1. Basic Statistics (for this class) Special thanks to Jay Pinckney (The HPLC and Statistics Guru) APOS

2. Some basic Definitions Population- the totality of individual observations about which inferences are to be made. Sample -collection of individual observations selected by a specified procedure. They are a subset of the population. – More observations are usually better than a few – Balanced designs (the same number of samples for all treatments) are preferred Variable (character) -the actual property measured by the individual observations Variance (aka Mean Square)- The mean of the squared deviations of observations from their arithmetic mean. Units are NOT the same as the original observations – AKA measures how far a set of numbers is spread out. Standard Deviation (SD)- The square root of variance. Units are the same as original observations 1 SD= 68.26%2 SD=95.46%3 SD=99.72%

3. Measurement Variables (3 Types) Categorical -observations in a limited number of categories which have no obvious scale (Diatom, dinoflagellate, cyanobacteria). Discrete - (discontinuous) a real scale, but not all values are possible. – Have only certain fixed numerical values with no intermediate values possible. (body segments, teeth, counts, number of offspring, eggs). – Discrete variables are usually integers. Continuous- any value is possible, only restricted by the measuring device (lengths, concentrations, etc.). – Includes areas, volumes, weights, angles, temperatures, time, percentages, rates, and sometimes behavior.

4. The Awesome Power of Statistics (APOS) Hypothesis Testing is the foundation for the scientific method Statistics is: the tool used to assign a level of confidence in hypothesis testing The Null Hypothesis (Ho) -The hypothesis that nothing is going on or that there is NO difference between the sets of observations – phytoplankton biomass is the same....or....pH is constant Statistics are used to test the null hypothesis....either accept or reject based on some predetermined level of confidence (usually p<.05) If you reject the null hypothesis, you must accept the alternate hypothesis (Ha) that there is a difference between the sets of observations example: phytoplankton biomass is different....or...pH is different

5. How confident are you in your decision? The P Value is the bottom line This is the probability that the null hypothesis is true A p-value of 0.05 (5% chance of the null hypothesis being true) is usually used as the critical level for the rejection of a null hypothesis. The smaller the p-value, the more confidence you can place in your conclusion. For example, a p-value of 0.001 means that there is a 1 in 1000 chance that the null hypothesis is true. What is the best p-value to use?

6. T-Test 2 Basic Types Type 1 – Group Comparisons (unpaired) t-Test Tests the null hypothesis that the two sets of data have the same mean (μ1 = μ2) – When the Data consist of 2 Groups (Unpaired Data) – There is no logical PAIR of samples – Use when you have 2 treatments and want to determine if the mean values of the two treatment groups are significantly different Example: Are boys taller than girls?

7. T-Test 2 Basic Types Type 2 – Paired Comparisons t-Test – When the data are “Paired” – Each observation for one treatment is paired with one for the other treatment (2 measurements made on the same EU) – Usually applied when an individual is tested for two different factors – Before and After Experiments (make a measurement, do something, make another after) – Also, two treatments are applied to the same EU – This test controls for variability between EU’s Examples: – Arm length in humans (left arm vs. right arm) (EU is the individual) – Two vertical profiles of pH in a sediment core (EU is the core) – Surface and bottom measurements of dissolved oxygen at the same location (EU is the location) – Collect a sample, split into two parts, apply different treatments to each half

8. Example Problem Galveston Bay is usually described as a shallow estuary (<3 m) with a well-mixed water column. An investigator conducted a preliminary study to try to confirm this statement. Starting in January 2006, the investigator went to a representative location in the bay and obtained measurements of various water quality parameters from two depths: 0.5 m (surface) and 2.5 m (bottom). These measurements were repeated at weekly/biweekly intervals for one year to gather enough data to make a general conclusion about the mixing status (mixed or stratified) of the water column at this location.

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11. Correlations Correlation and Regression are often misused terms that are usually assumed to be synonymous Correlation Concerned mostly with whether two variables are interdependent, vary together Cannot express one as a function of the other (i.e., no lines or equations) No distinction between dependent and independent variables Cannot assume or imply Causality (one variable does not cause the other, no cause and effect) More valid assumption is that the two variables are both effects of a common cause Purpose of the analysis is to estimate the degree to two variables vary together Regression Purpose is to describe the dependence of a variable (y) on an independent variable (x) Independent Variable - the variable under control of the investigator, fixed and known without error Dependent Variable - the variable that is measured (with error) Used to support hypotheses regarding possible causation of changes in y by changes in x Used to predict values of y given a value of x Used to explain variation in y due to x, using x as a statistical control Implies a Cause (x) and an Effect (y) relationship between two variables

12. Cont… Correlation analysis is used to determine the degree of association between two variables Determine whether two sets of observations are associated or correlated, the strength of the correlation, and whether it is significant or not Standard Correlation - Pearson’s Product-Moment Correlation The statistic is denoted as r (true correlation is ρ) and is termed the correlation coefficient r ranges from -1 (perfect negative correlation) to 0 (no correlation) to +1 (perfect positive correlation) The p-value indicates the significance for testing the hypothesis: H0: ρ = 0 Ha: ρ ≠ 0 Results are usually reported as “Pearson product-moment correlation analysis indicates a significant positive association between y1 and y2 (r = 0.51, n=22, p < 0.05)”

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15. Regression Analysis Used to determine a Cause and Effect relationship between two variables Useful for quantifying the form and strength of a relationship between 2 variables Data consist of a Y value (effect, dependent) and X value (cause, predictor, independent) May have multiple Y values for each value of X Least-Squares Linear Regression Analysis will supply the slope and y-intercept of a “best fit” line based on a minimization of the squared differences (least squares) Equation: Y = a + bX a = constant y-intercept b = slope

16. Other Formulas

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18. ANOVA –Analysis of Variance A Very Common parametric test used to determine if multiple means are significantly different Tests the H 0 that separate sets of data have the same mean Does this by testing if the variation within groups is the same as the variation between groups

19. Terminology Factor-The item that is being manipulated (e.g., nutrient concentration) Level-the different degrees of the factor (e.g., actual nutrient concentrations used in the experiment; control, 5 μM, 10 μM, 15 μM) Treatments-are the same as the factor levels

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