1. Measurement, Quantification and Analysis
Some Basic Principles
And
Some Basic Statistics

2. Three Major Issues
Biological and especially ecological data show high variability in quantitative traits
We almost never measure everything in field research; rather we sample from larger populations or data sets
Sampling leads to uncertainty about conclusions, so we always must estimate our uncertainty
Items 1) – 3) above are why we use Statistics!

3. Variability

4. All natural processes are variable, Whether continuous or discreet
Continuous data
All natural processes are variable,
Whether continuous or discreet
Discreet data
Plus, better sampling effort better describes distributions

5. In many processes, we observe characteristic distributions
37.5% A a AA Aa aa
(p + q)4 where p = q = 0.5
When (p + q) is raised to the 20th power, binomial is indistinguishable from the normal.
Binomial – Few interacting factors
Normal – Many interacting factors
2 factors: One way to get AA or aa, 2 ways to get Aa (p + q)2
4 factors: One way to have AAAA or aaaa, 4 ways to get AAAa or aaaA, and 6 ways to get AAaa

6. Major point: Biological processes are variable, but they often tend to vary in predictable of consistent patters.

7. Sampling and Estimation

8. To calculate the average in a sample:
A characteristic of field biology is the attempt to estimate parameters from highly variable populations of uncertain “true” value.
To calculate the average in a sample:
Mean = Sum of all observations/number of observation
To estimate the variability of the observations:
Variance = Sum of (individual observation – Mean of observations)2
_____________________________________________
Number of Individual Observations - 1 -1
Or to express this in the same units as the Mean:
Standard deviation = Square Root of the Variance

10. All natural processes are variable, Whether continuous or discreet
What happens when we estimate means? Select 5 observations at random. Then 10. Then 25.
Probability
Better sampled populations yield better distributions
Larger sample sizes yield better estimates
Means will also be variable, and will have a characteristic distribution
If you sample N means, they will have a variance as well

12. To estimate the variability of the means:
Divide the standard deviation (the square root of the variance)
by the square root of the sample size (why? Variability of the means is dependent upon sample size – decreases with large samples.)
Recall, To estimate the variability of the observations:
Variance = Sum of (individual observation – Mean of observations)2
_____________________________________________
Number of Individual Observations – 1
Divide the square root of the variance, the standard deviation,
by the square root of the sample size. The bigger the sample size,
the less variable the means
This is the Standard Error, which is used to calculate a Confidence Interval

13. If you want, for example, a 95% confidence that a mean measured was from this theoretical distribution, you want to multiply the standard error by a constant the grabs 95% of the distribution.
Note that plus or minus 2 SE gives you 4.26% (2.13 x 2), so it needs to be a little smaller, 1.96 to be precise.

14. Uncertainty

16. Confidence intervals represent a level of confidence about the true value of the mean. It is the mean, with the expected variability (standard error) combined with a constant derived from a theoretical distribution that defines how much uncertainty is acceptable.
Lets pick a constant that circumscribes 95% of the variability. In other words, if you sample repeated with a given sample size, a 95 % CI means that in 95% of the samples you collect, you will have the value of the true mean. The mean is either in or not!
No matter how well we sample, we will “miss-estimate” the population parameter a certain percentage.
What level of error are we willing to accept?
With a 95 % limit, 5 % of the time.
In theory, the tails are limitless, so we must set a criterion.
Decision rule – 5 % error.
Minimize this with replication

17. Importance of Replication?
One sample: Wrong 5% or 1/20 of the times you sample
Two replicated samples: Wrong 1/20 x 1/20 or 1/400
Three replicated samples: Wrong 1/20 x 1/20 x 1/20 or 1/8,000

18. What confidence do we want?
What error will we accept?
One things we do frequently in science is compare things.
For example, if one population bigger than another, which population are we sampling from? A B
What kinds of errors can we make?

19. Fundamental Principles
Have clearly defined hypotheses
Measure carefully
Sample intensively – large sample sizes reduce Beta-Error
Replicate – Replication reduces Alpha-Error

20. Samples of Data Sets from Previous Projects that required Quantification and Statistical Analysis

29. Sum of Squares df
Mean Square F
Sig.
Forearm (mm)
Between Groups 2
.000
Within Groups 43
3.557
Total 45
Foot (mm)
55.693
98.161
2.283

30. Table 2: Chi-Square Tests
Table 2: Chi-Square Tests
Value df
Asymp. Sig. (2-sided)
Pearson Chi-Square
15.181a 6
.019
Likelihood Ratio
16.437
.012
N of Valid Cases
150
2 cells (16.7%) have expected count less than 5. The minimum expected count is 3.87.
Table 2: Chi-Square Tests: The table above shows the Chi-Square value and level of significance.