Statistical Concepts Basic Principles

1. Statistical Concepts Basic Principles An Overview of Today’s Class What: Inductive inference on characterizing a population Why : How will doing this allow us to better inventory and monitor natural resources Examples Relevant Readings: Elzinga pp. 77-85 , White et al. Key points to get out of today’s lecture: Description of a population based on sampling Understanding the concept of variation and uncertainty

2. By the end of today’s lecture/readings you should understand and be able to define the following terms: Accuracy/Bias Precision Population parameters Sample statistics Standard Error Confidence Interval Mean Variance / Standard Deviation

3. Steps in Conducting an Assessment using Inventory and Monitoring • Develop Problem Statement—may include goals • Develop specific objectives • Determine important data to collect • Determine how to collect and analyze data • principles of statistics allows us to better plan how to collect the data • AND analyze it - they work in tandem • Collect data • Analyze data • Assess data in context of objectives

4. The Relation between Sampling and Statistics Can you make perfect generalizations from a sample to the population? There is uncertainty in inductive inference. The field of statistics provides techniques for making inductive inference AND for providing means of assessing uncertainty.

5. Why sample? Inductive inference: “…process of generalizing to the population from the sample..” Elzinga –p. 76 Target/Statistical Population Sample Unit Individual objects (in this case, plants) Elzinga et al. (2001:76)

6. We are interested in describing this population: • its total population size • mean density/quadrat • variation among plots At any point in time, these measures are fixed and a true value exists. These descriptive measures are called ? Population Parameters The estimates of these parameters obtained through sampling are called ? Sample Statistics

7. We are interested in describing this population: • its total population size • mean density/quadrat • variation among plots How did we obtain the sample statistics?

8. No Way! ALL sample statistics are calculated through an estimator “An estimator is a mathematical expression that indicates how to calculate an estimate of a parameter from the sample data.” White et al. (1982)

9. Is A sample statistic or population parameter ? is a sample statistic that estimates the population mean = population mean if all n units in the population are sampled You do this all the time! The Mean (average): What is the formal estimator you use? (standard expression, but often denoted by a some other character) Which states to do what operations?

10. Estimating the amount of variability Why? Recall: There is uncertainty in inductive inference. The field ofstatistics provides techniques for making inductive inference AND for providing means of assessing uncertainty. • Two key reasons for estimating variability: • a key characteristic of a population • allows for the estimation of uncertainty of a sample

11. Recall wedn lab: Each group collected data from 5 4m2 plots Did each group get identical results? Think about this conceptually, before mathematically: What characteristic of the population would affect the level of similarity among each groups’ samples? How about sampling method?

12. Estimating the Amount of Variation within a Population The true population standard deviation is a measure of how similar each individual observation (e.g., number of plants in a quadrat) is to the true mean

13. Populations with lots of variability will have a large standard deviation, whereas those with little variation will have a low value High or low? Counts of dock from wedn lab? What would the standard deviation be if there were absolutely no variability- that is, every quadrat in the population had exactly the same number ?

14. First, we calculate the population variance The Computation of the Standard Deviation • key is to get differences among observations, right? • then each difference is subtracted from the mean– • consistent with definition Does this make sense ? For the pop Std Dev, we take the SQRT of the Var

15. The Computation of the Standard Deviation The estimator of the variance – that is what produces the sample statistic, simply replaces N with the actual samples (n), and the true population mean with the sample mean The estimator of the standard dev is simply the SQRT of the var. Because of an expected small sample bias, n-1 is usually used rather than n as the divisor in both the var and stdev

16. Where Are We? We have computed a mean value of a population and a sample We have computed the variability of a population and a sample We now can use the variability of the sample to tell us something about uncertainty and the way we sampled to tell us something accuracy.

17. Bias vs Precision Bias (accuracy): Precision: Essentially, the “closeness” of a measured value to its true value; the average performance of an estimator The “closeness” of repeated measurements of the same quantity; the repeatability of a result.

19. The level of bias is a function of your sampling scheme and • estimator used. Your are in control of this! • Precision is a function of the variance of the population, and • How you sample: • Number of samples • Variability within samples (so quadrat SIZE and SHAPE • matters) compared to among samples • analytical techniques

20. Why does Bias and Precision matter in inventory and monitoring of natural resources? Lets imagine monitoring the density of dock in Ron’s pasture through time

21. The effect of sampling variation: a function of precision All estimates come from the same population

22. So how “good” are your parameter estimates? Lets examine this with the estimation of the population mean What influences the reliability of the estimate of the mean value?

23. Estimating the Reliability of a Sample Mean Standard error: the standard deviation of independent sample means Measures precision from a single sample (e.g., from a collection of quadrats) Quantified the certainty with which the mean computed from a random sample estimates the true population mean

24. Estimating the Reliability of a Sample Mean Formally, the SE is a function of the standard deviation of the sample and the number of samples SE=s/SQRT(n) Does this make sense?

25. Consider this example:

26. Communicating the Reliability of a Sample Mean Confidence Intervals Provides an estimate of precision around a sample mean or other estimated parameter Includes two components: confidence interval width confidence level: the probability that the interval includes the true value What’s the relation between the two?

27. Communicating the Reliability of a Sample Mean Estimating the Confidence Interval 95% CI = Mean +/- 1.96(SE) Intervals can be computed for any level of confidence desired in a particular study

28. How was this computed? The interpretation of this chart (p. 76) should now ( or soon!) be clear

29. Key points to get out of today’s lecture: Description of a population based on sampling Understanding the concept of variation and uncertainty Ability to define (and understand) the following terms: Accuracy/Bias Precision Population parameters Sample statistics Standard Error Confidence Interval Mean Variance / Standard Deviation