ENVE-GEOE 224 - 4 - Variability & Shape Statistics - Handout

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2021-01-09 1 ENVE/GEOE 224: Probability & Statistics Variability and Shape Statistics Prof. Philip J. Schmidt Department of Civil & Environmental Engineering, University of Waterloo Winter 2021 Learning Objectives Learn how to calculate the range, variance, standard deviation, coefficient of variation and Pearson’s second skewness coefficient → Be able to distinguish between a sample and a population when calculating the variance or standard deviation 2 of 11 ENVE/GEOE 224 (W2021) – P. Schmidt 1 2

2021-01-09 2 Example 1.3 So now we can summarize our data with measures of location. Is this sufficient? A cookie packaging plant is aiming to package 40 cookies per box. The number of cookies changes based on the speed of the conveyor belt. Speed 1 has a better mean, but unacceptably large variability! 3 of 11 ENVE/GEOE 224 (W2021) – P. Schmidt Speed 1 40 10 25 45 60 60 Speed 2 42 42 44 40 41 43 𝑥̅ = 40 𝑥̅ = 42 The Range Variability (or dispersion) indicates the extent to which individual samples vary within a specific population. The range of a set of data is simply the difference between the maximum and minimum values The wider the range is, the more variable the data are! 4 of 11 ENVE/GEOE 224 (W2021) – P. Schmidt 3 4

2021-01-09 3 Example 1.1 – Range 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: 𝑅𝑎𝑛𝑔𝑒 = 0.77 − 0.63 = 0.14 mg/L 5 of 11 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 The Variance Describes the degree of dispersion in the data about the mean For a sample of size 𝑛 from an For all members of a population infinitely large population: of size 𝑁 : 𝑠 ଶ = ∑ ௫ ೔ ି௫̅ మ ೙ ೔సభ ௡ିଵ 𝜎 ଶ = ∑ ௫ ೔ ିఓ మ ೙ ೔సభ ே 𝑥 ௜ − 𝑥̅ is called a deviation from the mean. There are 𝑛 − 1 independent deviations from the mean because ∑ 𝑥 ௜ − 𝑥̅ ௡ ௜ୀଵ = 0 , so 𝑥 ௞ can be determined from 𝑥̅ and the other data. The 𝑛 − 1 makes the 𝑠 ଶ an “unbiased” estimator of 𝜎 ଶ . 6 of 11 ENVE/GEOE 224 (W2021) – P. Schmidt 5 6

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2021-01-09 4 The Standard Deviation Describes the degree of dispersion in the data about the mean For a sample of size 𝑛 from an For all members of a population infinitely large population: of size 𝑁 : 𝑠 = ∑ ௫ ೔ ି௫̅ మ ೙ ೔సభ ௡ିଵ 𝜎 = ∑ ௫ ೔ ିఓ మ ೙ ೔సభ ே We will see how the variance and standard deviation are particularly useful statistics in the latter half of the course 7 of 11 ENVE/GEOE 224 (W2021) – P. Schmidt Coefficient of Variation The coefficient of variation (COV) describes the relative amount of variation in a population (also called the relative standard deviation) COV = 𝑠 𝑥̅ × 100% The variability is proportionately greater at monitoring well 1 8 of 11 ENVE/GEOE 224 (W2021) – P. Schmidt Monitoring Well Mean ( 𝒙 ഥ ) Standard Deviation ( 𝑠 ) 1 1.0 0.1 2 100.0 0.1 7 8

2021-01-09 5 Example 1.1 – Variance, Standard Deviation, and COV We already calculated 𝑥̅ = 0.695 mg/L ∑ 𝑥 ௜ − 𝑥̅ ଶ ௡ ௜ୀଵ = 0.0179 𝑠 ଶ = ∑ ௫ ೔ ି௫̅ మ ೙ ೔సభ ௡ିଵ = ଴.଴ଵ଻ଽ ଵଵ = 0.01627 mg 2 /L 2 𝑠 = ∑ ௫ ೔ ି௫̅ మ ೙ ೔సభ ௡ିଵ = 0.01627 ≅ 0.0403 mg/L COV = ௦ ௫̅ × 100% ≅ ଴.଴ସ଴ଷ ଴.଺ଽହ × 100% ≅ 18.4% 9 of 11 ENVE/GEOE 224 (W2021) – P. Schmidt Skewness Coefficient Pearson’s second skewness coefficient (SK) is a measure of the asymmetry of a dataset and its direction SK = 3 𝑥̅ − 𝑚 𝑠 This statistic isn’t numerically very useful, but it tells you whether the data are symmetrical, positively skewed, or negatively skewed 10 of 11 ENVE/GEOE 224 (W2021) – P. Schmidt mean = median mean < median mean > median 9 10

2021-01-09 6 Online Lecture In this week’s online lecture, we will apply these statistics to some real environmental data using features in Excel! → Learn how to use built-in functions to calculate basic statistics → Differentiate a statistical sample from a population → Discuss why the coefficient of variation is not a meaningful statistic for temperature data 11 of 11 ENVE/GEOE 224 (W2021) – P. Schmidt 11

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