1. Introduction to Statistics: Probability and Types of Analysis
Probability and parametric / non-parametric analyses

2. Glossary – a reminder….. Population – all members of a group
Sample – the subset that we study or measure
Random sample – data is sampled randomly
Variables – Factors that could influence the results
Treatments – What you are testing
Replication – repetition of the treatments / sampling
Levels – the levels at which the treatment is applied
Control – zero treatment; allows for comparisons
Statistic – value calculated for a specific statistical test

3. Probability - Rolling a Dice
The chance of something happening
Six possible outcomes, so the probability of rolling a 3?
And the probability of rolling an even number?
All of these probabilities can be written as a decimal (0 to 1)
Which leads us to p-values

4. The p-value
The probability that the observed result is consistent with the null hypothesis (i.e. no difference, association etc).
If the probability is below a predetermined level (usually 0.05) then reject the null hypothesis and accept the alternative hypothesis.
p-value of 0.05
Only 5% probability (p=0.05) that our results are due to chance.
Therefore a 95% probability that the difference we see is due to our experiment and not chance.

5. Why the magical p=0.05? It is a rather arbitrary value.
A standard value.
Any result with a p-value < 0.05 is said to be significant.
Are these therefore significant?
P=0.02
P=0.055
P=0.50

6. Critical Significance Values
P-value
Interpretation
Symbol
P>0.05
No evidence against null hypothesis
(No significant difference)
N.S.
P<0.05
Moderate evidence against null hypothesis (significant) *
P<0.01
Strong evidence against null hypothesis (very significant) **
P<0.001
Very strong evidence against null hypothesis (highly significant)
***

7. The Normal Distribution
The spread of the data is symmetrical.
Data is more concentrated in the middle than in the tails on either side.
Typical bell shaped curve.

8. Confidence Limits
How well do values computed from the sample reflect the trends in the entire population?
Use Confidence Limits – they are applied to frequency distributions either side (+/-) of the mean, and are based around the standard deviation estimations:
ONE standard deviation measure = 68 %;
TWO standard deviation measures = 95 %;
THREE standard deviation measures = 99 %.

9. 1 standard deviation from the mean
A normal distribution curve showing the proportion of values (%) which fall within a given distance from the mean
Mean
2 standard deviations from the mean
3 standard deviations from the mean
Freq.
34 %
34 %
13.5 %
13.5 %
2.25 %
2.25 %
Values

10. An example: The Galapagos Giant Tortoise!
Example: The mean life span for a Galapagos giant land tortoise is 100 years, and the standard deviation is 30.
There is a 34 % probability that other tortoises of the same species will be 70 to 100 years and 34 % probability they will be 100 to 130 years (68%, ).
There is a 95 % probability that other tortoises of the same species will be more than 40 years or less than 160 years;
There is a 0.5 % probability that other tortoises of the same species will be less than 10 years or more than 190 years.

11. Standard Error of Differences Between Means – the yardstick for deciding whether things are statistically ‘significant’.
Differences of up to 1 s.e.d.m between 2 samples would arise quite frequently by chance (up to 68%).
But differences exceeding 2 s.e.d.m would occur only 5% by chance – therefore it is unlikely (this is where the critical probability of 0.05 comes from).
In other words, we could conclude that the populations on test have different means.
At this stage, differences can be accepted as statistically different.

12. Statistical Testing Various statistical tests are available
Check the data meets the assumptions required by the test
Calculate the test statistic
Determine the probability or p-value
Reject or accept your null hypothesis

13. P=0.05
P=0.01

14. One or two tailed??? One-tailed test Two-tailed test
If we can predict an If we just predict some
increase or a decrease kind of change
If test statistic lies in the ‘red zone’ we reject the null hypothesis

15. Types of Statistical Analysis
Descriptive statistics
Non-parametric statistics
Parametric statistics

16. Parametric Statistics
Each test has certain parameters (assumptions) that the data must meet
For example
Must follow a known distribution (usually normal)
Each dataset must have an equal variance
If the data does not meet these parameters our results may result in errors
More powerful that non-parametric methods

17. Some other distributions
Skewed to the left
Bimodal

18. Non-Parametric Statistics
No particular distribution of the data needed
‘Distribution free’
Less powerful than parametric tests
‘Safer’
Cannot answer complicated questions
e.g.: complex interactions between variable, modelling associations etc.

19. When to Use What?
Computer software such as Minitab, Genstat and to some extent MS Excel mean that we don’t have to learn the complex maths!
But it is important that we use the computer to do the right test

20. Question Test Parametric Non-parametric
Is there a relationship between variables?
Correlation test
Pearson
Spearman
Is there a difference between independent groups?
Independent measures, 2 groups
Independent-measures t-test
Mann-Whitney test
Independent measures, > 2 groups
One-way, independent-measures ANOVA
Kruskal-Wallis test
Is there a difference between dependent groups?
Repeated measures, 2 conditions
Matched-pair t-test
Wilcoxon test
Repeated measures, >2 conditions
One-way, repeated measures ANOVA
Friedman's test

21. References
Dytham C Choosing and Using Statistics: A Biologist’s Guide. Oxford: Blackwell publishing.
The best for statistical computing.
Van Emden, HF Statistics for Terrified Biologists. Malden, MA: Blackwell Publishing.
The best for understanding how it all works.
Rowntree, D. (1981) Statistics without Tears. Penguin
Excellent little book and only a pound from ebay!
Morris, T.R. (1999) Experimental Design and Analysis in Animal Sciences. CAB International.
Townsend, J. (2002) Practical Statistics for Environmental and Biological Scientists. Wiley.
Zar, J.H. (1984) Biostatistical Analysis. 2nd Edition. Prentice Hall.
Statistics books in the LRC at 519.5
Minitab guidebooks at

22. Some useful websites

23. Are you able to …
Understand what a p-value is and describe what probability is?
Explore parametric and non-parametric analyses?
Understand why parametric and non-parametric analyses are required?
Review examples of parametric analyses and their non-parametric equivalent?

24. Testing for Normality on Minitab
Stat – Basic Statistics – Normality test
If p < 0.05 then the data is not normally distributed. This one is!