1. Frequency and Distribution
Statistics
Frequency and Distribution

2. Distribution Functions
Normal Distribution
Uniform Distribution
Skewed Distribution
Parametric statistics (continuous variables) assume normally distributed data

3. 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

4. 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).

5. Normal Distribution

6. Booooo!!!

7. Normal Distribution - Higher Variance

8. 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

9. 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.

10. Uniform Distributions
Forestry example – ages of pole sized timber.

11. CLT acting on a Uniform Distribution

12. 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

13. Skewed Descriptive Values

14. 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

15. 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%

16. Student t by confidence level