Chapter 3Graphical Methods - 1 VariableConstructing Pie ChartsPowerPoint PresentationConstructing Bar ChartsConstructing HistogramsSlide 7Interpreting HistogramsStem-and-Leaf PlotsTime Series PlotsSlide 11Numerical Descriptive MeasuresMeasures of Central Tendency - IExample - Philadelphia RainfallMeasures of Central Tendency - IIMeasures of Variability - IMeasures of Variability - IIExample - Philadelphia Rainfall (Population)BoxplotsSlide 20Summarizing Data of More than One VariableExample - Ginkgo and Acetazolamide for Acute Mountain Syndrome Among Himalayan TrekkersSlide 23Slide 24Slide 25ScatterplotsFrance August,2003 Heat Wave DeathsSlide 28Example - Pharmacodynamics of LSDManufacturer Production/Cost RelationSlide 31Chapter 3Summarizing DataGraphical Methods - 1 Variable•After data collected, sorted into categories/ranges of values so that each individual observation falls in exactly one category/range–Numeric Responses: Break “range” of values into non-overlapping bins and count number of units in each bin–Categorical Responses: List all possible categories (with “Other” if needed), and count numbers of units in each•Pie Chart: Displays percent in each category/range•Bar Chart: Displays frequency/percent per category•Histogram: Displays frequency/percent per “range”Constructing Pie Charts•Select a small number of categories (say 5 or 6 at most) to avoid many narrow “slivers”•If possible, arrange categories in ascending or descending order for categorical variablesPhilly Monthy Rainfall 1825-1869 (1/100 inches)1234567891011Category Range Count1 <100 172 100-199 783 200-299 1324 300-399 1155 400-499 866 500-599 557 600-699 278 700-799 179 800-899 610 900-999 311 >1000 4Monthly Philly Rainfall 1825-1869 (1/100 in)Constructing Bar Charts•Put frequencies on one axis (typically vertical, unless many categories) and categories on other•Draw rectangles over categories with height=frequency•Leave spaces between categoriesConstructing Histograms•Used for numeric variables, so need Class Intervals–Let Range = Largest - Smallest Measurement–Break range into (say) 5-20 intervals depending on sample size–Make the width of the subintervals a convenient unit, and make “break points” so that no observations fall on them–Obtain Class Frequencies, the number in each subinterval–Obtain Relative Frequencies, proportion in each subinterval•Construct Histogram–Draw bars over each subinterval with height representing class frequency or relative frequency (shape will be the same)–Leave no space between bars to imply adjacency of class intervalsHistogram020406080100120140rain100Frequency100200300400500600700800900100011001200MoreInterpreting Histograms•Probability: Heights of bars over the class intervals are proportional to the “chances” an individual chosen at random would fall in the interval•Unimodal: A histogram with a single major peak•Bimodal: Histogram with two distinct peaks (often evidence of two distinct groups of units)•Uniform: Interval heights are approximately equal•Symmetric: Right and Left portions are same shape•Right-Skewed: Right-hand side extends further•Left-Skewed: Left-hand side extends furtherStem-and-Leaf Plots•Simple, crude approach to obtaining shape of distribution without losing individual measurements to class intervals. Procedure:–Split each measurement into 2 sets of digits (stem and leaf)–List stems from smallest to largest–Line corresponding leaves aside stems from smallest to largest–If too cramped/narrow, break stems into two groups: low with leaves 0-4 and high with leaves 5-9–When numbers have many digits, trim off right-most (less significant) digits. Leaves should always be a single digit.Time Series Plots•Many datasets represent a single variable measured on a single unit at different time points•When measurements are made at equally spaced time points, goal is often to describe temporal variation•Annual measurements can reveal long-term trends•Sub-annual (weekly, monthly, quarterly) measurements can reveal long-term trends as well as seasonal fluctuations•Plots generally have measurement on vertical axis and time period on horizontal.•Some plots include bars around points to represent fluctuations within that time periodPhilly Rainfall 1/1825-12/1869010002000MonthRainfall (1/100th inches)Numerical Descriptive Measures•Numeric summaries of a set of measurements•Measures of Central Tendency describe the “location” or center of a set of measurements•Measures of Variability describe the “spread” or dispersion of a set of measurements•Parameters: Numeric descriptive measures based on Populations of measurements•Statistics: Numeric descriptive measures based on Samples of measurementsMeasures of Central Tendency - I•Mode: Most often occuring outcome (typically only of interest for variables taking on only “discrete” values)•Median: Middle value when measurements ordered from smallest to largest•Mean: Sum of all measurements, divided by total number of measurements (equal distribution of total)nyynNyNiiii :elements) ( Sample :elements) ( PopulationIn practice, we only observe sample, and use to estimate yExample - Philadelphia Rainfall 340341339 :Amounts Ordered68.367540198547198547)Population as (Treating Months 540)271()270(5401MyyyNiiNote: The mean is higher than median as a few very large amounts were observed.Measures of Central Tendency - II•Outlier: Individual measurement(s) falling far away from others. Can have large effect on mean, not median•Trimmed Mean (TM): Mean that is based on center measurements (deleting extreme measurements).•Mode: For continuous (smooth) distributions, mode is value corresponding to the peak of the frequency curve•Skewness: Shape of the distribution:–Mound-Shaped Distributions: Mode Median Mean TM–Right-Skewed Distributions: Mode < Median < TM < Mean–Left-Skewed Distributions: Mean < TM < Median < ModeMeasures of Variability - I•Variability: Magnitude of dispersion in data.•Range: Difference between largest and smallest measurements in a set.•pth-Percentile: Value that has at most p% of measurements below, and (100-p)% above it (0<p<100)–Lower Quartile = 25th Percentile (Q1)–Median = 50th Percentile (Q2)–Upper Quartile = 75th Percentile (Q3)•Interquartile Range: Difference between the upper and
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