Drennan R.D. Statistics for Archaeologists: A Common Sense Approach
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x Preface to the Second Edition
be appropriate) to put into use the appropriate tool to accomplish the task at hand.
The mechanics of complicated calculations and complicated computer software can
both divert attention away from central matters of principle concerning the work to
be done. In statistics, as in the several sports from which the clich´e is derived, it is
impossible to remind yourself too often to keep your eye on the ball.
ACKNOWLEDGEMENTS
The person most responsible for “infecting” me (his word, not mine) with the atti-
tude toward statistics represented here is Lee Sailer. Mark Aldenderfer and Doug
Price provided very helpful reactions to the manuscript of the first edition. I have
stubbornly refused to accept some of the advice generously offered by all three,
however, so they cannot be blamed for any deficiencies. Jeanne Ferrary Drennan
has put up with a lot of cursing as I tried to teach courses in archaeological data
analysis with texts I didn’t like, and she dedicated most of one December vacation
to helping put the first draft of this book in shape to use in class in January. She
pitched in once again with vital help in getting the manuscript for the second edi-
tion ready to send to the publisher, as did Adam Menzies and Scott Palumbo. My
most special thanks are reserved for the graduate and undergraduate students (and
the teaching assistants) who have struggled gamely along as I tried to give enough
coherence to this approach to data analysis in archaeology to use it in the courses
they took – sometimes using texts it contradicted, sometimes using no text at all,
and finally using successive versions of this book. They have contributed more than
they know to whatever clarity the exposition here may have.
Pittsburgh, PA Robert D. Drennan
Contents
Part I Numerical Exploration
1 Batches of Numbers
.......................................................... 3
Stem-and-Leaf Plots ........................................................... 4
Back-to-Back Stem-and-Leaf Plots ........................................... 9
Histograms ..................................................................... 11
Multiple Bunches or Peaks .................................................... 11
Practice ......................................................................... 14
2 TheLevelorCenterofaBatch ............................................. 17
The Mean....................................................................... 17
The Median .................................................................... 19
Outliers and Resistance ........................................................ 20
Eliminating Outliers ........................................................... 20
The Trimmed Mean ............................................................ 21
WhichIndextoUse............................................................ 23
Batches with Two Centers ..................................................... 23
Practice ......................................................................... 25
3 The Spread or Dispersion of a Batch ....................................... 27
The Range ...................................................................... 27
The Midspread or Interquartile Range ........................................ 28
The Variance and Standard Deviation ........................................ 29
The Trimmed Standard Deviation............................................. 32
WhichIndextoUse............................................................ 34
Practice ......................................................................... 36
4 Comparing Batches .......................................................... 37
The Box-and-Dot Plot ......................................................... 37
Removing the Level ........................................................... 42
Removing the Spread .......................................................... 42
Unusualness .................................................................... 45
Standardizing Based on the Mean and Standard Deviation .................. 48
Practice ......................................................................... 49
xi
xii Contents
5 The Shape or Distribution of a Batch ...................................... 51
Symmetry ...................................................................... 51
Transformations................................................................ 53
Correcting Asymmetry ........................................................ 56
The Normal Distribution ...................................................... 59
Practice ......................................................................... 61
6 Categories ..................................................................... 63
Column and Row Proportions ................................................. 69
Proportions and Densities ..................................................... 70
Bar Graphs ..................................................................... 71
Categories and Sub-batches ................................................... 73
Practice ......................................................................... 75
Part II Sampling
7 Samples and Populations .................................................... 79
What Is Sampling?............................................................. 80
Why Sample? .................................................................. 80
How Do We Sample? .......................................................... 82
Representativeness ............................................................. 85
Different Kinds of Sampling and Bias ........................................ 85
Use of Nonrandom Samples................................................... 88
The Target Population ......................................................... 93
Practice ......................................................................... 96
8 Different Samples from the Same Population ............................. 97
All Possible Samples of a Given Size......................................... 97
All Possible Samples of a Larger Given Size................................. 100
The “Special Batch” ........................................................... 103
The Standard Error............................................................. 104
9 Confidence and Population Means ......................................... 107
Getting Started with a Random Sample ...................................... 108
What Populations Might the Sample Have Come From? .................... 109
Confidence versus Precision................................................... 115
Putting a Finer Point on Probabilities – Student’s t .......................... 118
Error Ranges for Specific Confidence Levels ................................ 121
Finite Populations.............................................................. 123
A Complete Example .......................................................... 124
How Large a Sample Do We Need? .......................................... 126
Assumptions and Robust Methods ............................................ 128
Practice ......................................................................... 130
Contents xiii
10 Medians and Resampling .................................................... 133
The Bootstrap .................................................................. 136
Practice ......................................................................... 138
11 Categories and Population Proportions .................................... 139
How Large a Sample Do We Need? .......................................... 142
Practice ......................................................................... 143
Part III Relationships between Two Variables
12 Comparing Two Sample Means
............................................. 147
Confidence, Significance, and Strength ....................................... 151
Comparison by t Test .......................................................... 153
The One-Sample t Test ........................................................ 156
The Null Hypothesis ........................................................... 157
Statistical Results and Interpretations......................................... 160
Assumptions and Robust Methods ............................................ 161
Practice ......................................................................... 163
13 Comparing Means of More than Two Samples ........................... 165
Comparison with Estimated Means and Error Ranges ....................... 166
Comparison by Analysis of Variance ......................................... 168
Strength of Differences ........................................................ 174
Differences between Populations versus Relationships
between Variables
....................................................... 176
Assumptions and Robust Methods ............................................ 178
Practice ......................................................................... 179
14 Comparing Proportions of Different Samples............................. 181
Comparison with Estimated Proportions and Error Ranges ................. 181
Comparison with Chi-Square ................................................. 182
Measures of Strength .......................................................... 188
The Effect of Sample Size ..................................................... 189
Differences between Populations versus Relationships between Variables . 191
Assumptions and Robust Methods ............................................ 191
Postscript: Comparing Proportions to a Theoretical Expectation ........... 193
Practice ......................................................................... 196
15 Relating a Measurement Variable to Another Measurement Variable . 199
Looking at the Broad Picture .................................................. 200
Linear Relationships ........................................................... 201
The Best-Fit Straight Line ..................................................... 204
Prediction....................................................................... 207
How Good Is the Best Fit? .................................................... 209
Significance and Confidence .................................................. 211
xiv Contents
Analysis of Residuals .......................................................... 213
Assumptions and Robust Methods ............................................ 217
Practice ......................................................................... 220
16 Relating Ranks ................................................................ 223
Calculating Spearman’s Rank Correlation .................................... 224
Significance .................................................................... 226
Assumptions and Robust Methods ............................................ 228
Practice ......................................................................... 228
Part IV Special Topics in Sampling
17 Sampling a Population with Subgroups
.................................... 233
Pooling Estimates .............................................................. 234
The Benefits of Stratified Sampling ........................................... 236
18 Sampling a Site or Region with Spatial Units ............................. 239
Spatial Sampling Units: Points, Transects, and Quadrats .................... 240
Estimating Population Proportions............................................ 243
Estimating Population Means ................................................. 247
Densities........................................................................ 249
19 Sampling without Finding Anything ....................................... 251
20 Sampling and Reality ........................................................ 255
Part V Multivariate Analysis
21 Multivariate Approaches and Variables ................................... 263
A Sample Dataset .............................................................. 264
Kinds of Variables, Missing Data, and Statpacks ............................ 267
22 Similarities between Cases................................................... 271
Euclidean Distance ............................................................ 272
Euclidean Distance with Standardized Variables............................. 274
When to Use Euclidean Distance ............................................. 276
Presence/Absence Variables: Simple Matching and Jaccard’s Coefficients. 277
Mixed Variable Sets: Gower’s and Anderberg’s Coefficients................ 280
Similarities between Ixcaquixtla Household Units ........................... 281
23 Multidimensional Scaling.................................................... 285
Configurations in Different Numbers of Dimensions ........................ 286
Interpreting the Configuration................................................. 289
Contents xv
24 Principal Components Analysis............................................. 299
Correlations and Variables..................................................... 300
Extracting Components........................................................ 302
Carrying Out the Analysis ..................................................... 303
25 Cluster Analysis .............................................................. 309
Single Linkage Clustering ..................................................... 310
Complete Linkage Clustering ................................................. 312
Average Linkage Clustering ................................................... 313
Which Linkage Criterion to Choose .......................................... 315
How Many Clusters to Define ................................................. 316
Clustering by Variables ........................................................ 316
Clustering the Ixcaquixtla Household Data................................... 318
Index ................................................................................ 327
Chapter 1
Batches of Numbers
Stem-and-Leaf Plots................................................................................ 4
Back-to-Back Stem-and-Leaf Plots................................................................ 9
Histograms .......................................................................................... 11
Multiple Bunches or Peaks......................................................................... 11
Practice.............................................................................................. 14
A batch is a set of numbers that are related to each other because they are different
instances of the same thing. The simplest example of a batch of numbers is a set
of measurements of different examples of the same kind of thing. For example, the
lengths of a group of scrapers, the diameters of a group of post holes, and the areas
of a group of sites are three batches of numbers. In these instances, length, diameter,
and area are variables and each scraper, post hole, and site is a case.
The length of one scraper, the diameter of one post hole, and the area of one site
do not, together, make a batch of numbers because they are completely unrelated.
The length, width, thickness, and weight of one scraper do not, together, make a
batch because they are not different instances of the same thing; that is, they are
different variables measured for a single case. The length, width, thickness, and
weight of each of 20 scrapers make, not one batch of numbers, but four. These four
batches can be related to each other because they are four variables measured for
the same 20 cases. The diameters of a set of 18 post holes from one site and the
diameters of a set of 23 post holes from another site can be considered a single
batch of numbers (the variable diameter measured for 41 cases, ignoring entirely
which site each post hole appeared in). They can also be considered two related
batches of numbers (the variable diameter measured for 18 cases at one site and 23
cases at another site). Finally they can be considered two related batches of numbers
in a different way (the variable diameter measured for 41 cases and the variable site
classified for the same 41 cases). This last, however, carries us to a different kind
of batch or variable, and it is easier to stick to batches of measurements for the
moment.
R.D. Drennan, Statistics for Archaeologists, Interdisciplinary Contributions
to Archaeology, DOI 10.1007/978-1-4419-0413-3
1,
c
Springer Science+Business Media, LLC 2004, 2009
3
4 CHAPTER 1
STEM-AND-LEAF PLOTS
A list of measurements does not lend itself very well to making interesting obser-
vations, so the first step in exploration of a batch of numbers is to organize them. If
the batch is a set of measurements, the stem-and-leaf plot is the fundamental organ-
izational tool. Consider the batch of numbers in Table
1.1. Ordering them along a
scale can often help us to see patterns. Figure
1.1 shows how to produce a stem-
and-leaf plot that does exactly this for the numbers in Table
1.1. First, the numbers
are divided into a stem section and a leaf section. In the first case, for instance, 9.7
becomes a stem of 9 and a leaf of 7. The leaf for each number is placed on the stem
plot beside the stem for that number. The lines in Fig.
1.1 connect some of the num-
bers to the corresponding leaves in their final positions on the stem-and-leaf plot.
(Not all the connections are drawn in to avoid a hopeless confusion of lines.)
Several characteristics of this batch of numbers are immediately apparent in the
stem-and-leaf plot. First, the numbers tend to bunch together at about 9 to 12 cm.
Most fall in this range. Two more (14.2 and 7.6 cm) fall a little outside this range,
and one (44.6cm) falls far away from the rest. It is a fairly common occurrence for
batches of numbers to bunch together like this. It is also relatively common for one
or a few numbers in a batch to fall far away from the bunch where the majority of
the numbers lie. Such numbers that fall far from the bunch are often called outliers,
and we will discuss them in more detail later. For now it is sufficient to note that we
often examine such outliers with a skeptical eye. A post hole 44.6 cm in diameter
is certainly a very unusual post hole in this batch, and we might be suspicious that
someone has simply written the measurement down wrong. A quick check of field
drawings or photographs should be sufficient to determine whether such an error
has been made and, if so, to correct it. If, indeed, this measurement seems correct,
then one of the conspicuous features of this batch is that one post hole simply does
not seem to fit with the rest of the group.
Stem-and-leaf plots can be made at different scales (that is, using different inter-
vals on the stem), and the selection of an appropriate scale is essential to producing a
helpful stem-and-leaf plot. Table
1.2 shows another batch of numbers in a stem-and-
leaf plot at the same scale as in the previous example. The numbers here, however,
are spread out over such a large distance that the characteristics of the batch are not
Table 1.1. Diameters of 13
Post holes at the Black
Site (cm)
9.7 11.7
9.2 11.1
12.9 7.6
11.4 11.8
9.1 14.2
44.6 10.8
10.5
BATCHES OF NUMBERS 5
Figure 1.1. A stem-and-leaf plot of the numbers in Table 1.1.
clearly displayed. In Table 1.3 the same numbers yield a denser stem-and-leaf plot
when the stem is structured differently. In the first place, the numbers are broken
differently into stem and leaf sections – not at the decimal point but between the
units and tens. Since there are two digits for each leaf, commas are used to indi-
cate the separation between leaves. To avoid greatly increasing the density, two
positions are allowed on the stem for each stem section, the lower position cor-
responding to the lower half of the numbers that might fit that stem section and
the upper corresponding to the upper half (as indicated by the notations to the right
of the stem-and-leaf plot). The characteristics of the batch are much clearer in this
plot. The numbers bunch together from about 130 to 160. And one unusually light
6 CHAPTER 1
Table 1.2. Too Sparse Stem-and-Leaf Plot of Weights
of 17 Scrapers from the Black Site
Weight (g) Stems Leaves
169 5
168
148.7 148 7 167
166
154.5 154 5 165
164 7
169.5 169 5 163
162
145.1 145 1 161 2
160
157.9 157 9 159
158
137.8 137 8 157 9
156
151.9 151 9 155
154 5
146.2 146 2 153
152 0
164.7 164 7 151
9
150
149.3 149 3 149 3
148
7
141.3 141 3 147
146 29
161.2 161 2 145
1
144
146.9 146 9 143 0
142
152.0 152 0 141 3
140
143.0 143 0 139
138
132.6 132 6 137 8
136
115.3 115 3 135
134
133
132 6
131
130
129
128
127
126
125
124
123
122
121
120
119
118
117
116
115 3