1. Data Analysis

2. Basic Problem
There is a population whose properties we are interested in and wish to quantify statistically: mean, standard deviation, distribution, etc.
The Question – Given a sample, what was the random system that generated its statistics?

3. Central Limit Theorem
If one takes random samples of size n from a population of mean m and standard deviation s, then as n gets large, approaches the normal distribution with mean m and standard deviation
s is generally unknown and often replaced by the sample standard deviation s resulting in , which is termed the Standard Error of the sample.

4. Example

5. Normal distribution

6. Critical Values for Confidence Levels
.80
.90
.95
.99 a
0.20
0.10
0.05
0.01
a/2
.05
.025
0.005
za/2
1.28
1.64
1.96
2.58

7. Confidence Interval for Mean (large sample size)

8. Student’s t-distribution

9. Critical Values for Confidence Levels t-distribution
.80
.90
.95
.99 a
0.20
0.10
0.05
0.01
a/2
.05
.025
0.005
d.f. = 1
3.09
6.31
12.71
63.66
d.f. = 10
1.37
1.81
2.23
4.14
d.f. = 30
1.31
1.70
2.04
2.75
d.f. = ¥
1.28
1.65
1.96
2.58

10. Confidence Interval for Mean (small sample size, t-distribution)

11. Comparing Population Means
Unequal Variance
Pooled Variance

12. Hypothesis Testing (t-test)
Null Hypothesis – differences in two samples occurred purely by chance
t statistic = (estimated difference)/SE
Test returns a “p” value that represents the likelihood that two samples were derived from populations with the same distributions
Samples may be either independent or paired

13. Tails
One tailed test – hypothesis is that one sample is: less than, greater than, taller than,
Two tailed test – hypothesis is that one sample is different (either higher or lower) than the other

14. Paired Test Samples are not independent
Much more robust test to determine differences since all other variables are controlled
Analysis is performed on the differences of the paired values
Equivalent to Confidence interval for the mean

15. Paired Samples – New Site

16. TSS Concentrations vs. Time

17. BMP Performance Comparison
Commonly expressed as a % reduction in concentration or load
Highly dependent on influent concentration
Potentially ignores reduction in volume (load)
May lead to very large differences in pollutant reduction estimates
Preferable to compare discharge concentrations

18. Effect of TSS Influent Concentration

19. Sand Filter - TSS

20. Comparison of Effluent Quality

21. Exercise
Calculate average concentrations for each constituent for the two watersheds
Determine whether any concentrations are significantly different, report p value for null hypothesis
Calculate average effluent concentrations for the two BMPs and determine whether they are different from the influent concentrations – p values
Compare effluent concentrations for the two BMPs and determine whether one BMP is better than the other for a particular constituent.