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Statistics
Frequency and Distribution
Distribution Functions
1. Normal Distribution
2. Uniform Distribution
3. Skewed Distribution
Parametric statistics (continuous variables)
assume normally distributed data
Normal Distributions
• Many natural populations
are normal distributions –
tree heights in a mature
forest, weights of acorns
from the same tree, etc.
• Many non-normal
distributions can be
converted into normal
distributions via either
square root or logarithm
transformations
Normally Distributed Populations
can be mathematically described with two
parameters: a measure of central tendency, or
mu (the mean symbolized as µ), and a measure
of dispersion, or sigma (the standard deviation
symbolized as σ).
Normal Distribution
Booooo!!!
Normal Distribution - Higher Variance
Sampling from a normal distribution
Repeated
samples are
normally
distributed and
centered around
the parent
population mean
but have their
own deviations
based on the
number and type
of samples
Central Limit Theorem
states that the distribution of the means of a
sufficiently large number of independent samples
will be approximately normal, regardless of the
underlying distribution.
Uniform Distributions
• Forestry example – ages of pole sized timber.
CLT acting on a Uniform Distribution
Skewed Distributions
• Often observed in nature – i.e. tree ages
• Inverse J shaped curve
• Median is a better index of central tendency
than the Mean
Skewed Descriptive Values
Sample Size affects estimates
• Larger samples sizes
cause sample
parameters to more
closely estimate the
parameters of the
underlying population
• Standard error captures
this effect – decreasing
as sample size increases
Sample Size revisited
• Since t is very close to 2 for 95% confidence at
infinite sample size we will use it.
• E is the desired sampling error, we will use 10%
Student t by confidence level