3.2 Minimum ceramic sherd thickness (mm) from the Gallina region of New Mexico

3.3 Measurements of the maximum thickness of losange-shaped Early Upper Paleolithic projectile points

3.4 Summary feature information for four sites

3.5 The reorganized summary feature information

4.1 Number of residents in Alyawara camps

4.2 The sum of y, where = 23 people

4.3 Computations of the sample variance and standard deviation for the number of residents in Alyawara camps

4.4 Denominators for deriving an estimate of the standard deviation

4.5 Flake length (cm) by raw material

4.6 Frequency of ceramic vessels classified using cultural historical types

4.7 A second series of ceramic vessels classified using cultural historical types

5.1 Probability of rolling a given sum using a pair of dice

5.2 Probabilities of success or failure of finding a parking space at UNM during different times of the day

6.1 Lengths (mm) of unbroken adze rejects from three production locales on Hawaii

7.1 Possible outcomes of hypothesis testing

8.1 Summary statistics for the three series of samples of the Timberlake Ruin flake lengths

9.1 Coefficients of variation for ceramics produced by specialists and non-specialists

9.2 Power curve for H₀ : Y_i = μ₁, where σ = .5 mm

9.3 The true probability of committing a Type I error using paired t-tests

10.1 Maximum flake length (mm) for 40 flakes from Cerro del Diablo, Chihuahua, Mexico

10.2 Maximum flake lengths (mm) of four raw materials from Cerro del Diablo, Chihuahua, Mexico

10.3 Matrix Illustrating Y_ij

10.4 Measurements of the length of the same 30 points measured independently four times

10.5 Contrived measurements of the length of the 30 points listed in Table 10.4

10.6 Generalized ANOVA table

10.7 ANOVA analysis comparing the mean length of the 30 projectile points presented in Table 10.4

10.8 Summary information for the maximum sherd thickness of a sample of three southwestern pottery types

10.9 ANOVA analysis comparing the sherd thickness of three pottery types

10.10 Confidence intervals for the maximum sherd thickness of three pottery types

11.1 Information for the regression analysis of 10 projectile points from Ventana Cave, Arizona

11.2 Historic pueblo room counts

11.3 Regression analysis of historic pueblo population sizecompared to room counts

11.4 Test of significance for H₀ : b_Y*X = 0

11.5 Residuals d_Y*X calculated as Y −

11.6 Leverage coefficients for the data reported in Table 11.2

11.7 Standardized residuals for the pueblo data

11.8 Summary of confidence limits and measures of dispersion used in regression analysis

12.1 Stature and long bone lengths of 10 males from the University of New Mexico documented skeletal collection

12.2 Calculation of sums used to compute a Pearson’s correlation coefficient

12.3 ANOVA table for Pearson’s correlation coefficient

12.4 ANOVA analysis of the correlation between the humerus length and stature of 10 skeletons from University of New Mexico’s skeletal collection

12.5 The ranking of fish species by their abundance at the Newbridge and Carlin settlements

12.6 Critical values for Spearman’s r when sample size is equal to or smaller than 10

12.7 Paleobotanical information from the Newbridge and Weitzer sites

13.1 Pottery sherds classified by their surface treatment and temper type

13.2 Sherds classified by their surface treatment andtemper type in which there is a strong association between the two variables

13.3 An additional set of hypothetical frequencies of sherds classified by their surface treatment and temper type

13.4 Expected frequencies for the sherds in Table 13.3 assuming thatthere is no association between surface treatment and temper type

13.5 Chi-square test of the data presented in Table 13.3

13.6 A matrix of sherd frequencies in which the number of smooth surfaced sherds greatly outnumber the stamped surfaced sherds

13.7 Chi-square test of the data presented in Table 13.6

13.8 Frequencies of flaked stone artifacts grouped by provenience and raw material

13.9 Chi-square test comparing the frequencies of lithic raw materials from four areas of Galeana, Chihuahua, Mexico

13.10 Observed and expected values for the Galeana data

13.11 Adjusted residuals for the chi-square analysis of flaked stone raw materials recovered from Galeana, Mexico

13.12 Matrix explaining the symbolism for Fisher’s exact probability test

13.13 Frequency of burials from Aldaieta Cemetery, Spain, associatedwith weapons and utilitarian utensils

13.14 Length measurements (cm) for four classes of arrowheads from Ventana Cave, Arizona

13.15 Frequency distributions of the Ventana Cave projectile points

13.16 The direction of difference between each variate listed in Table 13.14 and the grand median

13.17 Number of variates greater than and less than the median

13.18 Chi-square test comparing the frequencies of observed and expected values of arrow points that are greater than and less than the grand median

13.19 Maximum length (mm) of Upper Paleolithic bone points

13.20 Frequency distribution of the maximum lengths of the Upper Paleolithic bone points

13.21 The number of Upper Paleolithic points greater than and less than the grand median

13.22 Chi-square test comparing the frequencies of observed and expected values of Upper Paleolithic bone points that are greater than and less than the grand median

14.1 Friction coefficients of test tiles with differing surface treatments

14.2 Frequency distribution for the friction coefficients of pottery test tiles with different surface treatments

14.3 Ranking of the pottery test tile data

14.4 Ratio of ash to original bone weight for burned bone

14.5 Frequency tables for the ash to total weight ratio for the various time periods

14.6 Rankings for the ash to total weight ratios

14.7 Cranial length for samples of Norse and Andaman Islander populations (mm)

14.8 Summary statistics for the data presented in Table 14.7

14.9 ANOVA analysis comparing the cranial lengths of Norse and Andaman Islander populations

14.10 The data in Table 14.7 reorganized to more clearly differentiate between the variables sex and culture area

14.11 A 2 × 2 matrix reflecting the sums of the data presented in Table 14.10

14.12 The structure of a two-way ANOVA table

14.13 Two-way ANOVA comparing the mean values for Norse and Andaman Islander males and females

14.14 The cranial lengths for three samples each of Norse males and females (mm)

14.15 Summary statistics of samples of Norse females and males presented in Table 14.14

14.16 The construction of a nested ANOVA table

14.17 Results of the nested ANOVA analysis comparing the cranial lengths of samples of Norse males and females

15.1 Metric variables measured for projectile points from Ventana Cave, Arizona

15.2 The amount of the variation in each variable extracted through the factor analysis

15.3 Total variance explained in the factor analysis of the Ventana Cave projectile points

15.4 Factor loadings for the first six factors

15.5 The squared factor loadings of each variable for the first six factors

16.1 A random numbers table

16.2 Changing confidence intervals as sample size increases

3.2 Frequency distribution resulting from grouping depths for Carrier Mills features into 11 classes: Interval = 2 cm

3.3 Frequency distribution resulting from grouping depths for Carrier Mills features into five classes: Interval = 5 cm

3.4 Frequency distribution of heights in your class

3.5 Frequency distribution of the Gallina ceramic minimum sherd thicknesses (mm)

3.6 Histogram of minimum sherd thicknesses

3.7 Stem and leaf diagram of Gallina ceramic data

3.8 Ordered stem and leaf diagram of Gallina ceramic data

3.9 Stem and leaf diagram created using increments of .5 cm

3.10 Stems that can be used to create a stem and leaf diagram of the data from Table 3.3

3.11 Cumulative frequency distribution of Carrier Mills feature depths (cm)

3.12 Plot of the cumulative frequency distribution of Carrier Mills feature depths

3.13 Plot of a symmetrical, or normal distribution

3.14 Plot of a left-skewed distribution

3.15 Plot of a right-skewed distribution

3.16 Plot of a bimodal distribution

3.17 Plot of a leptokurtic distribution

3.18 Plot of a platykurtic distribution

3.19 Plot of a mesokurtic distribution

3.20 Bar chart of the physiographic provenience of villages during the Early Monongahela period in the Lower Monongahela and Lower Youghiogheny river basins

3.21 Bar chart of the frequencies of villages during the Early Monongahela, Middle Monongahela, and Late Monongahela periods in various physiographic proveniences

4.1 Two distributions with identical means and sample sizes but different shapes

4.2 Box plots of flake length data presented in Table 4.5

5.1 The distribution of the probabilities of rolling a sum with a pair of dice

5.2 Distribution of the probability of finding a parking spot in UNM’s overcrowded parking lot

5.3 Venn diagram of probabilities of hearts and diamonds

5.4 Intersection of probabilities of obtaining a heart or an ace

5.5 Pascal’s triangle

6.1 The normal distribution

6.2 Two normal distributions with different means and the same standard deviation

6.3 Two normal distributions with the same mean and different standard deviations

6.4 Percentages of variates within one, two, and three standard deviations from μ

6.5 Areas under the normal distribution corresponding with the standard deviation

6.6 The relationship between the mean and Y_i = 6.4 mm

6.7 The area under the standardized normal distribution between Y₁ = 5.3 mm and Y₂ = 6.8 mm

7.1 Areas of rejection associated with α = .05

7.2 The areas under the normal distribution associated with Z = 1.46

7.3 Areas of a normal distribution of Thomomys bottae alveolar lengths associated with α = .05 and α = .01

8.1 Illustration of the confidence limits and critical region of Thomomys bottae alveolar length distribution

8.2 The distribution of 166 maximum flakes lengths

8.3 Distribution of the mean maximum flake lengths for 15 samples of three flakes

8.4 Distribution of the mean maximum flake lengths for 20 samples of 10 flakes

8.5 Distribution of the mean maximum flake lengths for 30 samples of 20 flakes

8.6 Comparisons of the t-distribution and the normal distribution

8.7 Different shapes of the t-distribution as determined by ν

8.8 Region of rejection for a one-tailed test corresponding with H₀ : ≤ μ

8.9 Critical value and area of rejection for H₀ : ≥ μ

9.1 β associated with H_a : Y_i = μ₂

9.2 Illustration of β decreasing as α increases

9.3 β associated with evaluating H₀ : Y_i = μ₁ where H₁ : Y_i = μ₂ and H₂ : Y_i = μ₃

9.4 Calculation of β for the alternate hypothesis H₁ : Yi = μ₂

9.5 Power for the null hypothesis H₀ : Y_i = μ₁ relative to H₁ : Y_i = μ₂

9.6 Distribution of coefficients of variation associated with pottery assemblages made by specialists and generalized producers

9.7 Illustration of β and the power associated with determining that an assemblage was made by specialists (H₀ : Y_i = μ_spec.)

9.8 Illustration of β and the power associated with determining that an assemblage was made by generalists (H₀ : Y_i = μ_gen.)

9.9 β and power for alternate means (μ_alt) less than the lower confidence limit

9.10 β and power for alternate means (μ_alt) larger than the upper confidence limit

9.11 β and power for an alternate mean (μ_alt) contained within the confidence limits

9.12 The power curve for the hypothetical alternative distributions to H₀ : Y_i = μ₁

10.1 Distribution of means in which the among-group variance is comparable to the within-group variance

10.2 Distribution of means in which the variance among means exceeds the variance within means

10.3 The F-distribution

11.1 The relationship between a tree’s age and number of tree rings

11.2 The relationship between the US dollar and Mexican peso in the spring of 2010

11.3 Average diastolic blood pressure of humans of various ages

11.4 Regression line describing the relationship between age and diastolic blood pressure

11.5 Scatter plot of the maximum length compared with weight for 10 projectile points from Ventana Cave, Arizona

11.6 An example of a functional relationship that meets the assumptions necessary for regression analysis

11.7 The effect on the likely range of regression lines as the inconsistent variation resulting from non-homoscedastic data allows the analysis to stray excessively from the mean for each X_i

11.8 An example of the effect on the likely range of regression lines resulting from skewed distributions

11.9 A relationship in which (X_i − ) and (Y_i − ) positively covary, in that as X_i − becomes large, so does Y_i −

11.10 A relationship in which (X_i − ) and (Y_i − ) negatively covary, in that as X_i − becomes larger, Y_i − becomes smaller

11.11 A relationship in which (X_i − ) and (Y_i − ) do not covary, in that as X_i − becomes larger, Y_i − does not consistently become larger or smaller

11.12 Regression line for the projectile point data presented in Table 11.1

11.13 Sources of variation at X_i, Y_i

11.14 The relationship between site population and the total number of rooms

11.15 Regression relationship between population size and the totalnumber of rooms

11.16 The range of regression coefficients that likely encompass thetrue regression coefficient

11.17 Confidence intervals for the regression coefficient illustrated in Figure 11.15

11.18 Regression residuals for the pueblo room data

11.19 A heteroscedastic distribution

11.20 Residual pattern indicative of heteroscedasticity with increasing variance as X_i increases

11.21 A curvilinear, as opposed to linear, distribution

11.22 A residual pattern indicative of a curvilinear distribution

11.23 A scatter plot in which the two end points contribute disproportionately to the regression relationship

11.24 The leverage coefficients reported in Table 11.6

11.25 Standardized residuals for the residuals reported in Table 11.5

11.26 A curvilinear relationship based on exponential growth

11.27 The logarithmical transformed relationship illustrated in Figure 11.26

12.1 The regression relationship between femur length and stature for 10 males from the University of New Mexico’s skeletal collection

12.2 A scatter plot depicting the relationship between humerus length and stature for 10 males from the University of New Mexico’s skeletal collection

12.3 % confidence intervals illustrating the variation in both dependent variables

12.4 An example of data that are consistent with a bivariate normal distribution without significant outliers

12.5 An example of clustered data for which Pearson’s correlation coefficient is inappropriate

12.6 Examples of possible relationships that can result in significant correlations

15.1 Examples of Ventana Cave projectile points and the measured variables

15.2 A traditional regression line organized using an x- and y-axis value of 0

15.3 The scatter plot illustrated in Figure 15.1 reorganized using the regression line as the axis

15.4 Scree plot for the factor analysis evaluating the Ventana Cave projectile point data

4.2 The variance

4.3 The standard deviation

4.4 Calculation formula for the sum of squares

4.5 The coefficient of variation

4.6 Correction formula for the coefficient of variation

4.7 The index of dispersion for nominal data

4.8 The index of qualitative variation

5.1 Calculation of an empirically derived probability

5.2 Calculation of the probability of repeated events

5.3 The binomial formula

5.4 Calculation formula for the binomial power terms

5.5 Calculation formula for the binomial coefficient

5.6 Unified formula for specific binomial terms

6.1 Calculation formula for a Z-score reflecting a standardized normal distribution

8.1 The standard error

8.2 The Z-score comparing a sample mean to μ

8.3 The standard error for a sample

8.4 The degrees of freedom

8.5 The t-score comparing a sample mean to a variate

8.6 The t-score comparing a sample mean to μ using s to approximate σ

10.1 The pooled variance

10.2 The variance among means

10.3 Calculation formula for the sum of squares of means

10.4 Variance within groups

10.5 Variance among groups

10.6 The F-test

11.1 The regression coefficient

11.2 The coefficient of determination

11.3 Standard error for a regression coefficient

11.4 Calculation of the variance of Y at X

11.5 t-test evaluating H₀ : b = 0

11.6 The standard error of for a given X_i

11.7 The standard deviation of Y_i around _i at a given X_i

11.8 The estimate of at Y_i

11.9 Calculation of D, one of the terms necessary to estimate

11.10 Calculation of H, one of the terms necessary to estimate

11.11 Lower confidence interval for estimate of at Y_i

11.12 Upper confidence interval for estimate of at Y_i

11.13 Calculation of the leverage coefficient

11.14 The standardized regression residual

12.1 Pearson’s product-moment correlation coefficient

12.2 The t-test for evaluating the significance of Pearson’s correlation coefficient

12.3 The standard error of the Pearson’s correlation coefficient

12.4 Explained sums of squares for Pearson’s product-moment correlation coefficient

12.5 Unexplained sums of squares for Pearson’s product-moment correlation coefficient

12.6 F-score comparing the explained and unexplained sums of squares for Pearson’s product-moment correlation coefficient

12.7 Spearman’s rank order correlation coefficient

13.1 The chi-square test

13.2 Degrees of freedom for the chi-square test

13.3 The expected values for the chi-square test

13.4 The chi-square residual

13.5 The adjusted chi-square residual

13.6 Fisher’s exact probability for 2 × 2 tables

13.7 Yate’s continuity correction

14.1 The Wilcoxon two-sample test

14.2 Approximating U using the normal distribution

14.3 The Kruskal–Wallis test

15.1 The eigenvalue

Most archaeologists are in the counting and measuring business not for its own sake, but to help us fashion a meaningful perspective on the past. Quantification isn’t required to back up every proposition that is made about the archaeological record, but for some propositions it is absolutely essential. For example, suppose we proposed an idea about differences in Hallstatt assemblages in Central Europe that could be evaluated by examining ceramic variation. Having observed hundreds of the pots, we could merely assert what we felt the major differences and similarities to be, and draw our conclusions about the validity of our original idea based upon our simple observations. We might be correct, but no one would take our conclusions seriously unless we actually took the relevant measurements and demonstrated that the differences and/or similarities were meaningful in a way that everyone agreed upon and understood. Quantification and statistics serve this end, providing us with a common language and set of instructions about how to make meaningful observations of the world, how to reduce our infinite database to an accurate and understandable set of characterizations, and how to evaluate differences and similarities. Importantly, statistics do this by using a framework that allows us to specify the ways in which we can be wrong, and the likelihood that we are mistaken. Statistics consequently provide archaeologists with a means to make arguments about cause that will ultimately help us construct explanations.

Statistical thinking plays an important role in archaeological analysis because archaeologists rely so heavily on samples. The archaeological record contains only the material remains of human activity that time and the vagaries of the environment (including human activity) have allowed to be preserved. The artifacts, features, and other material manifestations of human behavior that enter the archaeological record are only a small subset of those originally produced. Funding constraints, time limits, and our emphasis on conserving the archaeological record further dictate that archaeologists generally recover only a small subset of those materials that have been preserved. Thus, we have a sample of the material remains that have been preserved, which is only a sample of all of the materials that entered the archaeological record, which is only a sample of all of the materials that humans have produced.

Archaeologists are consequently forced to understand and explain the past using imperfect and limited data. Connecting our sample to a meaningful understanding of the past necessitates the application of a statistical framework, even when quantitative methods are avoided as part of a purportedly humanistic approach. It is only through statistical reasoning, no matter how implicit, that any form of general conclusion can be formed from the specifics of the archaeological record. Regardless of whether an archaeologist studies the social differentiation of Cahokia’s residents, subsistence shifts during the Mexican colonial occupation of New Mexico, or the religious systems of Upper Paleolithic cave dwellers, they are going to employ a statistical approach, even if they don’t acknowledge it. Quantitative methods allow us to make this approach explicit and make our arguments logically coherent and thereby facilitate their evaluation. Even the most ardent humanist should appreciate this.

As important as statistics are, we must remember that they are only tools, and subservient to theory. Our theoretical perspectives tell us which observations are important to make and how explanations are constructed. Statistics are useful only within this larger context, and it is important to remember their appropriate role. It is also important to recognize that the use of statistics does not equal science. The historical confluence of events that brought statistics, computers, the hypothetico-deductive method, and the theoretical advances of the New Archaeology to our discipline in a relatively brief span of time in the 1960s make it appear that they are inseparable. Nothing could be farther from the truth. While this might seem self-evident, at least one quite popular introductory archaeology textbook overstates the relationship, as a discussion of the role of science in archaeology begins with a brief discussion of statistics. Not the role of theory, not the scientific method, but statistics! Statistics do not a science make, and statistical analyses conducted in the absence of theory are merely vacuous description.

This book approaches quantification and statistics from the perspective that they are a simple set of tools that all competent archaeologists must know. Most readers will use statistics innumerable times throughout their career. Others may never calculate a mean or standard deviation willingly, but at least they will know the basics of the statistical tool kit. Choosing not to use a tool is fine. Remaining ignorant is unfortunate and unnecessary. At the very least, knowledge of statistics is needed to evaluate the work of others who do use them.

So, why should two archaeologists write a book about statistics when there are thousands of excellent statistics books in existence? Here are our reasons in no particular order. First, few of us entered archaeology because we wanted to be mathematicians. In fact, many archaeologists became interested in archaeology for very humanistic (or even romantic) reasons, and many avoided math in school like the plague. There definitely needs to be a book that is sympathetic to those coming from a non-quantitative background. We seek to achieve this goal by presenting the clearest description of techniques possible, with math no more complicated than simple algebra, but with enough detail that the reader will be able to actually understand how each technique operates.

Second, most statistics textbooks use examples that are not anthropological, and are very hard to relate to the archaeological record. While knowledge of dice examples is useful when playing craps in Las Vegas, the implications of these examples for archaeological studies are often difficult to decipher. Our examples are almost exclusively archaeological, and we hope that they provide good illustrations of how you might approach various archaeological data sets from a statistical perspective.

Third, archaeologists do not always need the standard set of statistics that are presented in popular textbooks. Some techniques of limited importance to archaeology are overemphasized in these texts, while other extremely important statistical methods are underemphasized or do not appear at all.

Fourth, it is our observation that many degree-granting programs in archaeology focus solely on computer instruction in quantitative methods rather than on the tried and true pencil and paper method. We have nothing against the use of computers and statistical software, as long as it is done by people who first learn statistical techniques by putting pencil to paper. However, our experience has shown us that when all training is focused on using a statistical package instead of learning a statistical method, the computer becomes a magic black box that produces “results” that students who don’t know what actually happened inside the box are (hopefully) trained to interpret. This lack of understanding causes confusion and, more importantly, embarrassment when insupportable or erroneous conclusions are drawn. These students need a friendly text to which they can refer to help clarify how the quantitative methods work and how their results should be understood.

Finally, many disciplines use samples, but few are as wholly reliant on them as is archaeology. This in turn means that the application of quantitative reasoning has special significance in archaeological research that needs to be explored if we are to produce the best archaeological analyses we can. This consideration is absent from statistical texts written for general audiences, but should be central to those specifically for archaeologists. It certainly will be central to the discussions that follow this chapter.

Ultimately, our goal is to illustrate the utility and structure of a quantitative framework to the reader (i.e., you), and to provide a full understanding of each statistical method so that you will understand how to calculate a statistical measure, why you would want to do so, and how the statistical method works mathematically. If you understand these issues, you will find each method to be intuitively meaningful and will appreciate the significance of its assumptions, limitations, and strengths. If you don’t understand these factors, your applications will be prone to error and misinterpretations, and, as a result, archaeology as a discipline will suffer. Hopefully, this text will serve to aid you, gentle reader, as we all work to accomplish our collective goals as a discipline.