1. BAE 5333 Applied Water Resources Statistics
Biosystems and Agricultural Engineering Department
Division of Agricultural Sciences and Natural Resources
Oklahoma State University
Source
Dr. Dennis R. Helsel & Dr. Edward J. Gilroy
2006 Applied Environmental Statistics Workshop
and
Statistical Methods in Water Resources

2. TEXTBOOK
Free on-line at:

3. SOFTWARE Minitab Version 15 Statistical Software

4. Choosing a Statistical Method Depends on:
Chapter 1
SUMMARIZING DATA
Numbers and Graphs
Choosing a Statistical Method Depends on:
Data characteristics
Study objectives

5. Characterizes of Environmental Data
Lower bound of zero
Presence of outliers, high values
Positive skewness
Non-normal distribution
High variance
Data below recording limits
Data collected by other people

6. Categories of Measured Data
Continuous: 1.10, 2.56, ….
Discrete: 1, 2, 5, 15
Qualitative, Grouped, Categorical
Site 1, Site 2, Site 3
Below Detection Limit, Above Detection Limit

7. Histograms Show how many times Y occur in several groups of X.
Require grouping of a continuous variable
Y-axis: frequency or relative frequency
X - Independent Variable
Y - Dependent Variable

8. Box Plots Good for continuous data Based on percentiles
50th percentile (median)
50 percent of data below or equal to median

9. Inter-quartile Range (IQR) A Measure of Variability
1st Quartile (Q1) = 25% data ≤ this value
2nd Quartile (Q2) = Median 50% data ≤ this value
3rd Quartile (Q3) = 75% data ≤ this value
IQR = Q3 - Q1
IQR
IQR = 15 – 2.5 = 12.5
Example 2
1,2,3,……..97,98,99
Q1 = 25
Q3 = 75
IQR = = 50
25th
Quartile
(2.5)
75th
Quartile
(15)

10. Box Plots

11. Box Plots Outliers Ends of Vertical Lines - Whiskers
Whisker – extends to highest or lowest data value within the limit.
Upper Limit = Q (Q3 - Q1)
Lower Limit = Q (Q3 - Q1)
Q1 = First Quartile, 25th percentile
Q3 = Third Quartile, 75th percentile

12. Population vs. Sample
Data are samples that we assume represent the characteristics of a population.

13. Mean and Standard Deviation
Summary Statistics
Mean and Standard Deviation

14. Mean vs. Median Effect of Outliers
Suppose an error is made, and Median Mean
Becomes:
The mean is NOT a “resistant” measure of the center
Median and percentiles are generally not sensitive to outliers

15. Symmetric vs. Skewed Data
Box Plots
Approximate
Normal
Distribution
Non-normal
Distribution

16. Common in Environmental Data
Positive Skewness
Common in Environmental Data

17. Probability Density Function (pdf) Normal (Gaussian) Distribution
X = continuous variable
μ = mean
σ2 = variance

18. Cumulative Density Function
Cumulative Density Function, F(b)
Area under the probability density function fx(x) from a to b.

19. Calculating Probabilities

20. Example Distributions
Uniform
Triangular

21. Example Distributions
Lognormal
Exponential

22. Boxplot vs. Probability Density Function
pdf
Normal
μ=0
σ2=1

23. Symmetric vs. Skewed Data Histograms and Box Plots
Symmetrical Data
Approximates a Normal Distribution

24. Symmetric vs. Skewed Data Histograms and Box Plots
Box plot is compressed due to outliers.

25. (top half box width increases)
Increasing Skewness
(top half box width increases)

26. Cumulative Distribution Functions
Histogram of natural log of loads
and the resulting empirical
cumulative density function (CDF)
Blue – best fit normal distribution
Red – Empirical CDF

27. If data are also straight, they follow a normal distribution.
Probability Plots
Theoretical normal distribution plots as a straight line on normal probability paper.
If data are also straight, they follow a normal distribution.

28. Not Normally Distributed
Concentrations are
Not Normally Distributed

29. Logs of Concentration are
Normally Distributed

30. What to do with skewed data?
Data with outliers have a mean that may be larger than 75% of the data
If we want a more “typical” measure of the center, we have two choices:
Use a different method, i.e. use the median or geometric mean
Transform the data

31. Purpose of Transformations
Make data more normal
Make data more linear
Make data more constant variance

32. Positive and Negative Skew
Source:

33. Transformations Using Ladder of Powers

34. Geometric Mean Mean of the natural logs of the data
If the logs are normally distributed, the geometric mean is:
An estimate of the MEDIAN
NOT the mean
Mean = 7.40
Median = 0.50
Geometric Mean = 0.62

35. Outliers
Observations that are different from the rest of the observations in the data set
May be the most important observations in the data set
Example: Antarctic ozone data
NEVER throw way an outlier(s)
Use an alternate method or transform the data

36. Cause of Outliers Measurement or recording error Skewed data
Solution: identify and fix problem
Skewed data
Solution: use alternate method or transformation
Data from a different population
Solution: split into two groups based on science and analyze separately

37. MINITAB Laboratory 1 Reading Assignment
Chapter 1 Summarizing Data (pages 1-12)
Chapter 2 Graphical Data Analysis (pages 17-64)
Statistical Methods in Water Resources
by D.R. Helsel and R.M. Hirsch
MINITAB
Laboratory 1