# Introduction to Environmental Statistics. The Big Picture. Populations and Samples. Sample Data. Examples of sample data

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1 A Few Sources for Data Examples Used Introduction to Environmental Statistics Professor Jessica Utts University of California, Irvine 1. Statistical Methods in Water Resources by D.R. Helsel and R.M. Hirsch (H&H). Statistical Methods and Pitfalls in Environmental Data Analysis, Yue Rong, Environmental Forensics, Vol 1,, pgs 13-. (Rong) 3. Data provided by Steve Saiz (Saiz) 4. Introduction to Probability and Statistics, Mendenhall and Beaver (M&B) 5. Occasionally, my own data because I understand it! Day 1, Morning, Slide The Big Picture Why use statistical methods? Describe features of a data set Make inferences about a population using sample data Estimate parameters (such as means) with a certain level of confidence Test hypotheses about specific parameters or relationships Predict future values based on past data Day 1, Morning, Slide 3 Populations and Samples Population data: Measurements available on all units of interest. Example: Annual peak discharge for Saddle River, NJ from 195 to (H&H) Can be considered as population data if only those years are of interest. Can be considered as sample data and used for inference about all possible years. Day 1, Morning, Slide 4 Sample Data Sample data used for two purposes: Describing that sample only Making inferences to a population Ideal is a random sample but almost impossible to get. Instead: Fundamental Rule for Using Data for Inference: Available data can be used to make inferences about a much larger group if the data can be considered to be representative with regard to the question(s) of interest. Day 1, Morning, Slide 5 Examples of sample data Nickel effluent data, City of San Francisco (data from Saiz) Grab samples from 1999 to Representative of a larger population of nickel concentration data; what population? Groundwater monitoring data for benzene concentrations for 16 quarters, 1996 to 1999 (data from Rong) Representative of a larger population? Day 1, Morning, Slide 6

2 Independent vs Paired Samples For comparing two situations, data can be collected as independent samples or as matched pairs. Examples: Independent samples: Compare wells upgradient and downgradient from a toxic waste site for a certain chemical. Matched pairs (H&H): Compare atrazine concentrations before (June) and after (Sept) application season in 4 shallow groundwater sites (same sites both times). Day 1, Morning, Slide 7 Class Input and Discussion Share examples of data you have collected and/or dealt with in your job: How were the data collected? What was the question of interest? Paired data or independent samples? Population or sample? If sample, what larger population do they represent? Day 1, Morning, Slide 8 Types of Data There are various ways to classify data, but probably not too useful for your data: Nominal: Name only, also called categorical data Example: Classify wells by land use in area (residential, agricultural, industrial, etc) Ordinal: Ordered, but numbers may not have much meaning. Example (H&H): = concentrations below reporting limit, 1 = above rl but below a health standard, = above health standard. Day 1, Morning, Slide 9 Types of Data, Continued Interval: Numbers have meaning, but ratios do not. There is no absolute ( none ). Example: Temperature Ratio: There is an absolute Ratios have meaning. Example: Nickel concentration (it makes sense to talk about a sample having twice the concentration of another sample). Day 1, Morning, Slide 1 Categorical vs Quantitative Data Categorical data Nominal, and sometimes ordinal For a single variable, summaries include frequencies and proportions only Quantitative data Interval, ratio, sometimes ordinal Summarize with numerical summaries Multiple variables (same or different types). For instance, categorize wells by land use, and compare a quantitative measure across uses. Day 1, Morning, Slide 11 Types of Statistical Procedures Graphical summaries Provide visual information about the data set Provide guidance on what statistical methods are appropriate Numerical summaries Provide information about specific features of the data set Inference (parametric, nonparametric) Infer things about a population, often to answer a yes/no question Day 1, Morning, Slide 1

3 Summary Features of Quantitative Data 1. Location (Center, Average). Spread (Variability) 3. Shape (Normal, skewed, etc) 4. Outliers (Unusual values) We use pictures and numerical information to examine these. Day 1, Morning, Slide 13 Questions about quantitative variables: One Quantitative Variable Question 1: What interesting summary measures, like the average or the range of values, can help us understand the data? Example: What is the average nickel concentration in the SF effluent data, and how much variability is there? Question : Are there individual data values that provide interesting information because they are unique or stand out in some way (outliers)? Example: (M&B) Data on mercury concentration in livers of 8 dolphins were all over 1 micrograms/gram except 4 of them, which were all under 1. Explanation: 4 dolphins under 3 years old, others all more than 8 years old. Day 1, Morning, Slide 14 One Categorical, One Quantitative Variable (Comparing across categories) Question 1: Are the quantitative measurements similar across categories of the categorical variable? Example: (H&H) Do wells upgradient and downgradient of a hazardous waste site have the same average concentration of a toxic compound? Question : When the categories have a natural ordering (an ordinal variable), does the quantitative variable increase or decrease, on average, in that same order? Example: Do low, medium and high flow areas of a stream have an increasing (or decreasing) average amount of a certain type of vegetation? Pictures for Quantitative Data Look at shape, outliers, center (location), spread, gaps, any other interesting features. Four common types of pictures: Histograms: similar to bar graphs, used for any number of data values. Stem-and-leaf plots and dotplots: present all individual values, useful for small to moderate sized data sets. Boxplot or box-and-whisker plot: useful summary for comparing two or more groups. Day 1, Morning, Slide 15 Day 1, Morning, Slide 16 Histogram: SF Nickel Effluent Data (Saiz) Frequency Histogram of Nickel 3. Nicke l Values are centered at about.6 or.7 (µg/l) Shape is skewed to the right (more on this later) Values range from about 1.9 to Notes about histograms Intervals are equally spaced. Example: Each interval has width.5. One goal is to assess shape. Between 6 and 15 intervals is a good number (may need to use more if there are gaps and/or outliers). 11 in nickel example. Some authors suggest using smallest k with k n, but not good for small n. Ex: n =39, so would use only k = 6. Decide where to put values that are on the boundary. For instance, would go in the interval from to, or from to 4? Need to be consistent. (Not relevant in this example.) Can use frequencies (counts) or relative frequencies (proportions) as vertical axis. Example uses frequencies. Day 1, Morning, Slide 18

4 Frequency Number of bins can change picture Ex: Peak discharge, Saddle River, NJ (H&H) 8 1 intervals PeakDischarge, Saddle River, NJ 4 Frequency intervals 1 3 PeakDischarge, Saddle River, NJ Even a small change in number of intervals made a difference. 4 Creating a Dotplot These can be useful for comparing groups Ideally, number line represents all possible values and there is one dot per observation. Not always possible. If dots represent multiple observations, footnote should explain that. As with histogram, divide horizontal axis into equal intervals, then put dots on it for each individual in each interval. Example (next slide): Compare ln(specific capacity) for wells in Appalachians of Pennsylvania, 4 rock types. [ln(x) = natural log of x] Day 1, Morning, Slide Dotplots of Ln(specific capacity), H&H Each dot represents one observation. Note different ranges, and possibly different centers. Dolomite Limestone Siliclastic Metamorphic Log of Specific Capacity (gal/min/ft) Creating a Stemplot (stem and leaf plot) Ex: SF nickel effluent data.8, 3., 3.3,.5,.3,.4,.7,.8,.6, 3.9, 3.5,.5, 3.7, 4.4,.3,.6,.5,.,.6, 3., 3., 1.9,.3,.3, 3.5,.4,.,.4,.4,.,.,.5,.8,.7,.8,.1,.6, 3.3,.1 Step 1: Create the Stem Divide range of data into equal units to be used on stem. Have 6 to 15 stem values, representing equally spaced intervals. Here, we could use or 5 for each digit from 1 to 4. Example: Each of the 6 stem values represents a possible range of.5 First one represents 1.5 to 1.9, then. to.4, then.5 to.9, and so on, up to 4. to Creating a Stemplot Step : Attach the Leaves Attach a leaf to represent each data point. Next digit in number used as leaf; drop any remaining digits. Example: First 5 values are.8, 3., 3.3,.5,.3. The numbers after the decimal point are the leaves Optional Step: order leaves on each branch. Day 1, Morning, Slide 3 Further Details for Creating Stemplots Reusing digits two or five times. Goal: assess shape. Stemplot A: EX: 1 9 = 1.9 Two times: 1 st stem = leaves to 4 nd stem = leaves 5 to 9 Stemplot B: Five times: 1 st stem = leaves and 1 nd stem =leaves and 3, etc.

5 Nickel Example: Shape Frequency Describing Shape Symmetric, bell-shaped Stemplot B: Histogram of Nickel Symmetric, not bell-shaped Bimodal: Two prominent peaks (modes) Skewed Right: On number line, values clumped at left end and extend to the right (Very common in your data sets.) Nickel Skewed Left: On number line, values clumped 4 4 at right end and extend to the left (Ex: Age at 4 4 This shape is called death from heart attack.) skewed to the right. Day 1, Morning, Slide 6 Bell-shaped example: Heights of 94 females Heights of 94 female college students. Bell-shaped, centered around 64 inches with no outliers. Bimodal Example: Old Faithful Geyser, time between eruptions, histogram from R Commander Times between eruptions of the Old Faithful geyser, shape is bimodal. Two clusters, one around 5 min., other around 8 min. Source: Hand et al., 1994 Day 1, Morning, Slide 7 Day 1, Morning, Slide 8 Boxplots, based on Five Number Summary : The five-number summary display Median Lower Quartile Upper Quartile Lowest Highest Lowest = Minimum Highest = Maximum Median = number such that half of the values are at or above it and half are at or below it (middle value or average of two middle numbers in ordered list). Quartiles = medians of the two halves. Day 1, Morning, Slide 9 Boxplots Visual picture of the five-number summary Example: How much do statistics students sleep? 19 statistics students asked how many hours they slept the night before (a Tuesday night). Five-number summary for number of hours of sleep (details of how to find these a little later) Two students reported 16 hours; the max for the remaining 188 students was 1 hours. Day 1, Morning, Slide 3

6 Creating a Boxplot 1. Draw horizontal (or vertical) line, label it with values from lowest to highest in data.. Draw rectangle (box) with ends at quartiles. 3. Draw line in box at value of median. 4. Compute IQR = distance between quartiles. 5. Compute 1.5(IQR); outlier is any value more than this distance from closest quartile. Draw line (whisker) from each end of box extending to farthest data value that is not an outlier. (If no outlier, then to min and max.) 6. Draw asterisks to indicate the outliers. Creating a Boxplot for Sleep Hours 1. Draw horizontal line and label it from 3 to 16.. Draw rectangle (box) with ends at 6 and 8 (quartiles). 3. Draw line in box at median of Compute IQR (interquartile range) = 8 6 =. 5. Compute 1.5(IQR) = 1.5() = 3; outlier is any value below 6 3 = 3, or above = Draw line from each end of box extending down to 3 and up to Draw asterisks at outliers of 1 and 16 hours. Day 1, Morning, Slide 31 Day 1, Morning, Slide 3 Interpreting Boxplots Divides the data into fourths. Easily identify outliers. Useful for comparing two or more groups. Outlier: any value more than 1.5(IQR) beyond closest quartile. ¼ of students slept between 3 and 6 hours ¼ slept between 6 and 7 hours ¼ slept between 7 and 8 hours ¼ slept between 8 and 16 hours Sometimes boxplots are vertical instead of horizontal; also, useful for comparisons Ln (Specific Capacity) Dolomite Boxplot of ln(spec, Cap.) of Wells in PA Limestone Siliclastic Metamorphic Outliers and How to Handle Them Outlier: a data point that is not consistent with the bulk of the data. Look for them via graphs. Can have big influence on conclusions. Can cause complications in some statistical analyses. Cannot discard without justification. May indicate that the underlying population is skewed, rather than one unique outlier (especially with small samples) Day 1, Morning, Slide 35 Possible reasons for outliers and what to do about them: 1. Outlier is legitimate data value and represents natural variability for the group and variable(s) measured. Values may not be discarded. They provide important information about location and spread.. Mistake made while taking measurement or entering it into computer. If verified, should be discarded or corrected. 3. Individual observation(s) in question belong(s) to a different group than bulk of individuals measured. Values may be discarded if summary is desired and reported for the majority group only.

7 Example: Sleep hours Two students were outliers in amount of sleep, but the values were not mistakes. Example: Students gave mother s height Dotplot of momheight momheight Reason 1: Natural variability, it is not okay to remove these values. Day 1, Morning, Slide 37 Height of 8 inches = 6 ft 8 inches, almost surely an error! Reason #, investigate and try to find error; remove value. Example: Weights (in pounds) of 18 men on crew teams: Cambridge:188.5, 183., 194.5, 185., 14., 3.5, 186., 178.5, 19. Oxford: 186., 184.5, 4., 184.5, 195.5,.5, 174., 183., 19.5 Note: last weight in each list is unusually small.??? Day 1, Morning, Slide 39 Example: Weights (in pounds) of 18 men on crew teams: Cambridge:188.5, 183., 194.5, 185., 14., 3.5, 186., 178.5, 19. Oxford: 186., 184.5, 4., 184.5, 195.5,.5, 174., 183., 19.5 Note: last weight in each list is unusually small.??? They are the coxswains for their teams, while others are rowers. Reason 3: different group, okay to remove if only interested in rowers. Day 1, Morning, Slide 4 Numerical Summaries of Quantitative Data Notation for Raw Data: n = number of individual observations in a data set x 1, x, x 3,, x n represent individual raw data values Example: Nickel effluent data: So n = 39, and x 1 =.8, x = 3., x 3 = 3.3, x 4 =.5 etc... Day 1, Morning, Slide 41 Describing the Location of a Data Set Mean: the numerical average Median: the middle value (if n odd) or the average of the middle two values (n even) Symmetric: mean = median Skewed Left: usually mean < median Skewed Right: usually mean > median Day 1, Morning, Slide 4

8 Pictures from Helsel &Hirsch: Data values skewed to the right Bell-shaped distribution Day 1, Morning, Slide 43 Day 1, Morning, Slide 44 Determining the Mean and Median xi The Mean x = n where xi means add together all the values The Median If n is odd: Median = middle of ordered values. Count (n + 1)/ down from top of ordered list. If n is even: Median = average of middle two ordered values. Average the values that are (n/) and (n/) + 1 down from top of ordered list. Day 1, Morning, Slide 45 The Mean, Median, and Mode Ordered Listing of 16 Benzene values (Rong) 3.8, 35, 38, 55, 11, 1, 13, 18,, 3, 3, 34, 48, 81, 98, 1 Mean (numerical average): 37. Median: 19 (halfway between 18 and ) Mode (most common value): no single mode Day 1, Morning, Slide 46 Frequency Median (19, half of area) vs Mean (37, balance point) Skewed to Right Histogram of Benzene Benzene 1 1 The Influence of Outliers on the Mean and Median Larger influence on mean than median. High outliers and data skewed to the right will increase the mean. Low outliers and data skewed to the left will decrease the mean. Ex: Suppose ages at death of your eight greatgrandparents are: 8, 4, 75, 78, 8, 8, 81, 8. Mean age is 544/8 = 68 years old Median age is (78 + 8)/ = 79 years old Day 1, Morning, Slide 48

9 Caution: Confusing Normal with Average Common mistake to confuse average with normal. Is woman 5 ft. 1 in. tall 5 inches taller than normal?? Example: How much hotter than normal is normal? October came in like a dragon Monday, hitting 11 degrees in Sacramento by late afternoon. That temperature tied the record high for Oct. 1 set in 198 and was 17 degrees higher than normal for the date. (Korber, 1, italics added.) Article had thermometer showing normal high for the day was 84 degrees. High temperature for Oct. 1 st is quite variable, from 7s to 9s. While 11 was a record high, it was not 17 degrees higher than normal if normal includes the range of possibilities likely to occur on that date. Day 1, Morning, Slide 49 Describing Spread (Variability): Range, Interquartile Range and Standard deviation Range = high value low value Interquartile Range (IQR) = upper quartile lower quartile = Q 3 - Q 1 (to be defined) Standard Deviation (most useful for bell-shaped data) Day 1, Morning, Slide 5 Benzene Example, n = , 35, 38, 55, [Q 1 = (55+11)/ = 8.5] 11, 1, 13, 18, [Median = 19], 3, 3, 34, [Q 3 = (34+48)/ = 41] 48, 81, 98, 1 Five number summary Median = 19 has half of the values above, half below Two extremes describe spread over 1% of data Range = = Two quartiles describe spread over middle 5% of data Interquartile Range = = 37.5 Finding Quartiles by hand Split the ordered values at median: half at or below the median ( at if ties) half at or above the median Q 1 = lower quartile = median of data values that are (at or) below the median Q 3 = upper quartile = median of data values that are (at or) above the median Day 1, Morning, Slide 5 Hands-On Activity #1 Data and details on activity sheet For the San Francisco effluent nickel data: Find a 5-number summary Draw a boxplot What can be concluded about shape from the boxplot? Results given in class Five number summary: Boxplot: Shape: Day 1, Morning, Slide 53 Day 1, Morning, Slide 54

10 Percentiles The k th percentile is a number that has k% of the data values at or below it and (1 k)% of the data values at or above it. Lower quartile: Median: Upper quartile: 5 th percentile 5 th percentile 75 th percentile Day 1, Morning, Slide 55 Describing Spread (Variability): Range = high value low value Interquartile Range (IQR) = upper quartile lower quartile = Q 3 - Q 1 Standard Deviation most useful for bell-shaped data Day 1, Morning, Slide 56 Describing Spread with Standard Deviation Standard deviation measures variability by summarizing how far individual data values are from the mean. Think of the standard deviation as roughly the average distance values fall from the mean. Day 1, Morning, Slide 57 Describing Spread with Standard Deviation: A very simple example Numbers Mean Standard Deviation 1,1,1,1,1 1 9, 9,1,11, Both sets have same mean of 1. Set 1: all values are equal to the mean so there is no variability at all. Set : one value equals the mean and other four values are 1 points away from the mean, so the average distance away from the mean is about 1. Day 1, Morning, Slide 58 Calculating the Standard Deviation Formula for the (sample) standard deviation: ( x x) i s = n 1 The value of s is called the (sample) variance. An equivalent formula, easier to compute, is: xi nx s = n 1 Day 1, Morning, Slide 59 Calculating the Standard Deviation Example: 9, 9, 1, 11, 11 Step 1: Calculate x, the sample mean. Ex: x = 1 Step : For each observation, calculate the difference between the data value and the mean. Ex: -1, -1,, 1, 1 Step 3: Square each difference in step. Ex: 1, 1,, 1, 1 Step 4: Sum the squared differences in step 3, and then divide this sum by n 1. Result = variance s Ex: 4/(5 1) = 4/4 = 1 Step 5: Take the square root of the value in step 4. Ex: s = standard deviation = 1 =1

11 Population Standard Deviation Data sets usually represent a sample from a larger population. If the data set includes measurements for an entire population, the notations for the mean and standard deviation are different, and the formula for the standard deviation is also slightly different. A population mean is represented by the Greek µ ( mu ), and the population standard deviation is represented by the Greek sigma (lower case) σ = ( x i µ ) n Day 1, Morning, Slide 61 Bell-shaped distributions Measurements that have a bell-shape are so common in nature that they are said to have a normal distribution. Knowing the mean and standard deviation completely determines where all of the values fall for a normal distribution, assuming an infinite population! In practice we don t have an infinite population (or sample) but if we have a large sample, we can get good approximations of where values fall. Day 1, Morning, Slide 6 Examples of bell-shaped data Women s heights mean = 64.5 inches, s =.5 inches Men s heights mean = 7 inches, s = 3 inches IQ scores mean = 1, s = 15 (or for some tests, 16) Women s heights, n = 94 students Note approximate bell-shape of histogram Normal curve with mean = 64.5, s =.5 superimposed over histogram Frequency Histogram of Women's Heights Mean = 64.5 Mean 64.5 StDev.5 N 94 4 Day 1, Morning, Slide Height Interpreting the Standard Deviation for Bell-Shaped Curves: The Empirical Rule For any bell-shaped curve, approximately 68% of the values fall within 1 standard deviation of the mean in either direction 95% of the values fall within standard deviations of the mean in either direction 99.7% (almost all) of the values fall within 3 standard deviations of the mean in either direction Day 1, Morning, Slide 65 Ex: Population of women s heights 68% of heights are between 6 and 67 inches 95% of heights are between 59.5 and 69.5 inches 99.7% of heights are between 57 and 7 inches

12 Women s Heights: How well does the Empirical Rule work? Mean height for the 94 students was 64.5, and the standard deviation was.5 inches. Let s compare actual with ranges from Empirical Rule: Range of Values: Mean ± 1 s.d. Mean ± s.d. Mean ± 3 s.d. Empirical Rule 68% in 6 to 67 95% in 59.5 to % in 57 to 7 Actual number Actual percent 7/94 = 74.5% 89/94 = 94.7% The Empirical Rule, the Standard Deviation, and the Range Empirical Rule tells us that the range from the minimum to the maximum data values equals about 4 to 6 standard deviations for data sets with an approximate bell shape. For a large data set, you can get a rough idea of the value of the standard deviation by dividing the range by 6 (or 4 or 5 for a smaller dataset) Range s 6 94/94 = 1% Day 1, Morning, Slide 68 Standardized z-scores Standardized score or z-score: Observed value Mean z = Standard deviation Example: UCI Verbal SAT scores had mean = 569 and s = 75. Suppose someone had SAT = 674: Verbal SAT of 674 is 1.4 standard deviations above mean. To find proportion above or below, use Excel or R Commander Verbal SAT scores for UCI students Normal, Mean=569, StDev= z = = Verbal SAT of 674 for UCI student is 1.4 standard deviations above the mean for UCI students. Day 1, Morning, Slide 69 About 9% below Verbal SAT Score 674 About 8% above 674 Day 1, Morning, Slide 7 The Empirical Rule Restated for Standardized Scores (z-scores): For bell-shaped data, About 68% of the values have z-scores between 1 and +1. About 95% of the values have z-scores between and +. About 99.7% of the values have z-scores between 3 and +3. Day 1, Morning, Slide 71

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