sampling, where the site is divided into its natural zones (or strata) and like
before, random numbers are chosen. However, in this method the squares are proportional to the natural zones; 85%
forest equals 85% of the samples are taken from that area (Renfrew 2000: 77).
Judgmental sampling, which is not representative of the whole, is a non-probabilistic form of sampling and is
associated with similar methods including haphazard sampling, grab sampling, and purposive sampling. “Nonprobabilistic sampling is regarded as intuitive, inductive, and unstated” (Hole 1980: 219). All of these terms
represent a method referring to “explicit or implicit application of a variety of nonrandom selection criteria”
(Drennan 1996:88). In judgmental sampling, samples are “selected by looking over the range of elements in a
population and specifically deciding to include certain elements in the sample and exclude others” (Drennan
1996:88). Obviously, this method has a large degree of bias. It was important to include due to the current
instinctive feeling of archaeological sampling.
METHODOLOGY
The first step of this process was to become familiar with the literature concerning sampling. This was an
important step because it helped to frame the concerns and debates within sampling and helped to provide a general
understanding of the subject. Another familiarization step was to become comfortable with the computer programs
necessary for this project, including Microsoft Access and Excel, and Surfer by Golden Software, a map generating
program. After sufficient skills with these programs had been developed the sampling experiments could be carried
out.
Sampling experiments were carried out at three percent intervals (10%, 30%, and 50%) with five trials each.
Originally, these experiments were to be carried out for the total ceramics data and for the identified Tiwanaku
phase ceramics. However, a different approach has been considered and decided on instead. Each of the trials will
take into account the total ceramics data, yet the data will show the percent levels for three identified ceramic
phases: Formative, Early Intermediate, and Tiwanaku. These percents will be compared to the total percents for each
of the ceramic phases. For example, if a 10% random sample shows 13% of sherds belonging to the Tiwanaku phase
ceramics, this will be compared to the percent of Tiwanaku phase ceramics for the entire site.
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UW-L Journal of Undergraduate Research X (2007)
The sampling experiments focus on four types of sampling commonly relied upon by archaeologists: random
sampling, systematic sampling, stratified sampling, and judgmental sampling. For the random sampling
experiments, Microsoft Access was used to randomly sample units according to the percent level specified. For the
systematic sample, the site is divided into as many 25 x 25 meter groups as possible. The units within these groups
are then selected randomly according to percent. See Figure 2 for systematic group distributions.
Systematic Group Distributions
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Figure 2. Systematic Group Distributions
For the stratified random sample, the site is broken up into three areas related to their relative elevation: highest
part of the sites, middle and lowest. Grids are then selected randomly according to percent level. See Figure 3 for the
stratified group distributions.
Stratified Group Distributions
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Figure 3. Stratified Group Distributions
Since the judgmental sample is a non-probabilistic method, it was carried out differently than the previous
methods. Three students in the Senior Thesis class (ARC 499) who did not attend the 2005 field school at Pirque
Alto were given a site map with general information such as the physical layout of the site, areas of higher artifact
concentration (like one would see on a pedestrian survey). They were then asked to indicate which areas based on
the information they had been given would be their choice for sampling. Size of the sampled areas was equal to the
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percent levels used for the other three sampling methods. For example, the first student was given enough units to
equal a 10% sample of the site to lay out as they see fit; for the second student, 30% and so on. Though this method
differs significantly and will not be able to be compared in exactly the same way, it is an important avenue to
explore.
After the experiments were carried out, the information was plotted onto a site map using Surfer by Golden
Software. This allows for a clear illustration of the differences between the sampling methods and their visual
results. A series of descriptive statistics (mean and standard deviation) were generated and will be used in
comparison to the visual results. T-tests were also carried out to determine if there was a statistically significant
difference between the samples and the full dataset in terms of the relative proportions of different periods of
ceramics. It was determined that there was not a statistically significant difference; meaning all samples may be
considered numerically representative of the whole.
Armed with both the visual representations of the sampling experiments created using Surfer and the statistics
generated from the samples, a comprehensive comparison of the methods will then occur. The relative success or
failure of the various sampling methods will be viewed in regards to their implications for interpretations of the site
and then applied on a larger scale to view the effect of sampling on the field of archaeology.
RESULTS
In addition to the main goal of determining the validity and precision of common archaeological sampling
methods, this thesis also sought to address four specific questions:
1. Do results from the sampling methods equal or come close to the spatial (ex. identified ceramic clusters)
and numeric results indicated from surface collection?
2. What method gives the most spatially and numerically accurate results for this site and why?
3. At what percentage of unit samples taken per method do the results show the greatest spatial and numerical
accuracy, (10%, 30%, or 50%)?
4. In what way are the methods of sampling biased, and how can these biases be addressed or corrected?
These questions will be addressed throughout this portion of the paper, which is itself broken up into three
different sections. In the first section, results in terms of the individual methods (random, systematic, etc) will be
discussed. This will be followed by a section describing the results on each percent interval, including the spatial
representations of each sample method and the statistics and ceramic phase percentages. This is done to illustrate the
benefits and problems associated with each method and provide more detailed information at the sample size level.
General comments about the various methods and their results will comprise the last section. On the following page,
Figure 4 shows the overall distributions of all ceramics over the entire site for reference. Maps for the Formative,
Early Intermediate, and Tiwanaku Ceramics can be found at the end of the document.
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Total Ceramics
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64 35 136 125 68 79 79 19 60 70 44 47
54 81 145 132 117 109 75 139 155 117 104 51 163 230
103 182 159 102 213 261 275 156 448 162 104 245 97 224 197 267 196 133 124
19 78 117 147 104 185 274 93 128 169 205 53 36 26 84 217 88 40 249
22 65 99 161 93 47 121 95 211 170 131 249 46 79 129 150 157 255 150 147
27 20 114 234 116 74 112 241 200 236 117 0
42 90 204 107 65 88 178 249
23 40 41 135 140 112 128 112 65 161 150 148 122 134 113 68 109 122 199 296 323
14 32 125 140 203 242 165 95 45 64 145 189 200 259 178 124 62 89 259 172 198
28 30 86 264 277 143 148 135 145 143 8
95 44 212 97 59 87 43 235 219 555 400
55 33 123 132 179 264 164 141 197 333 283 145 41 69 48 109 59 156 620 146 328 123 468
67 19 74 101 78 161 139 99 212 221 299 78 75 28 87 157 239 120 181 125 0 263 185
21 53 92 121 108 339 129 289 230 198 695 135 118 20 69 212 177 266 184 81 48 340 284 0
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23 0
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26 31 53 44 74 38 82 137 58 127 37 21 16 11 97 342 309 179 119 100 51 53
20 89 80 134 122 130 202 102 137 171 233 315 239 239 101 47 89 139 70 69 47 83
54 92 105 135 151 168 87 180 82 0 162 190 267 216 216 159 157 61 26 51 20
88 53 129 65 243 197 145 162 194 348 317 71 101 90 119 136 12
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Figure 4. Total ceramics distribution
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UW-L Journal of Undergraduate Research X (2007)
Individual Sampling Method Results
Random Sampling. Without a doubt, random sampling produced the most varied results, most notably within the
spatial distributions. At odds with this is that the statistical and phase percentage results gave a representative
sample. The circled areas on the maps indicate areas of high frequency identified ceramic phase clusters that would
have been missed by the sample. For the 10% random sample (see Figure 5), the circled upper right-hand and lower
section contain the two largest clusters of Tiwanaku ceramics on the site. As one of the goals of the Pirque Alto
project was to gather information about the Tiwanaku influence at the site, missing these clusters would have
drastically affected our understanding of the site.
Random Sample 10% (2)
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Figure 5. Random Sample 10% (Trial 2)
The above map was generated from the 10% sample level, and it can be seen that is performs poorly in locating
the major clusters of identified ceramics. For this sample type, this trend continues through the 30% samples. Figure
6 shows that even at three times the previous sample size a large chunk of the site was not sampled.
Random Sample 30% (4)
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Figure 6. Random Sample
30% (Trial 4)
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For the random sample, the most reliable sample size level was 50%, (see Figure 7).
Random Sample 50% (2)
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Figure 7. Random Sample 50% (Trial 2)
Table 1 shows the numeric results for all three maps shown above. Despite the large spatial differences between
all three intervals, the random samples produced the most reliable and representative numeric results compared to
both the systematic and the stratified samples. However, when viewed in conjunction with their spatial
distributions, random sampling comes up quite short. For a statistician random sampling would be the most ideal,
but for the archaeologist who deals spatially and numerically, random sampling has little to offer. “In an
archaeological context, simple random sampling will not work” (Rootenberg 1964:182).
Table 1. Random Sample Numeric Results
Number of Units Sampled
Mean ceramics per unit
Formative Phase %
Early Intermediate Phase %
Tiwanaku Phase %
Diagnostic Ceramics %
Non-Diagnostic Ceramics %
Whole Site
406
128.81
4.46%
0.90%
18.99%
25.24%
74.76%
RS 10% (Trial 2)
41
119.22
3.85%
0.94%
19.33%
24.96%
75.04%
RS 30% (Trial 4)
121
138.66
5.26%
0.84%
18.47%
25.30%
74.70%
RS 50% (Trial 2)
203
126.98
3.99%
0.96%
18.70%
24.54%
75.46%
Systematic Random Sampling. The systematic random sampling method provided the most consistent results of all
three probabilistic sampling methods spatially and numerically, and therefore it was difficult to choose certain trials.
Of all the percent levels, 50% provides the most solid results, but logistically speaking a 30% sample is more
attainable. However, even at the 10% level the systematic random sample still performs beautifully. At the
beginning stage of an archaeological investigation, surface sampling is crucial to provide instruction as to which
areas of the site may be significant. Due to this importance, systematic random sampling dominates its competitors
as a way to provide archaeologists with a far-reaching distribution of samples, providing an overall better knowledge
of the site. “This procedure has a number of advantages. Classes may be established with regard to different
variables that one whish to control, which makes possible the reliable evaluation of variability in other phenomena
with respect to the class-defined variables” (Binford 1964: 142).
At the 10% level, the phase percentage information is rather representative (see Figure 8). The 10% sample also
provides a rather good physical distribution of units throughout the site (see map on following page with circled
Tiwanaku ceramic clusters). Compared to the 10% random sample results, systematic random sampling provides a
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tremendously improved method. For the site of Pirque Alto, a 10% systematic sample returned results representative
to the whole and would have been a useful method to employ.
Systematic Random Sample 10% (1)
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27 20
320
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147
90
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32
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Grid North
11 to 38
38 to 89
89 to 127
127 to 194
194 to 340.1
79
54
259
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300
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89
283
35
99
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88
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66
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38
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127 37
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250
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0
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20 m
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310
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330
340
350
118
360
370
380
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410
Grid East
Figure 8. Systematic Random Sample 10% (Trial 1)
The 30% systematic random sample does an excellent job of displaying representative numerical results and
gives excellent physical coverage. The circled areas on Figure 9 represent key Tiwanaku ceramic clusters.
Systematic Random Sample 30% (1)
61
350
117
340
330
Grid North
75 139 155 117
162 104
19 78 117
97
205 53
65 99
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26
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183 to 620.1
47
102 213 261
320
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88 40
129
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88 178
65 161 150
140
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33 123
143 148
30
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173 145
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69
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11 97
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89 139
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135 151 168
88
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219 555
156 620
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113 26
208 39
85
5
118
128
0
10
20 m
300
310
320
330
340
350
360
370
23
380
390
400
410
Grid East
Figure 9. Systematic Random Sample 30% (Trial 1)
At the 50% level, the systematic random sample provides extremely close numeric results for all five trials.
When spatially represented, this sample gives a very clear picture of areas of ceramic occupation on the site. In
Figure 10, the Tiwanaku clusters have been circled.
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UW-L Journal of Undergraduate Research X (2007)
Systematic Random Sample 50% (5)
350
340
330
320
Grid North
310
300
61
64
125
79 79 19
54 81 145
75 139 155 117
103 182
97
93 47
20 114
140 203 242
86 264 277
55 33 123
259
83 110
25
20
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54 92 105
102 137
168
82
340 284 0
162 80 150
106
97
179 119 100
89 139 70
267
145 162 194
148
80 181
136 12
189 145 35
153 72 96 208 39
290
0
18
14
6
73
N
Mag
109 59
0
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66 118
98
Grid
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157 61
71
N
66
36 272
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0 263 185
177
87
21 16
129 65
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156 620 146 328 123
69
49 139 118
38
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695 135
44 94 44 67
31 53 44
172
59 87 43
69 48
173 145 324 155
323
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95 44
212 221 299 78 75
289 230
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109 122
189
143 8
264 164
92 121
280
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45
148 135
101 78
30 30
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255
117 0
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250
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131
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23 40
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169 205 53 36 26 84
22 65 99
0 to 48
48 to 88
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130 to 182
182 to 695.1
230
213 261 275 156
19
290
0
70 44 47
20 m
310
320
330
340
350
360
370
380
390
400
410
Grid East
Figure 2. Systematic Random Sample 50% (Trial 5)
Table 2 shows the numeric results for all the systematic random trials shown above.
Table 2. Systematic Random Sample Numeric Results
Number of Units Sampled
Mean ceramics per unit
Formative %
Early Intermediate %
Tiwanaku %
Diagnostic Ceramics %
Non-Diagnostic Ceramics %
Whole Site
406
128.81
4.46%
0.90%
18.99%
25.24%
74.76%
SRS 10% (Trial 1)
49
119.92
4.61%
0.82%
19.84%
25.85%
75.80%
SRS 30% (Trial 1)
130
128.56
4.28%
0.94%
18.11%
24.24%
75.76%
SRS 50% (Trial 5)
211
124.25
3.96%
0.76%
19.00%
24.67%
75.33%
Stratified Random Sampling. If random samples were the least spatially reliable in terms of finding identified
clusters of ceramics, and systematic random samples were the best overall, stratified random sampling was firmly in
the middle. Visually, the stratified samples performed better than the random samples (except at the 50% level,
where they are roughly equal), but significantly less well than the systematic samples. You can see that, like the
random sample at 10%, the stratified random sample tends to miss key clusters or cover them only very slightly. It is
not until the 30% and especially 50% stratified sample level does this tend to correct itself slightly. Despite a fairly
good coverage, there are still clusters of sampled units. However, Figure 11 for the most part offers a good spatial
distribution.
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UW-L Journal of Undergraduate Research X (2007)
Stratified Random Sample 10% (5)
350
14 to 47
47 to 70
70 to 112
112 to 150
150 to 468.1
60
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330
131 249 46 79 129
20
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64
Grid North
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20 m
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370
14
66
380
390
400
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Grid East
Figure 11. Stratified Random Sample 10% (Trial 5)
Figure 12 displays the visual distribution of the 30% sample level. The 30% sample level performed better in all
trials than did the 10%.
Stratified Random Sample 30% (4)
350
340
330
104 51
182 159
448
19 78 117
104 185 274 93 128
93
Grid North
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170 to 558.1
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65
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20 m
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Grid East
Figure 12. Stratified Random Sample 30% (Trial 4)
At the 50% level, (see Figure 13) the stratified sample, much like the systematic and the random become
somewhat the same. While the high percentage guarantees a broad ranging recovery, it also starts to become
redundant.
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UW-L Journal of Undergraduate Research X (2007)
Stratified Random Sample 50% (5)
0
350
340
330
136 125 68 79
54
Grid North
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236 117 0
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162 80
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127 37 21
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239 101 47 89
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145 162
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244 172 106 117 61
11
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125 0
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180 82
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20 114 234 116 74 112 241
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169
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0 to 47
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180 to 468.1
44 47
109 75 139
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60
109 59
39 15
67 128
0
10
20 m
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98 23
29
330
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350
360
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380
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410
Grid East
Figure 13. Stratified Random Sample 50% (5)
Table 3 displays the numeric results for all of the samples shown above.
Table 3. Stratified Random Sample Numeric Results
Number of Units Sampled
Mean ceramics per unit
Formative Phase %
Early Intermediate Phase %
Tiwanaku Phase %
Diagnostic Ceramics %
Non-Diagnostic Ceramics %
Whole Site
406
128.81
4.46%
0.90%
18.99%
25.24%
74.76%
Strat. 10% (Trial 5)
41
109.24
6.30%
1.05%
18.15%
26.35%
73.65%
Strat. 30% (Trial 4)
122
121.28
4.85%
0.95%
21.85%
28.47%
85.00%
Strat. 50% (Trail 5)
203
121.39
5.18%
0.98%
22.82%
29.91%
70.09%
Judgmental Sampling. Numeric figures from all three judgmental samples (10, 30, and 50%) all provided
representative results. (See table 4).
Table 4. Judgmental Sample Numeric Results
Number of Units Sampled
Mean ceramics per unit
Formative Phase %
Early Intermediate Phase %
Tiwanaku Phase %
Diagnostic Ceramics %
Non-Diagnostic Ceramics %
Whole Site
406
128.81
4.46%
0.90%
18.99%
25.24%
74.76%
Judgmental 10%
41
130.63
4.66%
0.93%
19.02%
25.54%
74.46%
Judgmental 30%
122
148.63
4.66%
0.93%
19.02%
25.54%
74.46%
Judgmental 50%
203
131.04
4.94%
0.86%
18.89%
25.59%
74.41%
The judgmental samples had extremely varied spatial results. At the 10% level, the student chose a systematic
sample approach and managed to cover the site rather uniformly. (See Figure 14).
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UW-L Journal of Undergraduate Research X (2007)
Judgmental Sample 10%
350
139
340
102
205
330
320
121
26
88
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255 150
116
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296
242
310
Grid North
0 to 47
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150 to 219
219 to 328.1
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20 m
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370
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Grid East
Figure 14. Judgmental Sample 10%
At the 30% level, the student chose a transect approach and cut the site into sections using three long corridors
of sampled units. This student arranged the transects to intersect high, middle and low artifact areas, as well as to
cover the visible features. (See Figure 15).
Judgmental Sample 30%
61
350
340
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Grid North
139 155
163 230
156 448
97 224
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93 128
36 26
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68 109 122 199
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124 62 89 259
143 148 135 145
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83 110
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44 94
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74 38
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270
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348
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20 m
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Grid East
Figure 15. Judgmental Sample 30%
At the 50% level, the student again took a systematic approach; however, she avoided a property divider lined
with cactus and thorny shrubbery. (See Figure 16).
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UW-L Journal of Undergraduate Research X (2007)
Judgmental Sample 50%
0
350
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Grid North
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333 283
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53 44
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20 m
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36 272 285 244 172 106
58 127
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235 219
20 69 212 177 266
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241 200
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47 121
116 74
40 41 135 140
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147 104 185 274 93
27 20
0 to 54
54 to 93
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148 to 200
200 to 448.1
70 44
139 155
15
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390
400
410
Grid East
Figure 16. Judgmental Sample 50%
Visually, the 10% and 50% samples are quite similar. The 30% sample, featured in Figure 5 again with
Tiwanaku clusters circled, presents a completely new method of sampling — transecting. Judgmental sampling has
an extremely useful function in archaeology as it allows the field to become more flexible. Depending on the results
of a systematic sample for example, judgmental sampling allows the archaeologist to redirect the project and
investigate potential areas otherwise excluded through sampling.
Percent Interval
The most effective percent interval: 50%
The most realistic percent interval: 30%
Despite claims in the literature to the contrary, it is this author’s belief that a 10% sample with any method is
normally not conclusive or representative enough. What is being studied is comprised of many parts, some more
measurable than others, and a 10% sample does not allow much leeway for the discovery of lesser reflected
materials. However, should a 10% sample be the only available course of action, a systematic random sample should
be the used, as at the 10% level its results were the most acceptable.
General Comments
Unfortunately, due to the nature and scale of this project, the role statistics and statistical estimations play in
conjunction with sampling was not able to be explored. If someone were to continue the work begun by this project,
the next logical step would be to conduct various statistical calculations (such as Chi square testing) to fully
determine the statistical extent of the sample procedures.
CONCLUSIONS
At the start of this project, I found myself trying to straddle a line between the hardcore samplers with very
complex math equations and those like Hole who saw sampling and statistics in archaeology as something that had
focused too much energy on “probing the method instead of the ground” (Hole 1980: 219). After conducting my
experiments and taking into account numeric and spatial representations and keeping in mind the various opinions
within the literature, I still found myself straddling the line.
Through the process of conducting the experiments for this project and reading through the literature discussing
the problems, concerns and benefits of sampling, it was possible to see how these factors played out within my
experiments. At the end of this process, I had several more questions than answers and had grown immensely more
cynical about math and sampling in regards to archaeology. I wrestled (on a much smaller scale) with all the issues
mentioned in most of the journals and books I had read.
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UW-L Journal of Undergraduate Research X (2007)
First, how does one go about choosing the appropriate sample size? “The optimum sample can be summarized
as being small enough to yield statistically representative, significant, and accurate results at the minimum cost of
labor” (Rootenberg 1964:186). For me, the choice was a relatively arbitrary one. I wanted to choose values that were
both realistically applied archaeologically and that would give a varied enough result to draw conclusions about the
various methods employed. However, when I examined the literature on determining appropriate sample size, there
were many conflicting methods. Many were somewhat arbitrary, such as my own, i.e. believing or deciding that a
10% sample would be representative or a 20% sample, etc.
On the other end of the spectrum, I found some archaeologists recommended very complex mathematical or
statistical equations in order to figure out the appropriate size. Looking at the math required, it seems likely that few
archaeologists, without the help of a professional mathematician or statistician, will be able to do (and understand
the implications of) the math required to determine sample size through these means. Approaching the topic as a
realist, we need to work within our field to create methods that are logistically and practically applicable for the
majority of individuals within our discipline.
The second issue I dealt with was if there is really any way to know that your sample is representative since
archaeology deals with a seemingly indefinite population. Due to the nature of my thesis, I knew what my
population was; I knew how many sherds total, I knew what phases, and I knew where. In the real world, no
archaeologist has this advantage. The question that arose in my mind was this: is it worse to base your idea of what
is representative on a complex mathematical equation, or draw upon similar archaeological sites combined with your
knowledge of the area? Or is it better instead to use a combination, to rely upon your own judgment and knowledge
of the area and then relate that to sampling and statistics as a way to aid you through the rest of the process, while
still not being afraid to change your approach based on judgment?
The last of my four specific questions mentioned in the introduction dealt with the bias in sampling and if this
was something able to be addressed or corrected. At the completion of this project, it occurred to me that the most
damaging bias within sampling currently is the belief that with the right sampling method and the right sample size,
a region or a site will be able to tell all. It is reflected in the literature that many put faith into these sampling
methods that they no longer put into themselves. While Orton does allude to the field having returned to a more
instinctive style of sampling, the effect that the explicit samplers has had on the field is irreversible (Orton 2000:6).
Is the way we conduct sampling intrinsically biased? To this I have to answer a resounding yes. Deciding the
size of sample, the size of a unit, the boundaries of a site – all are typically made based upon the archaeologist’s
judgment. Even if they are based with the most stringent of mathematics and statistics, you cannot simply cut and
paste methods from one discipline into another. Statistics and sampling from any other field cannot be expected to
operate in the same capacity within archaeology. “None of the sampling and estimation schemes currently in
common use in archaeology take into account fully the spatial component in the data” (Hole 1980: 230). This is
obviously something that needs to change. Bias is not necessarily a horrible thing, as long as we acknowledge it as a
field, and adapt our methodology to take this into consideration.
For example, many site reports will simply state what percent sample was taken of the site, with very little
regard to the process leading to that decision. If that discussion was included in the final site report, so that anyone
who read it could understand why a 10%, or a 30% sample was chosen, or why a stratified approach was taken, then
the harmful bias of sampling would be lessened. If you can legitimately explain your sampling strategy and how you
came to that decision, the bias you have worked within becomes objective; consistency and specificity are crucial.
Based on the results of the sampling experiments conducted during this project and a careful review of the
literature on the subject, the following conclusions have been reached: the most spatially and numerically
representative sampling method is a systematic sample of at least 10% (though this author prefers a 30% sample).
Numerically this method was not always as accurate as the random samples, however; it consistently at all percent
levels provided a very even coverage of the site. As the results of surface recovery can be somewhat dubious
(though it was decided at Pirque Alto the surface was a good indication of the subsurface) a systematic random
sample approach on the subsurface would also prove extremely beneficial. At the beginning of an archaeological
investigation, it would give you the best vantage point through which to view the site as a whole.
It is from this point that a judgmental sample is the next logical step. After the systematic random sample has
provided wide-spread even coverage, the judgmental sample allows for the archaeologist to reevaluate the site, his
or her research questions, and allows them the flexibility to decide where and how to proceed. This method should
not be looked down upon or regarded as less effective or legitimate. The idea that all archaeology must be backed up
by sophisticated mathematics and estimation statistics to be considered legitimate is rather insulting. “Without
accurate assessment of the personal element, archaeology will be reduced to a set of mechanically applied
approaches practiced by a series of competent but disinterested researchers, a grim prospect for the discipline”
(Redman 1987: 262).
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UW-L Journal of Undergraduate Research X (2007)
Sampling (and to some extent, statistics) within archaeology should be used as follows:
1. A method through which to locate archaeological sites on a regional scale.
2. A method through which to perform archaeology realistically, taking into account the current curation
crisis, as well as monetary, labor and time constraints
3. An aid in the analysis of collected data, not as a means to do far too much with way too little. “Given the
usual money and time restrictions, the minimum is far too often taken to be the maximum” (O’Neil 1993:
523).
4. A way to operate both spatially and numerically in a way to produce the most contextual results.
Though it is displeasing to end this thesis in a contradictory fashion, I find that it is where my experimental
results and own personal conclusions lead. Sampling within archaeology is both at once a useful and dangerous tool.
It allows us to work realistically within a world of constraints; however, it also has the propensity for great abuse.
While my experiments concluded that a certain method of sampling at a certain percent was the most accurate and
beneficial approach for the site of Pirque Alto, this does not mean that this is the approach that will work for every
site; it would be naïve to think so.
Instead, the new direction of sampling in archaeology should be one of judgmental enforced flexibility.
Sampling should continue to be relied upon, and the basic guidelines and the statistical analysis associated with it
should be followed. However, it is extremely important that archaeologists expand on their methods, become far
more specific about their decision making process, and not be afraid to rely on their own judgment. Archaeology is a
complex, multifaceted field which requires more than an explicit methodology.
ACKNOWLEDGEMENTS
I would first like to thank Drs. Tim McAndrews and Connie Arzigian for their guidance throughout this project.
Without them I would not have had a clue where to begin or how to proceed. I would also like to thank Dr. Jim
Theler for redirecting my thoughts and questions. Thanks to the other members of the 2005 Bolivia Field School for
helping to collect the raw data. Thanks to the Undergraduate Research Committee for funding this project. I would
like to thank my mother and father Denise and Al Green for listening to me describe this project and giving me their
support even though they are interested not in the slightest. I would like to acknowledge my friends, in particular
Beth Plunger, Chris Salinas and Beth Boe for keeping me company throughout this process and offering their advice
over much needed drinks. Lastly, I’d like to thank Phil Lee for his amazing proof reading abilities.
REFERENCES
Binford, Lewis. 1964. A Consideration of Archaeological Research Design. American Antiquity 29:425-441.
Burkholder, JoEllen. 1997. Tiwanaku and the Anatomy of Time: A New Ceramic Chronology From the Iwawi Site,
Department of La Paz, Bolivia. New York: Binghamton University.
Drennan, Robert D. 1996. Statistics for Archaeologists: A Common Sense Approach. New York: Plenum Press.
Flannery, Kent. 1970. The Early Mesoamerican Village. New York: Academic Press.
Hole, Bonnie Laird. 1980. Sampling in Archaeology: A Critique. Annual Review of Anthropology 9:217-234.
Janusek, John Wayne. 2004. Identity and Power in the Ancient Andes: Tiwanaku Cities Through Time. New York:
Routledge.
McAndrews, Timothy. 2006. Preliminary Results from the Multicomponent Site of Pirque Alto in Cochabamba,
Bolivia. Poster presented at the 71st annual meetings of the Society for American Archaeology in San Juan,
Puerto Rico. Tim McAndrews, Claudia Rivera, Carla Jaimes.
Mueller, James. 1975. Sampling in Archaeology. Tucson: The University of Arizona Press.
Nance, Jack. 1981. Statistical Fact and Archaeological Faith: Two Models in Small-Sites Sampling. Journal of Field
Archaeology 8:151-165.
O’Neil, Dennis. 1993. Excavation Sample Size: A Cautionary Tale. American Antiquity 58:523-529.
Orton, Clive. 2000. Sampling in Archaeology. Cambridge: Cambridge University Press.
Plog, Steven and Michelle Hegmon. 1993. The Sample Size-Richness Relation: The Relevance of Research
Questions, Sampling Strategies, and Behavioral Variation. American Antiquity 58:489-496.
Plog, Steven, Fred Plog, and Walter Wait. 1978. Decision Making in Modern Surveys. In Advances in Archeological
Method and Theory. Volume 1, edited by Michael Schiffer. pp. 383-421. Academic Press, Inc., New York.
Redman, Charles. 1987. Surface Collection, Sampling and Research Design: A Retrospective. American Antiquity
52:249-265.
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UW-L Journal of Undergraduate Research X (2007)
Renfrew, Colin, and Paul Bahn. 2000. Archaeology: Theories, Methods and Practice. Third ed. New York: Thames
& Hudson.
Rootenberg, S. 1964. Archaeological Field Sampling. American Antiquity 30:181-188.
Schiffer, Michael, Alan P. Sullivan, Timothy Klinger. 1978. The Design of Archaeological Surveys. World
Archaeology 10:1-28.
Thompson, S.K. and G.A.F. Seber. 1996. Adaptive Sampling. (New York: John Wiley and Sons).
Vescelius, G.S. 1960. Archaeological Sampling: A Problem of Statistical Inference. In Essays in the Science of
Culture: In Honor of Leslie White, edited by Gertrude E. Dole and Robert Carnerio. pp. 457-470. Crowell, New
York.
Young – Sanchez, Maraget. 2004. Tiwanaku: Ancestors of the Inca. University of Nebraska Press, Lincoln.
18
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APPENDI
Total Ceramics
350
340
330
320
Grid North
310
300
290
61 0
0 to 53
53 to 93
93 to 135
135 to 196
196 to 695.1
64 35 136 125 68 79 79 19 60 70 44 47
54 81 145 132 117 109 75 139 155 117 104 51 163 230
103 182 159 102 213 261 275 156 448 162 104 245 97 224 197 267 196 133 124
19 78 117 147 104 185 274 93 128 169 205 53 36 26 84 217 88 40 249
22 65 99 161 93 47 121 95 211 170 131 249 46 79 129 150 157 255 150 147
27 20 114 234 116 74 112 241 200 236 117 0
42 90 204 107 65 88 178 249
23 40 41 135 140 112 128 112 65 161 150 148 122 134 113 68 109 122 199 296 323
14 32 125 140 203 242 165 95 45 64 145 189 200 259 178 124 62 89 259 172 198
28 30 86 264 277 143 148 135 145 143 8
95 44 212 97 59 87 43 235 219 555 400
55 33 123 132 179 264 164 141 197 333 283 145 41 69 48 109 59 156 620 146 328 123 468
67 19 74 101 78 161 139 99 212 221 299 78 75 28 87 157 239 120 181 125 0 263 185
21 53 92 121 108 339 129 289 230 198 695 135 118 20 69 212 177 266 184 81 48 340 284 0
280
270
30 30 12 54 83 110 18 173 145 324 155 94 87 224 194 162 80 150 59 75 66 180 75
23
0 25 57 44 94 44 67 11 49 139 118 106 67 36 272 285 244 172 106 117 61
26 31 53 44 74 38 82 137 58 127 37 21 16 11 97 342 309 179 119 100 51 53
20 89 80 134 122 130 202 102 137 171 233 315 239 239 101 47 89 139 70 69 47 83
54 92 105 135 151 168 87 180 82 0 162 190 267 216 216 159 157 61 26 51 20
88 53 129 65 243 197 145 162 194 348 317 71 101 90 119 136 12
260
148 127 80 181 183 189 145 35 113 26 14
6
207 153 72 96 208 39 95 73 85
5
N
N
Mag
250
54 40 109 59 128 66 118 39 15
67 128 159 98 23
Grid
240
290
0 18
0
300
20 m
310
29
320
330
340
350
360
Grid East
Figure 17 Total Ceramics Distribution
19
370
380
390
400
410
Green
UW-L Journal of Undergraduate Research X (2007)
Formative Phase
350
340
330
320
Grid North
310
300
290
2
0
0
1
3
5
9
to
to
to
to
to
1
3
5
9
92.01
2
2
7
5
2
4
1
1
2
5
4
0
4
4
1 11
9
4
2
0
0
0
2
0
0
4
8
6
3
3
5
3
6
0
10 3
3
4
4
10 6
7
4
2
4
3
1
0
3
3
0
0
4
7
6
9
0
2
2
3
3
1
0
5
4
6
5
5
5
3
3
0
2
7
1
1
0
0
2
3
4
3
6
4
1
1
1
4
5
0
4
3
2
3
3
0
0
2
5
1
0
1
1
8
0
1
1
0 11 3
3
0
1
4
3
5
4
1
0
5
6
0
4 15 31
0
1
1
3
7
2
2
0
5
8
3
4
7
8
3
5
5
5 18
0
0
1
1
5
1
4
3
3
1
0
0
0
2
0
2
0
2
5 14 37 62
3
0
0
5
4 10 0
5
15 5
0
2
1
2
0
4
2
1
0
6
4 12 31
2
1
6
1
1
3
0
2
3
8
1
6
0
2
2
3
6
7
9
1
0
4
2
1
0
5
4
9
7
0
4
4
4
7
0
4
2
1
13 14 35 12
4
1
9
5
1
5
1
1
1
5
2
4
0
12 6
2
7
2
12 12
3
4
2 16 2
2
0
4
1
2
5
3
2
1
3
3
4
3
3
3 46 20 26 29 12
4
6
1
3
2
1
6
2
5
6
2
11 0
1
0
1
6 92 23 19 12
6
2
5
0
2
1
3
7
1
7
9
2
5
11 13 0
10
6
5
6
0
8
7 10 2
14 3
0
2
4 23 8
17 11 7
6
280
270
260
2
8
5
7
4 11
6
11 12 13 26 23 17
5
7
1
1
9
20 13 3
23 12 30 15
0
0
3
6 26
5
9
0
1
2
0
0
7
9
0
6 18 14 16
2
0
15 17 8
N
11 1
6
10 27
2
2
10
0 16
Grid
240
290
0
N
Mag
250
10 4
2
7 17
9 59
0
300
20 m
310
3
3
2
0
320
330
340
350
360
370
Grid East
Figure 18 Formative Phase Ceramic Distribution
20
380
390
400
410
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UW-L Journal of Undergraduate Research X (2007)
Early Intermediate Phase
350
340
330
320
Grid North
310
300
290
5
0
0
1
2
3
4
to
to
to
to
to
1
2
3
4
20.01
0
0
2
1
1
0
0
0
0
0
0
1
0
0
1
4
0
3
1
3
0
4
0
0
6
1
1
0
4
2
1
0
3
3
1
3
3
2
1
0
0
0
2
2
0
0
1
0
6
2
6
2
1
7
10 16 0
0
0
2
0
1
0
3
0
0
1
1
0
5
0
1
2
6
4
4
2
1
6
0
0
0
5
1
2
0
2
1
0
1
4
3
5
5
2
0
0
0
0
1
1
4
4
1
2
0
0
2
2
2
3
2
0
0
2
1
0
0
3
2
0
2
2
4
1
0
0
2
2
2
0
0
0
0
1
2
3
3
1
0
0
0
0 20
3
3
1
1
2
0
5
4
1
0
4
1
0
3
1
6
2
1
0
0
4
0
2
2
0
0
0
1
0
8
3
3
5
1
1
0
4
0
0
1
3
0
1
1
1
1
2
1
0
1
0
1
0
1
2
3
0
1
1
0
1
4
2
2
2
1
2
0
1
1
0
1
1
1
1
0
4
0
3
3
3
0
1
0
1
0
0
0
0
0
0
0
0
0
2
0
0
0
1
0
0
1
0
2
0
0
0
3
0
0
0
0
1
0
0
1
0
0
0
1
3
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
2
0
2
0
0
1
0
2
0
1
1
0
0
1
0
0
0
4
0
1
0
0
0
0
0
0
0
1
0
0
0
2
1
0
6
1
2
0
0
0
0
0
0
0
0
0
0
0
0
1
0
2
2
5
2
3
0
0
4
0
0
0
0
0
0
0
0
0
0
0
2
0
0
1
0
1
1
3
0
0
0
0
0
0
0
0
0
0
0
0
4
0
0
0
0
0
0
0
0
2
0
0
0
1
0
0
0
0
0
0
0
0
0
0
2
1
0
0
0
0
0
1
0
280
270
260
N
N
Mag
250
Grid
240
290
0
300
20 m
310
0
320
330
340
350
360
370
Grid East
Figure 19 Early Intermediate Phase Ceramic Distribution
21
380
390
400
410
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UW-L Journal of Undergraduate Research X (2007)
Tiwanaku Phase
350
340
330
320
Grid North
310
300
290
10 13 46 42
6
7
6
0
24
8
3
8
24 37
7
8
9
13 1
9 15 26 22 16 13 9
22 34 40 18 27 42 24 24 49 31 9
0 to 7
7 to 15
15 to 24
24 to 39
39 to 115.1
29 44
21 15 36 49 31 41 33 38
2
6 17 18
9 39 48 19 19 43 19 8
7
1
17 52 25
5
4 11 11
9 14 26 24 41 46 24 50
7
9
20 55 46 47 32 40
0
0 23 18 11 13 15 34 48 53 31 0
2
6
2
5 26 24 38 52 42 33 10 16 53 32 30 26 23 16 10 12 30 27 35
2
6 19 29 43 26 36 22 34 39 1
7
0 29 27 23 75 42 32 41 66 76 36
1
17
4
17 6
6
7 25 19 18 43 38 40 43 52 94 41 31 6
280
270
6
9
25 22 24 26 22 16 47 30 23 16 32 25 18 18 47 31 48 42
11 15 17 15 11 21
16 20 31 34 27 46 47 65 8
4
8
21 35 46 29 32 32
2 10 16 7
24 14 79 44 14 24 30 24 10
5
0
1
8
11
6
2
5
4 21 3
7
4
6 11 18 15 23 13 15 39 50 105 84 80 24
18 5
3
9
2 11 16 20 36 37 17 26 28 0
8
2
9
28 29 33 11
19 17 10 65 4
3
5
2
5
8 17 63 22
3 15 15 32 43 20 21 12
25 33 33 27 14 13 20 8
3
7 29 18 21 11 4
4
3
2
11 43 24 65 62 33 104 108 10 19 16 30 30
3
0
0
6
1
1
0 12 12
2
42 22 29 64 60 21 21
65 42 34 12 10
7 20
0
N
Mag
290
4 15
50 88 115 77 33 25 37 15
22 12
Grid
240
0 90 39
14 43 39 49 18 16 17 72 80 0
2
9
8 45 35 67 47
33 19 10 60 30 99 40 87
7
N
250
11 12 72 25 13 16 32 44
4
260
0 43
0
300
6
7
22 17 21
7
9
19 10
20 m
310
2
1
2
5
320
330
340
350
360
370
Grid East
Figure 20 Tiwanaku Phase Ceramic Distribution
22
380
390
400
410