1. Introduction Osborn

2. Daubert is a benchmark!!!: Daubert (1993)- Judges are the “gatekeepers” of scientific evidence. Must determine if the science is reliable Has empirical testing been done? Falsifiability Has the science been subject to peer review? Are there known error rates? Is there general acceptance? Frye Standard (1928) essentially Federal Government and 26(-ish) States are “Daubert States” “Legal” Science

3. Any time an observation is made, one is making a “measurement” Measurement and Randomness 1.Experimental error is inherent in every measurement Refers to variation in observations between repetitions of the same experiment. It is unavoidable and many sources contribute 2.Error in a statistical context is a technical term BHH

4. Experimental error is a form of randomness Randomness: inherent unpredictability in a process The the outcomes of the process follow a probability distribution Measurement and Randomness Statistical tools are used to both: Describe the randomness Make inferences taking into account the randomness Careful!: Bad data, assumptions and models lead to garbage (GIGO)

5. Frequency: ratio of the number of observations of interest (n i ) to the total number of observations (N) Probability (frequentist): frequency of observation i in the limit of a very large number of observations We will almost always use this definition It is EMPIRICAL! Probability

6. Belief: A “Bayesian’s” interpretation of probability. An observation (outcome, event) is a “measure of the state of knowlege” Jaynes. Bayesian-probabilities reflect degree of belief and can be assigned to any statement Beliefs (probabilities) can be updated in light of new evidence (data) via Bayes theorem. Probability

7. Study of relationships in data Descriptive Statistics – techniques to summarize data E.g. mean, median, mode, range, standard deviation, stem and leaf plots, histograms, box and whiskers plots, etc. Inferential Statistics – techniques to draw conclusions from a given data set taking into account inherent randomness E.g. confidence intervals, hypothesis testing, Bayes’ theorem, forecasting, etc. What is Statistics??

8. For the Sciences, we ask: Are the differences in measurements characterizing two (or more) objects real or just due to (the characteristic) randomness? Why do we use statistical tools? Furthermore, for the Forensic Sciences we ask: Do two pieces of evidence originate from a common source? For this, we must at least answer the above.

9. Almost all of statistics is based on a sample drawn from a population. Population: The totality of observations that might occur as a result of repeatedly performing an experiment Why not measure the whole population? Usually impossible Likely wasteful Population should be relevant. Part logic Part guess Part philosophy…. Population and Sample

10. Sampling: Sample: a few observations that are made from a population Draw members out of as population with some given probability Random sample: if all observations have an equal chance of being made and no observation affects any other Also called independent and identically distributed random sample (I.I.D.) Want a random sample to be representative of the population Biased sample if not the case* Population and Sample

11. Sample Representations: Data and Sampling Population Representative Sample Biased Samples Population Sample Population Sample

12. Types of sampling: (Simple) Random Sampling Every data item is selected independently of every other. Every member of a population has an equal chance of being selected Systematic Sampling Pick every k th data item to be in the sample Easier to conduct but risk getting a biased sample Stratified Sampling Partition population into disjoint groups containing specific attributes of a particular category (strata) Random sample from the groups Data and Sampling

13. Parameter: any function of the population Statistic: any function of a sample from the population Statistics are used to estimate population parameters Statistics can be biased or unbiased Sample average is an unbiased estimator for population mean We may construct distributions for statistics Populations have distributions for observations Samples have distributions for observations and statistics Parameters and Statistics

14. Univariate Statistics: Statistical tools used to analyze one random variable Univariate vs. Multivariate Statistics Random variable could be raw observation or a statistic Common tools are: (univariate) hypothesis testing, ANOVA, linear regression Multivariate Statistics: Statistical tools used to analyze many random variables Random variables can also be raw observations (often encountered in chemometrics) or statistics (currently popular in marketing, finance, surface metrology)

15. Don’t if you can see clear differences/similarities in your data and can clearly articulate how in court! If you can’t differentiate or want to study/search for differences within a well defined population AND univariate methods don’t do the trick: Why Use Multivariate Statistics? A linear (or non-linear) combination of many experimental variables (multivariate) may do the trick!