1. Why the Normal Distribution is Important
Calculating probabilities for events
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
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)

2. Normal Distribution
Many measurements we make on anthropological sample units (such as…) produce distributions similar to the theoretical normal distribution.
Therefore, we can often use the normal distribution to calculate probabilities for
empirically derived distributions
Single variates in distributi0ns
Variate ranges in distributions
Why is the world this way?
Random variation
Central Limit Theorem

4. Properties of the theoretical normal distribution
Symmetry
Highest point is the mean (and…)
Area under the curve sums to one.
Distribution is asymptotic at either end
Distribution of means from multiple randomly drawn samples will also be normal.

5. Calculating probabilities with the normal distribution
Different normal distributions have different means (µ) and standard deviations (σ)
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
These and other variate ranges easily translated into probabilities
x = 7.14, sd = 2.30, so, e.g., probability of measuring a sherd greater than 11.4 mm is 4.6%

6. Comparing Probabilities from Different Normal Distributions
If we have normal distributions of different µ and σ can’t meaningfully compare the probability of single variate across them.
We need some standardized normal distribution to make these comparisons.
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

7. Putting Z-scores to Work
Using data on pocket gopher mandibles, µ = 5.7 mm and σ = mm
With mandible of length 6.4 mm = z- score of 1.46
What area to calculate if we want to know:
Probability of mandible smaller than mm?
Probability of mandible bigger than mm?
Using a table of z-scores