ENVE-GEOE 224 - 2 - Graphical Presentation of Data - Handouts

2021-01-09 1 ENVE/GEOE 224: Probability & Statistics Graphical Presentation of Data Prof. Philip J. Schmidt Department of Civil & Environmental Engineering, University of Waterloo Winter 2021 Week 1: Presenting & Describing Data This week will include three modules → Graphical presentation of data → Location statistics → Variability and shape statistics We’re starting with some easier concepts before diving into probability theory, probabilistic modelling, and statistical inference from data Review all three pre-recorded modules before our live online lecture that will focus upon application of these concepts to real data using features in Excel 2 of 14 ENVE/GEOE 224 (W2021) – P. Schmidt 1 2

2021-01-09 2 Learning Objectives Learn how to plot histograms and relative frequencies → Plot a frequency histogram → Plot a relative frequency polygon → Plot a cumulative relative frequency graph Learn how to plot an empirical cumulative density function → Be able to explain the difference from a cumulative relative frequency polygon 3 of 14 ENVE/GEOE 224 (W2021) – P. Schmidt Plotting Data There is almost a countless number of types of plots, of which we will learn just a few → Plots can be exploratory to learn from data or expository to present the data or results of an analysis in a compelling way 4 of 14 ENVE/GEOE 224 (W2021) – P. Schmidt Images from Schmidt, Emelko, Thompson (2020) and Emelko, Schmidt, Borchardt (2019) 3 4

2021-01-09 4 Example 1.1 Consider an evaluation of the sensitivity of several analytical techniques to determine various water quality parameters (ion concentrations) such as NO 3 - (nitrate). Using one analytical technique, you measure NO 3 - concentration 12 times. Your recorded data (in mg/L) are: We will plot these data and compute several statistics describing them! 7 of 14 ENVE/GEOE 224 (W2021) – P. Schmidt 0.63 0.65 0.71 0.69 0.73 0.68 0.66 0.68 0.68 0.77 0.72 0.74 Data range from 0.63–0.77 mg/L → four intervals of width 0.05 mg/L Clearly define the interval in which values on a boundary are included! Class Intervals Frequency 0.60 < NO 3 - ≤ 0.65 2 0.65 < NO 3 - ≤ 0.70 5 0.70 < NO 3 - ≤ 0.75 4 0.75 < NO 3 - ≤ 0.80 1 Example 1.1 – Histogram & Frequency Polygon 8 of 14 ENVE/GEOE 224 (W2021) – P. Schmidt 7 8

2021-01-09 5 Example 1.1 – Histogram & Frequency Polygon The number, width, and location of bins in a histogram is arbitrary Changing these choices can change the appearance of the plot 9 of 14 ENVE/GEOE 224 (W2021) – P. Schmidt Class Intervals Frequency 0.56 < NO 3 - ≤ 0.60 0 0.60 < NO 3 - ≤ 0.64 1 0.64 < NO 3 - ≤ 0.68 5 0.68 < NO 3 - ≤ 0.72 3 0.72 < NO 3 - ≤ 0.76 2 0.76 < NO 3 - ≤ 0.80 1 0.80 < NO 3 - ≤ 0.84 0 Relative & Cumulative Relative Frequency Relative frequency: class frequency divided by total number of data → To plot a relative frequency polygon , plot a point at the midpoint of each interval and the corresponding relative frequency, then connect these points Cumulative relative frequency: sum of relative frequencies up to and including the class in question → To plot a cumulative relative frequency graph , plot a point at the upper end of each interval and the corresponding cumulative relative frequency, then connect these points 10 of 14 ENVE/GEOE 224 (W2021) – P. Schmidt 9 10

2021-01-09 7 13 of 14 ENVE/GEOE 224 (W2021) – P. Schmidt Example 1.1 – Empirical Distribution Function Strictly speaking, the empirical distribution function is a step function. However, simply connecting the dots is often adequate. Online Lecture In this week’s online lecture, we will apply these techniques to some real environmental data using features in Excel! → Use the built-in histogram feature in the Analysis ToolPak → What will this histogram teach us about the annual pattern of daily maximum temperatures at the University of Waterloo Weather Station? 14 of 14 ENVE/GEOE 224 (W2021) – P. Schmidt 13 14