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

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

4. 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)

5. 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

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

7. 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)

8. 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?

9. 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

10. Recall lab: Each group collected data from 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?

11. 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—the sample unit) is to the true mean Can we develop a mathematical expression for this?

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

13. First, we calculate the population variance 2  The Computation of the Population Variance and Standard Deviation • key is to get differences among observations, right? • then each difference is subtracted from the mean– • consistent with definition = Var= Does this make sense ? For the pop Std Dev, we take the SQRT of the Variance, std==SQRT(var)

14. The Computation of the Sample Variance and 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 estimated variance. Because of an expected small sample bias, n-1 is usually used rather than n as the divisor in both the var and stdev

15. Estimating the Sample Standard Deviation Worksheet: compare the sample variation of mass of deer mice to mass of bison; which is more variable?

16. _ CV= s/X * 100 Coefficient of Variation: A measure of relative precision “The coefficient of variation is useful because, as a measure of variability, it does not depend upon the magnitude and units of measurements of the data.” Elzinga et al: 142 Usually expressed as a percent, Using the coefficient of variation, what is more variable, mass of deer mice or bison?

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

18. 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 Coefficient of variation Population parameters Sample statistics Mean Variance / Standard Deviation Friday’s class: from sampling variability to confidence intervals