1. Understanding Your Data Set Statistics are used to describe data sets Gives us a metric in place of a graph What are some types of statistics used to describe data sets? –Average, range, variance, standard deviation, coefficient of variation, standard error

2. Length NumberPondLake 13438 27882 34858 42476 56460 65870 73499 86640 92268 104491 Average=47.268.2 Table 1. Total length (cm) and average length of spotted gar collected from a local farm pond and from a local lake.

3. Are the two samples equal? –What about 47.2 and 47.3? If we sampled all of the gar in each water body, would the average be different? –How different? Would the lake fish average still be larger? Length NumberPondLake 13438 27882 34858 42476 56460 65870 73499 86640 92268 104491 Average=47.268.2

4. Range Simply the distance between the smallest and largest value Length (cm) Figure 1. Range of spotted gar length collected from a pond and a lake. The dashed line represents the overlap in range.

5. Length (cm) Does the difference in average length (47.2 vs. 68.2) seem to be much as large as before?

6. Variance An index of variability used to describe the dispersion among the measures of a population sample. Need the distance between each sample point and the sample mean.

7. Figure 2. Mean length (cm) of each spotted gar collected from the pond. The horizontal solid line represents the sample mean length.

8. We can easily put this new data set into a spreadsheet table. By adding up all of the differences, we can get a number that is a reflection of how scattered the data points are. –Closer to the mean each number is, the smaller the total difference. After adding up all of the differences, we get zero. –This is true of all calculations like this What can we do to get rid of the negative values? #LengthMeanDifference 13447.2 -13.2 27847.2 30.8 34847.2 0.8 42447.2 -23.2 56447.2 16.8 65847.2 10.8 73447.2 -13.2 86647.2 18.8 92247.2 -25.2 104447.2 -3.2 Sum =0

9. Sum of Squares #LengthMeanDifferenceDifference 2 13447.2 -13.2174.24 27847.2 30.8948.64 34847.2 0.80.64 42447.2 -23.2538.24 56447.2 16.8282.24 65847.2 10.8116.64 73447.2 -13.2174.24 86647.2 18.8353.44 92247.2 -25.2635.04 104447.2 -3.210.24 Sum = 0 3233.6 Now 3233.6 is a number we can use! This value is called the SUM OF SQUARES.

10. Back to Variance Sum of Squares (SOS) will continue to increase as we increase our sample size. –A sample of 10 replicates that are highly variable would have a higher SOS than a sample of 100 replicates that are not highly variable. To account for sample size, we need to divide SOS by the number of samples minus one (n-1). –We’ll get to the reason (n-1) instead of n later

11. Calculate Variance (σ 2 ) σ 2 = S 2 =  (X i – X m ) 2 / (n – 1) SOS Degrees of Freedom Variance for Pond = S 2 = 3233.6 / 9 = 359.29

12. More on Variance Variance tends to increase as the sample mean increases –For our sample, the largest difference between any point and the mean was 30.8 cm. Imagine measuring a plot of cypress trees. How large of a difference would you expect (if measured in cm)? The variance for the lake sample = 400.18.

13. Standard Deviation Calculated as the square root of the variance. –Variance is not a linear distance (we had to square it). Think about the difference in shape of a meter stick versus a square meter. By taking the square root of the variance, we return our index of variability to something that can be placed on a number line.

14. Calculate SD For our gar sample, the Variance was 359.29. The square root of 359.29 = 18.95. –Reported with the mean as: 47.2 ± 18.95 (mean ± SD). Standard Deviation is often abbreviated as σ (sigma) or as SD. SD is a unit of measurement that describes the scatter of our data set. –Also increases with the mean

15. Standard Error Calculated as: SE = σ / √(n) –Indicates how close we are to estimating the true population mean –For our pond ex: SE = 18.95 / √10 = 5.993 –Reported with the mean as 47.2 ± 5.993 (mean ± SE). –Based on the formula, the SE decreases as sample size increases. Why is this not a mathematical artifact, but a true reflection of the population we are studying?

16. Normal Distribution Most characteristics follow a normal distribution –For example: height, length, speed, etc. One of the assumptions of the ANOVA test is that the sample data is ‘normally distributed.’

17. Sample Distribution Approaches Normal Distribution With Sample Size

20. Sample Size The number of individuals within a population you measure/observe. –Usually impossible to measure the entire population As sample size increases, we get closer to the true population mean. –Remember, when we take a sample we assume it is representative of the population.

21. Effect of Increasing Sample Size I measured the length of 100 gar Calculated SD and SE for the first 10, then included the next additional 10, and so on until all 100 individuals were included.

22. Sample Size

23. SD = Square root of the variance (Var =  (X i – X m ) / (n – 1))

24. Sample Size SE = SD / √(n)

26. MEAN ± CONFIDENCE INTERVAL When a population is sampled, a mean value is determined and serves as the point-estimate for that population. However, we cannot expect our estimate to be the exact mean value for the population. Instead of relying on a single point-estimate, we estimate a range of values, centered around the point-estimate, that probably includes the true population mean. That range of values is called the confidence interval.

27. Confidence Interval Confidence Interval: consists of two numbers (high and low) computed from a sample that identifies the range for an interval estimate of a parameter. There is a 5% chance (95% confidence interval) that our interval does not include the true population mean. y ± (t  /0.05 )[(  ) / (  n)] 28.17 ± 2.29 25.88    30.45 (use 1.96) (SE)