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Why the Normal Distribution is
Important
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Calculating probabilities for events
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Why do we do this?
Theoretically & empirically derived
probabilities
Probabilities for complex events
For same reasons, we want to know
the probabilities of obtaining
particular distributions
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There are two ways probabilities can be
determined for continuous distributions.
First, we can use (re)sampling from a larger
distribution to empirically determine
probabilities. This is called “ Monte Carlo
simulations ” , and is somewhat common in
archaeology. A second, more traditional
(and easier) approach is to cluster the
unique distributions into different “ types ”
based on their similarities in important
characteristics. These groups of similar
distributions can be further characterized
by an ideal (i.e., theoretical) distribution
that typifies the distributions ’ important
characteristics. Statistics for measuring
probabilities can then be developed based
upon our knowledge of the ideal
distribution and applied to the real
distributions by extension (VP&L:87)
Normal Distribution
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Many measurements we make on
anthropological sample units (such
as…) produce distributions similar to
the theoretical normal distribution.
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Therefore, we can often use the
normal distribution to calculate
probabilities for
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empirically derived distributions
Single variates in distributi0ns
Variate ranges in distributions
Why is the world this way?
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Random variation
Central Limit Theorem
Properties of the theoretical
normal distribution
1.
Symmetry
2.
Highest point is the
mean (and…)
3.
Area under the
curve sums to one.
4.
Distribution is
asymptotic at
either end
5.
Distribution of
means from
multiple randomly
drawn samples will
also be normal.
Calculating probabilities with the
normal distribution
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Different normal distributions have
different means (µ) and standard
deviations (σ)
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However, because of the five
properties of all normal distributions:
µ ± 1 σ comprises 68.26% of all variates
• µ ± 2 σ comprises 95.44% of all variates
• µ ± 3 σ comprises 99.73% of all variates
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These and other variate ranges easily
translated into probabilities
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x = 7.14, sd = 2.30, so, e.g., probability of
measuring a sherd greater than 11.4 mm
is 4.6%
Comparing Probabilities from
Different Normal Distributions
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If we have normal distributions of
different µ and σ can’t meaningfully
compare the probability of single
variate across them.
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We need some standardized normal
distribution to make these
comparisons.
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Standardize different normal
distributions by the µ and σ to create a
single distribution with µ = 0 and σ ± 1
a
From previous: Yi = 7.14, sd = 2.30, so
sherd 11.4 mm thick has z score of 1.85
Putting Z-scores to Work
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Using data on pocket gopher
mandibles, µ = 5.7 mm and σ = 0.48
mm
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With mandible of length 6.4 mm = zscore of 1.46
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What area to calculate if we want to
know:
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Probability of mandible smaller than
6.4 mm?
Probability of mandible bigger than
6.4 mm?
Using a table of z-scores