Drennan R.D. Statistics for Archaeologists: A Common Sense Approach
Подождите немного. Документ загружается.
THE SHAPE OR DISTRIBUTION OF A BATCH 59
Figure 5.3. Transformation rulers: a “normal” ruler (above); a ruler that would give length mea-
surements with a square transformation (center); the same square transformation ruler with tick
marks every five units instead of every 1/5 of the length (bottom).
deviation to be useful indexes of the center and the spread. The mean and stan-
dard deviation are fundamental to many of the techniques discussed farther on
in this book, and if the mean and standard deviation are not telling us the truth
about the center and spread of the batch, then those other techniques will not
work well. Transformation can be thought of as measuring with special rulers.
Fig.
5.3 shows three rulers. At the top is a “normal” ruler that we might use
to measure the length of some object. In the middle is a “square transformation
ruler.” If we measured the same object with this ruler, the result would be the
square of the “normal” measurement. The bottom ruler is a square transforma-
tion ruler just like the middle ruler, except that the tick marks, instead of being
evenly spaced along the ruler, are placed every five units. This shows the way the
units of measurement are distributed differently along the square transformation
ruler than they are along the “normal” ruler, and may provide a better common-
sense feel for just how it is that the square transformation shifts numbers along the
scale to change the shape of a batch. The same is true of all the other transfor-
mations we have discussed. Using them amounts to nothing more than measuring
with a peculiar ruler. Although it certainly seems strange at first thought, there
is no reason that we couldn’t use rulers like the square transformation rulers in
Fig.
5.3. We would just have to measure everything we wanted to compare with
the same peculiar ruler. That is exactly what we are doing when we transform a
batch of numbers – we are measuring with a peculiar ruler, and we must use the
same peculiar ruler (or transformation) on all the batches we want to be able to
compare.
THE NORMAL DISTRIBUTION
The single-peaked symmetrical shape we have been pursuing with transformations
in this chapter has the essential characteristics of the (in)famous normal distribution.
Actually, the normal distribution implies some other more specific characteristics,
but in practical terms, a batch that is single peaked and symmetrical can be taken
as close enough to a normal distribution to apply statistical techniques suitable for
batches that are normally distributed.
60 CHAPTER 5
The requirement of normality in the distributions of batches that are to be ana-
lyzed in certain ways is no deep mathematical mystery. It is not a question of abiding
by some secret and sacred principle understandable only to the high priests of
statistics. Understanding the importance of the normal distribution begins with such
simple and intuitively intelligible notions as the ways in which numerical indexes
of level and spread work on asymmetrical batches. We have seen that these numer-
ical indexes simply do not produce sensible results when applied to a batch that is
not single peaked and symmetrical. This is the starting point for understanding why
some statistical techniques must only be applied to normal distributions. Many of
them begin by characterizing the batch to be analyzed with the mean and standard
deviation. If these numerical indexes do not provide accurate and meaningful mea-
sures of the center and spread of the batch, then no technique that takes them as a
starting point can be expected to produce accurate and meaningful results.
To summarize, then, if we wish to study a batch of numbers that is not single
peaked and symmetrical, we must often take special action. This consists, first, of
splitting a batch with multiple peaks into multiple separate batches, each with a sin-
gle peak. Second, we can use transformations to make the shape of a single-peaked
batch more symmetrical and/or deal with any outliers we observe. These initial data
preparation steps are important to the success of many statistical techniques and
must not be overlooked. Analysis of a batch of numbers should always begin by
exploring the batch with a stem-and-leaf plot, and taking whatever corrective action
may be indicated to deal with multiple peaks, asymmetrical shapes, and outliers.
Picking the best transformation to correct asymmetry is a question of subjective
judgment. It requires a bit of practice to look at the distributions produced by dif-
ferent transformations and decide which is most symmetrical. These judgments are
especially difficult in small batches of numbers where displacing a single number
in the stem-and-leaf plot can have a strong effect on the apparent symmetry. It is a
good idea not to be too strongly swayed by appearances that could be changed if
only one or two numbers in the batch were slightly different. It is better, instead,
to concentrate more heavily on major trends that would only be altered if many
numbers were changed.
Picking the best transformation is also a process of trial and error. Although
Table
5.3 might help you guess which transformation to try, it is almost always
necessary to try several transformations and look at the results (by examining stem-
and-leaf and box-and-dot plots of the transformed batches) in order to decide which
produces the most symmetrical shape. Compromises are often required, especially
when transformations are being applied to two or more batches that eventually are
to be compared. The same transformation may not produce the most symmetrical
shape for each of the batches involved, but the same transformation must be applied
to all the batches if they are to be compared after transformation.
An alternative to correcting asymmetry through transformations is to use statisti-
cal techniques based not on the mean and standard deviation but on other indexes –
like the trimmed mean and trimmed standard deviation – that are more resistant
to the effects of outliers and asymmetry. Such approaches will be discussed where
relevant in the following chapters. Generally, if the presence of outliers presents a
THE SHAPE OR DISTRIBUTION OF A BATCH 61
problem for the use of means and standard deviations, use of the trimmed mean and
standard deviation is a good solution. If pervasive asymmetry in the distribution is
the problem, applying an appropriate transformation is more effective.
PRACTICE
1. Look carefully at the shapes of the batches of areas of Early and Late Bronze Age
sites near Nanxiong from Table
3.5. In the practice questions from Chapters 3 and
4, you have already made stem-and-leaf and box-and-dot plots of these batches.
Does either batch have a skewed shape? If so, is it skewed upward or downward?
2. If either Early or Late Bronze Age site areas are skewed, use your statpack to
experiment with transformations to correct the asymmetry. Which transforma-
tion would you choose for each batch individually? Why? Which transformation
would you use for both if you intended to compare the transformed batches?
Why?
Chapter 6
Categories
Column and Row Proportions ..................................................................... 69
Proportions and Densities .......................................................................... 70
Bar Graphs .......................................................................................... 71
Categories and Sub-batches ........................................................................ 73
Practice.............................................................................................. 75
The batches of numbers that we have discussed in Chapters1–5 have all consisted
of a set of measurements of some kind. There is a fundamentally different kind of
batch that we must discuss before continuing with the second section of this book.
This other kind of batch results from observations of characteristics that are not
measured exactly, but instead are grouped into different categories. Archaeologists
are quite accustomed to the notion of categorizing things. We usually discuss it
under the heading of typology, and the definition of typologies (or sets of categories)
for artifacts is widely recognized as a fundamental initial step in description and
analysis. Much has been written about the “correct” way to define pottery types in
particular. Our concern here is not how to define categories, but rather what to do
with the result once we have defined them and counted up how many things there
are in each one. When we classify the ceramics from a site as Fidencio Coarse,
Atoyac Yellow-White, and Socorro Fine Gray, we are dealing in categories. When
we count the number of flakes, blades, bifaces, or debitage from a site, we are also
dealing in categories. When we add up the number of cave sites and open sites in
a region, we are once again dealing in categories. When we divide the sites in a
region into large sites, medium-sized sites, and small sites, we are still once more
dealing in categories. Data recorded in terms of such categories comprise batches
just as do data recorded as true measurements (for example, in centimeters, grams,
hectares, etc.).
Table
6.1 provides an example of such categorical information for a set of 140
pottery sherds. One observation made about each sherd is the site where it was
recovered. There are three categories here: the Oak Grove site, the Maple Knoll
site, and the Cypress Swamp site. A second observation concerns incised decora-
tion, with two categories: each sherd is either incised or unincised. This may seem
an unusual way to present such information, and indeed it is. The presentation is
R.D. Drennan, Statistics for Archaeologists, Interdisciplinary Contributions
to Archaeology, DOI 10.1007/978-1-4419-0413-3
6,
c
Springer Science+Business Media, LLC 2004, 2009
63
64 CHAPTER 6
Kinds of Data
Some statistics books begin with a chapter about different kinds of data as if
the recognition of a standard set of several fundamentally different kinds of
data were the rock upon which all statistical analysis was built. The “funda-
mentally” different kinds, however, are defined in different ways by different
authors, and many books don’t make much of such distinctions at all. There
are almost as many sets of terms as there are authors, and some of the same
terms are used in contradictory ways by different authors. The point is that
there are a number of characteristics of batches of numbers that vary. You can
analyze batches with certain characteristics in one way, but batches that lack
those characteristics can be analyzed in a different way. The most important
distinction made in this book is between what are here called measurements
and categories.
Measurements are things like lengths, widths, areas, weights, and so on—
quantities we measure along a scale of appropriate units. True measurements
come in fractional values as well as whole-number values (difficult numbers
to work with like 3-
13
/
16
inches, or much easier numbers to work with like
9.68 cm), and there is, in principle at least, an infinite number of potential
values along the scale. Measurements, as the term is used in this book, can
also include numbers of things, like number of inhabitants in different regions,
number of artifacts in different sites, and so on. Measurements can also be
derived from other measurements arithmetically. Densities of artifacts in dif-
ferent excavation units, for example, are measurements derived by dividing
the number of artifacts encountered in each excavation unit by the volume of
deposit excavated to arrive at the number of artifacts per cubic meter for each
excavation unit. In exploratory data analysis, measurements are sometimes
called amounts (measurements along a scale) and counts (counted numbers
of things) and balances (which can have positive and negative values). Mea-
surements are made along what are sometimes called ratio scales or interval
scales. A ratio scale has a meaningful zero point, as, for example, a length
or weight. An interval scale has an arbitrary zero point, which prevents some
kinds of manipulations. The usual example of a scale with an arbitrary zero
point is temperature. The fact that the zero point is arbitrary means that you
really can’t say that 60
◦
is twice as hot as 30
◦
(on either Fahrenheit or Celsius
scales, although you can do such things with the temperature scale measured
from absolute zero).
Categories are essentially groups of things, and we count the number of
things in each group. We ordinarily work with sets of categories that are mutu-
ally exclusive and exhaustive. That is, each thing in the set of things we are
studying must fit into one category and only one category. Pottery types are a
CATEGORIES 65
common kind of category in archeology, and we recognize that pottery types
need to be defined so that each sherd can be placed in one and only one type.
Colors represent another set of categories. We may sort things out as red, blue,
or green. If we find bluish-green things, we may need to add a fourth category
to the set. Categories are sometimes called nominal data. If the categories can
logically be arranged in a specific order, then they form ordinal data or ranks.
Pottery types do not have this property. Categories like large, medium, small,
and tiny do; we recognize that to say small, large, tiny, medium is to put these
categories out of order. If there are very many categories in the set, ranks begin
to act like true measurements in some ways.
The most important distinction between kinds of data for the organiza-
tion of thoughts in this book is between what we will call measurements
and categories, but we will also consider some special treatments that can be
applied to ranks—treatments that relate strongly to things we can do with true
measurements.
cumbersome and tells us virtually nothing about patterns. In an instance like this,
we would much more likely present the information as a tabulation, and that is what
we will do shortly. It is often convenient to manage categorical information in the
manner in which it is presented in Table
6.1, however, especially with computers.
Thus it is important to recognize Table
6.1 as one means of organizing the same
information we will see in more familiar form in the following tables. Table
6.1
is the most complete and detailed way of recording this information and the most
similar to the way in which batches of measurements were initially presented in
previous chapters.
Table
6.2 presents the information about where the sherds were recovered in a
more compact, familiar, and meaningful way. This simple tabulation of frequencies
(or counts)andproportions (or percentages) immediately tells us something about
how much pottery came from where – something that was not at all apparent in
Table
6.1. More pottery came from Oak Grove than any other site and less from
Maple Knoll. Table
6.3 performs the same task for the information about decoration
for the same 140 sherds. Most of the sherds are unincised, but the difference in
proportions between incised and unincised is not extreme.
In effect, Table
6.1 contains sets of related batches, like the related batches that
have been discussed in previous chapters. In this case, we could divide the sherds
into three related batches, as in Table
6.2: sherds from the Oak Grove site, sherds
from the Maple Knoll site, and sherds from the Cypress Swamp site. Or we could
divide them in a different way into two related batches, as in Table
6.3:incised
sherds and unincised sherds. Each set of categories is simply one way of dividing
the whole set of sherds into different batches. We might well want to compare the
first three batches (the sherds from each of the three sites) in regard to the other
set of categories (incised decoration). Table
6.4 extends the tabulations of Tables6.2
and 6.3 to accomplish this comparative goal, by simultaneously dividing the sherds
66 CHAPTER 6
Table 6.1. Information about 140 Pottery Sherds
Oak Grove Unincised Maple Knoll Unincised Cypress Swamp Incised
Maple Knoll Incised Oak Grove Incised Cypress Swamp Unincised
Cypress Swamp Unincised Oak Grove Unincised Cypress Swamp Unincised
Cypress Swamp Incised Oak Grove Unincised Oak Grove Incised
Cypress Swamp Incised Maple Knoll Incised Oak Grove Unincised
Cypress Swamp Unincised Cypress Swamp Unincised Maple Knoll Incised
Cypress Swamp Incised Cypress Swamp Unincised Maple Knoll Unincised
Oak Grove Incised Oak Grove Incised Oak Grove Incised
Oak Grove Unincised Oak Grove Unincised Oak Grove Unincised
Maple Knoll Unincised Maple Knoll Unincised Maple Knoll Incised
Oak Grove Incised Cypress Swamp Incised Cypress Swamp Incised
Oak Grove Unincised Cypress Swamp Incised Cypress Swamp Unincised
Maple Knoll Incised Oak Grove Incised Oak Grove Incised
Maple Knoll Unincised Oak Grove Unincised Oak Grove Unincised
Cypress Swamp Unincised Maple Knoll Unincised Oak Grove Unincised
Cypress Swamp Incised Cypress Swamp Unincised Maple Knoll Incised
Oak Grove Unincised Cypress Swamp Incised Cypress Swamp Unincised
Maple Knoll Incised Oak Grove Incised Cypress Swamp Incised
Maple Knoll Unincised Oak Grove Unincised Oak Grove Incised
Cypress Swamp Incised Maple Knoll Unincised Oak Grove Unincised
Oak Grove Incised Cypress Swamp Unincised Maple Knoll Unincised
Oak Grove Unincised Cypress Swamp Incised Maple Knoll Incised
Maple Knoll Unincised Oak Grove Unincised Oak Grove Incised
Cypress Swamp Unincised Maple Knoll Incised Oak Grove Unincised
Oak Grove Incised Cypress Swamp Unincised Cypress Swamp Incised
Oak Grove Unincised Oak Grove Unincised Cypress Swamp Unincised
Maple Knoll Incised Maple Knoll Incised Cypress Swamp Unincised
Maple Knoll Unincised Cypress Swamp Unincised Oak Grove Incised
Oak Grove Incised Cypress Swamp Incised Oak Grove Unincised
Oak Grove Unincised Oak Grove Incised Maple Knoll Unincised
Maple Knoll Incised Oak Grove Unincised Cypress Swamp Incised
Cypress Swamp Incised Oak Grove Unincised Oak Grove Incised
Cypress Swamp Unincised Maple Knoll Incised Oak Grove Unincised
Oak Grove Incised Cypress Swamp Unincised Oak Grove Unincised
Oak Grove Unincised Oak Grove Incised Maple Knoll Incised
Maple Knoll Incised Oak Grove Unincised Cypress Swamp Unincised
Maple Knoll Unincised Oak Grove Unincised Oak Grove Incised
Cypress Swamp Unincised Maple Knoll Incised Oak Grove Unincised
Maple Knoll Incised Cypress Swamp Unincised Maple Knoll Unincised
Cypress Swamp Unincised Oak Grove Incised Cypress Swamp Unincised
Oak Grove Incised Oak Grove Unincised Cypress Swamp Incised
Oak Grove Unincised Maple Knoll Incised Cypress Swamp Unincised
Maple Knoll Unincised Cypress Swamp Incised Oak Grove Incised
Cypress Swamp Unincised Cypress Swamp Unincised Oak Grove Unincised
Maple Knoll Incised Oak Grove Incised Maple Knoll Incised
Oak Grove Incised Oak Grove Unincised Maple Knoll Unincised
Oak Grove Unincised Maple Knoll Incised
CATEGORIES 67
Calculating Percentages and Rounding Error
We have noted rounding error before, but Tables 6.2 and 6.3 provide an oppor-
tunity to clear up this little mystery completely. We know that 140 sherds
are 100%, and the percentages in Table 6.3 add up to 100.0, but the per-
centages of the three categories in Table 6.2 add up to only 99.9%. In both
tables the percentages have been rounded off to one digit following the deci-
mal point. For Table 6.3, the full calculations of the percentages are 64 / 140 =
.4571428571428571428. . . and 76 / 140 = .5428571428571428571. ..
Both of these numbers will continue to repeat the same sequence of
digits (. .. 142857...) forever. The division will never come out even, no
matter how far out it is carried. To change .4571428571428571428.. . and
.5428571428571428571. . . from ordinary decimal fractions into percent-
ages, of course, we multiply them by 100: 45.71428571428571428...%and
54.28571428571428571...%. (And while we’re on the subject, it is worth
emphasizing that 0.45 and 45.0% are the same number. There is a big dif-
ference between 0.45 and 0.45%—0.45 means .45 out of 1.00 or 45 out of
100 or 4,500 out of 10,000 [that is, almost half], but 0.45% means 0.45 out
of 100 or 45 out of 10,000 [or far less than half]. It is essential to be care-
ful with a decimal point and the % symbol.) Clearly the percentages we have
here must be rounded off. If we want one digit after the decimal place in
the percentage, 45.71428571428571428. . . % rounds down to 45.7% and we
lose the extra .01428571428571428...%, and 54.28571428571428571...%
rounds up to 54.3%. In rounding this number up we have actually added
.01428571428571428.. . to it, which is exactly the amount that we lost in
rounding the other percentage down. Since one percentage has been raised
by the same amount that the other has been lowered by, the amounts cancel
each other out when we add the percentages together, and the total is 100.0%.
In Table 6.2, however, all three percentages turn out to round down:
• 59/140 = .42142857142857. .. which rounds down to .421 (or 42.1%),
losing .042857142857...%;
• 37/140 = .26428571428571. .. which rounds down to .264 (or 26.4%),
losing .028571428571...%;and
• 44/140 = .31428571428571. .. which rounds down to .314 or 31.4%,
losing .028571428571...%.
If we add up what we have lost in rounding all three percentages down, we
get almost exactly the 0.1% that is missing from the 99.9% total of the three
rounded off percentages:
(.042857142857...%+ .028571428571...%+ .028571428571.. . %
= .099999999999...%)
68 CHAPTER 6
Precisely the same thing can happen in the other direction if more is gained by
rounding some percentages up than is lost by rounding others down. Thus the
total of a set of percentages can be slightly more than 100%. Sometimes, doing
percentage calculations with more decimal digits of precision will remove
rounding error. In a case like this example, however, where the quotient of
the division repeats infinitely, it doesn’t matter how far out we carry the cal-
culations. They will never come out exactly even. Sooner or later we have to
round off, and accept a little rounding error.
Table 6.2. Sherds from Three Sites
Oak Grove Maple Knoll Cypress Swamp Total
Frequency 59 37 44 140
Proportion 42.1% 26.4% 31.4% 99.9%
Table 6.3. Pottery Decoration
Incised Unincised Total
Frequency 64 76 140
Proportion 45.7% 54.3% 100.0%
Table 6.4. Incised and Unincised Sherds from Three Sites
a. Frequencies
Oak Grove Maple Knoll Cypress Swamp Total
Incised 25 21 18 64
Unincised 34 16 26 76
Total 59 37 44 140
b. Column Proportions
Oak Grove Maple Knoll Cypress Swamp Average
Incised 42.4% 56.8% 40.9% 45.7%
Unincised 57.6% 43.2% 59.1% 54.3%
Total 100.0% 100.0% 100.0% 100.0%
c. Row Proportions (not useful in this instance)
Oak Grove Maple Knoll Cypress Swamp Total
Incised 39.1% 32.8% 28.1% 100.0%
Unincised 44.7% 21.1% 34.2% 100.0%
Average 42.1% 26.4% 31.4% 99.9%
CATEGORIES 69
by site and by incised decoration. Such a tabulation is sometimes called a cross
tabulation or two-way table, because it divides the entire set of sherds into categories
in two different ways simultaneously. In this kind of table there are also two different
ways to use percentages in comparing these batches.
COLUMN AND ROW PROPORTIONS
Following the frequencies in the two-way table are column proportions (Table 6.4b).
Each of the column proportions is a proportion of the total number of sherds at
the corresponding site, so the proportions for each site add up to 100% (within
rounding error). These proportions are similar to those in Table
6.3, but now they
are calculated separately for each of the three sites. The average column proportions
at the extreme right of Table
6.4b are not simply the averages of the individual site
proportions. They are actually the same as the proportions in Table
6.3.Thatis,they
are the proportions for the complete set of sherds considering all sites together.
Column proportions are useful for comparing columns to each other – in this
instance comparing the three sites to each other with regard to their relative pro-
portions of incised and unincised sherds. The assemblage of sherds from the Maple
Knoll site stands out in regard to incised pottery, since 56.8% of the sherds from the
Maple Knoll site are incised. At the Oak Grove and Cypress Swamp sites, incised
pottery is considerably less abundant, amounting to 42.4% and 40.9% of the sherds
at the two sites, respectively.
Table
6.4c provides row proportions for the table. These are proportions, not of
the total number of sherds from each site, but rather of the total number of sherds in
each decoration category (incised and unincised). This is not the way we would want
to calculate proportions in this instance. The highest proportion for incised sherds,
for example, in Table
6.4c is for the Oak Grove site, but this is at best uninteresting,
and at worst misleading. The highest proportion for unincised sherds is also at the
Oak Grove site. These high proportions simply reflect the fact that we made a larger
collection at the Oak Grove site than at any other (59 sherds, as opposed to 37
and 44 at the other sites, as seen in Table
6.4a). As a consequence we came home
with more incised sherds and more unincised sherds than from either of the other
sites. The prevalence of incised decoration in the assemblage at the Maple Knoll
site (reflected meaningfully in the column proportions) is completely obscured in
the row proportions because of the probably entirely meaningless circumstance that
the collection from Oak Grove consists of more sherds.
We could, of course, have set the two-way table up the other way in the first
place, with rows corresponding to sites and columns to decoration categories instead
of columns corresponding to sites and rows to decoration categories. If we had done
that, then we would have wanted row proportions, not column proportions. Whether
to choose row proportions or column proportions for a particular table seems
intuitively obvious to many people, but not to everyone. Occasionally there is a
situation in which either row proportions or column proportions might be meaning-
ful, depending on which point needs to be made. In the prototypical archaeological