# PowerPoint Presentation - Parameters and Statistics

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```					Why Anthropological
Statistics?

Increasing reliance on quantification in
much anthropological research
Ability to successfully complete an
independent research project
Ability to critically analyze the published
literature of our field
Parameters and Statistics

“A parameter is any quantitative measure
which seems to characterize a population”
(Thomas, 1986:35).
“A statistic is any quantitative measure
which seems to characterize a sample”
(ibid.).
note that this “statistic” is different than
“statistics” in the normal sense.
Anthropological Data

Anthropological data are “counts,
people, objects, & things…there are no
data until an anthropologist generates
them. Data do not passively exist. Data
must be generated” (Thomas, 1986: 7).
Constants,Variables, &
Symbols

A constant (represented by a symbol) is a
quantity that can assume only one value.
A variable (represented by a symbol) is a
quantity that can assume more than one
value.
A variate is an individual measurement of a
variable.
Cranial Capacity (X) of
Neanderthals

Specimen        Symbol          Variate
La Ferrasie     X1              1641cc
Spy 1           X2              1525cc
Spy 2           X3              1425cc
Gibraltar       X4              1300cc

• The variates (Xi) are the real data (i.e., observations,
counts, measurements) of anthropology, while the
variable (X) is an abstraction.
Kinds of Variables
Discrete variables can assume only fixed,
predetermined values.
Continuous variables can assume any
interval of measurement, determined
solely by the precision of the
measurement apparatus. These
measurements are only approximations of
the true value.
All variables must be clearly “operationally
defined” to insure repeatability or objectivity.
Kinds of Variables (2)
Dependent Variables are said to be dependent
upon some other variable, often called the
Independent Variable
Independent Variables are often used as
“grouping” variables in statistical analyses, while
it is the Dependent Variables that the researcher
is interested in studying.
Age and blood pressure
Sex and longevity
Femur length and locomotor behavior
Raw material and tool type
Levels of Measurement
Nominal Scale
simple, unranked categorical scale using
exhaustive and mutually exclusive
categories.
Ordinal Scale
ranked scale or ordered pattern of discrete
categories in a meaningful sequence
exhaustive, mutually exclusive, and
asymmetrical categories.
no indication of the magnitude of
difference.
Levels of Measurement
Interval Scale
possesses all the properties of an interval
scale but also equal distances between
groups.
zero point is arbitrarily set.
Ratio Scale
possesses all the attributes of interval
scale but also has a true, fixed zero point.
allows a wide range of mathematical
operations.
Populations and Samples
“A statistical population is a set of variates
(counts, measurements, or
inquiries are to be made” (Thomas,
1986:35).
arbitrary, but can be operationally defined.
not to be confused with a biological
population.
“A sample is any subset of a population”
(ibid.).
What are Statistics?
Descriptive statistics reduce masses of data to
manageable proportions.
measures of central tendency indicate where and
how the data cluster (mean, mode, median)
measures of dispersion summarize the variation
(variance, standard deviation, coefficient of
variation)
Inferential statistics involve the “process of
reasoning from a sample statistic to a
population parameter using the principles of
probability” (Thomas, 1986:38).
Data Tables, Frequency
Distributions, & Histograms
Tables are used for summarizing and
presenting data in understandable format
Frequency distributions are used for
seeing trends and distributions within a
dataset
create class boundaries and midpoints
calculate frequencies & cumulative
frequencies
use histograms for graphic presentation
Graphs

Bar graphs
used when one of the variables is nominal
Broken-line graphs (line graphs)
both variables must surpass the nominal
level
Circle graphs (pie charts)
compare parts of a whole
Ogives (cumulative curves)
Bar graph with error bars

60
% Blood
Type A          40

Error bars are plus
and minus one         20
standard deviation

Africa     Asia       Europe

Remember to always label the axes and define the error bars
Pie Chart

Average
subdisciplinary
Sociocultural
proportions in
American                              50%
Anthropology             Biological
Departments               20%

Archaeology
30%
Used to distinguish
the parts of a whole
Line Graph

200k                                             Depopulation of
California
Indians
100k

50k

1700   1750   1800   1850   1900   1950     1990
Cumulative frequency
graph

% epiphyses
complete

Age
Measures of Central
Tendency

Mean: expresses the average or “center
of gravity” of a set of variates.
mean is summation of the variates (Xi)
divided by the sample size (n)
most efficient and most stable measure of
central tendency
can be badly skewed by extreme variates
know the symbols for sample & population
mean
Measures of Central
Tendency

Median: the middle term of a set of
variates
exactly the middle term if n is odd
mean of 2 middle terms if n is even
less informative & less stable than mean
less likely to be skewed by extreme variates
Measures of Central
Tendency

Mode: the most common variate
uninfluenced by extreme variates
not unique (e.g., bimodal distribution)
lack of computational potential
more useful to a gambler than the other two
measures of central tendency, since it is the
most likely outcome of a single trial
Measures of Dispersion
Range: difference between highest and
lowest variates
Variance: sum of squared deviations
from mean, divided by sample size
note that units are squared
Standard deviation: square root of
variance
Know the symbols and formulas for
sample & population variance &
standard deviation
Coefficient of Variation

When comparing dispersion between two
samples with very different means, the CV is
more useful than the standard deviation
Standard Deviation is an absolute measure of
dispersion (or homogeneity) and thus is a direct
function of mean
CV = 100*S/mean and is a relative (rather than
an absolute) measure of dispersion
unlike the standard deviation, the CV is
independent of the mean
CV or SD ?

 Which population is more variable?
Mean1 = 10, SD1 = 2, N1 = 1000
Mean2 = 1000, SD2 = 100, N2 = 1000

 Compute CV to answer this question
CV1 = (100 x 2)/10 = 20
CV2 = (100 x 100)/1000 = 10

 While pop 2 has a SD 50 times as large as pop 1 in
absolute terms, pop 1 is twice as variable as pop 2,
as measured by CV.
Moral: don’t be misled by absolute size of standard deviation
Normal Distributions

 An infinity of normal
distributions can exist, each
one described by its mean
and standard deviation
 If we know the height of
the curve (Yi) for any
variate (Xi), we can
calculate the probability
associated with that variate
More properly, we calculate
the probability to the left or
right of any variate by the
area under the curve
Distributions

Mean +/- 1 sd contains 68.26% of all
variates
Mean +/- 2 sd contains 95.44%
95% confidence interval
Mean +/- 3 sd contains 99.74%
99% confidence interval
Standard Normal Distribution

50%                    50%
below                  above
mean                   mean
2.5%                   2.5%
below                  above
-2 SD                  2 SD

34%                    34%
between                between
mean and               mean and
-1 SD                  +1 SD
Normal Distributions and
z-scores

A standardized normal distribution allows us to
use a single table of values for areas under the
normal curve associated with all values of Xi
Mean of zero and standard deviation of one
z-score is a “standardized normal deviate”
z = (Xi - mean)/standard deviation
Ex. Calculate z score for an Xi of 20 when mean
is 35 and standard deviation is 5.
z = (20-35)/5 = -3
Consult statistical table to determine probability
associated with this z-score (area left of the curve)
Standard Normal
Distribution

Probability of Xi of 20 or
less in this distribution is
equal to the area under
the curve to the left of
z = -3
Statistical table of z-
scores tells us the
probability is .0013

50% of all variates fall within mean and
z = +/- 0.674
95% of all variates fall within mean and
z = +/- 1.960
99% of all variates fall within mean and
z = +/- 2.575
Anthropological Problems

What is the likelihood that a new skull with
cranial capacity of 657 cc belongs to a taxon
with mean of 521 and sd of 67?
If average student at field school can excavate a
10cm level in a 1 m square test unit in 1.3 hours
with sd of 15 minutes, what is the probability
that a random student can excavate the unit in:
Longer than 1 hour?
At least 2 hours?
Twice as fast as the average?
Steps Towards Solving
These Problems

Draw and label the normal distribution in
the problem (mean=521, sd - 67)
Calculate the z-score (Xi = 657)
Check the statistical table to determine
the probability associated with the
calculated z-score
Sampling Error

Error involved in inferring population
mean based on calculated sample mean
Two determinants of sampling error are
Sample size
Population variability
Sampling Error is measured by Standard
Error of the Mean* (SEM)
SEM = SD/√n

*Sanocki refers to it as “Standard Error” (SE)
Sampling Distribution of
the Mean

Repeated sampling (with replacement) of
a population and repeated calculation of
sample means of a given n yields a
“sampling distribution of the mean”
The resulting distribution is normal
Mean of this distribution is equal to the
population mean
Standard deviation of this distribution is equal
to the SEM
Hypothesis Testing

A critical part of inferential statistics
Begin with Null Hypothesis (Ho)
Hypothesis of no effect of IV/no difference between
samples or populations
Alternative to null is Research Hypothesis
Hypothesis of significant difference between groups,
or significant effect of IV
Goal of analysis is to “falsify the Ho”, leaving
one with the Research Hypothesis
Hypothesis Testing

Can the differences between groups be
explained by…
Chance or error (accept Ho)
Real effect of the IV (reject Ho)
This is essentially a problem in probability
How likely is it that group differences are due to
chance (or due to effect of IV)?
Statistics allows us to answer these questions in
an objective and repeatable manner
Graphical Look at
Hypothesis Testing

Control Group            Experimental Group

Xi X
i X
i        Xi

-2    -1 0    1      2   The same situation occurs
when comparing a single
So, when do we reject and       variate (Xi) to a group.
when do we accept the Ho?
Region of Rejection

“unlikely region”                 “unlikely region”

“likely region”
Arbitrarily set at 95% confidence interval,
beyond 2 standard deviations, p = 0.05
If difference between means falls within 95%
confidence interval, we accept Ho: if it falls
outside the 95% CI, we reject Ho.
What if We’re Wrong?

Type 1 error
Reject a True Ho
Type 1 errors occur because in 5% of the
cases, the difference really will fall in the
region of rejection, beyond 2 SD (z=1.96)
Type 2 error
Accept a False Ho
Type 2 errors occur when the IV effect is too
small for our analysis to catch.
Statistical Decision Box

IV has no effect   IV has effect

Accept Ho     Correct          Type 2 Error
Decision

Correct
Reject Ho   Type 1 Error         Decision
Significance of Type 2 Errors

We can never definitively accept Ho, but we can
definitively reject it (at an agreed level of
probability, p=.05)
We either reject or fail to reject the Ho
Power of a statistical test is the ability to detect
an IV effect, if present, or
The ability to avoid a Type 2 error
Statistical Power can be calculated
Increased sample sizes can increase Power
Probability of Type 1 and 2 errors are inversely
proportional to each other
Student’s t-Test

For comparison of means of 2
independent samples
Did IV have a significant effect?
Conceptually, t-score is IV effect/Error
Error is represented by standard error of the
differences between means (SEdiff)
t-score standardizes the IV effect by
determining how many SEdiffs it is from zero
t value is same as number of SEdiffs
Testing for Significance

95% confidence interval is standard for
determining region of rejection
p = .05
Roughly 2 SEdiffs from zero in 2-tailed test
Find t-critical in Table
t-distribution varies based on degrees of freedom
DF = n1 + n2 - 2
Compare tcritical to tobtained
Reject Ho if tobtained is larger than tcritical
One or Two Tailed Tests

Only use 1-tailed t-test when there is
strong a priori reason to expect a
directional result
tcritical is smaller for 1-tailed test and larger for
2-tailed, so
2-tailed test is therefore more conservative
or harder to falsify
Default should always be 2-tailed test
t-Tests for Paired or
Dependent Samples

Sometimes you want to compare pairs of
individuals, for example
Husbands and wives
Same individuals before and after some treatment
t-tests can be run, using a slightly different
formula, for paired samples
The SEdiff is based on the mean difference
between each paired variate
Conceptually, it is exactly the same as for
independent samples
Assumptions of the t-test

Variable must be measured on interval or
ratio scale
Variables must exhibit independent errors
Sample variates are randomly selected
from a normally distributed population
In 2 sample comparison, the 2 parent
populations have homogenous variances
ANOVA and the F-statistic

Analysis of Variance expands on t-test
Allows comparisons of more than two groups
Allows experimental designs with more than one
independent variable
Allows experimental designs with more than one
level per independent variable
Conceptually, F-statistic is very similar to t
It is also based on ratio of IV effect/error
More formally, F = MSbetween/MSwithin
ANOVA and the F-statistic

F distribution is not normal (skewed to right)
Region of rejection is in right tail
Shape of F distribution varies based on two DF
measures
DFIV = number of levels-1
Dferror = number scores - number of levels
Both DF measures are needed to determine Fcritical
ANOVA: Factorial Design

Factorial designs involve two or more IV’s
“main effects” of the 2 or more IV’s
“interaction effect” also occurs in which the effect of
one IV differs based on the level of the other
Four different DF measures occur with 2 IV’s
DFA = A-1 (A = number of levels in IV A)
DFB = B-1 (B = number of levels in IV B)
DFAxB = (A-1)(B-1)
DFerror = N - AB (N is total number of scores)
DFtotal - N - 1

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 views: 12 posted: 9/15/2012 language: English pages: 48
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