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Methods for Describing Sets of Data
Chapter 2 Methods for Describing Sets of Data Slides for Optional Sections Section 2.8 Methods for Detecting Outliers Slides 31-34 Section 2.9 Graphing Bivariate Relationships Slide 35 Section The Time Series Plot Slide 36
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Objectives Describe Data using Graphs Describe Data using Charts
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Describing Qualitative Data
Qualitative data are nonnumeric in nature Best described by using Classes 2 descriptive measures class frequency – number of data points in a class class relative = class frequency frequency total number of data points in data set class percentage – class relative freq. x 100 Add discussion of class percentage
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Describing Qualitative Data – Displaying Descriptive Measures
Summary Table Class Frequency Class percentage – class relative frequency x 100
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Describing Qualitative Data – Qualitative Data Displays
Bar Graph
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Describing Qualitative Data – Qualitative Data Displays
Pie chart
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Describing Qualitative Data – Qualitative Data Displays
Pareto Diagram
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Graphical Methods for Describing Quantitative Data
The Data
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Graphical Methods for Describing Quantitative Data
For describing, summarizing, and detecting patterns in such data, we can use three graphical methods: dot plots stem-and-leaf displays histograms
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Graphical Methods for Describing Quantitative Data
Dot Plot
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Graphical Methods for Describing Quantitative Data
Stem-and-Leaf Display
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Graphical Methods for Describing Quantitative Data
Histogram
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Graphical Methods for Describing Quantitative Data
More on Histograms Number of Observations in Data Set Number of Classes Less than 25 5-6 25-50 7-14 More than 50 15-20
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Summation Notation Used to simplify summation instructions
Each observation in a data set is identified by a subscript x1, x2, x3, x4, x5, …. xn Notation used to sum the above numbers together is
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Summation Notation Data set of 1, 2, 3, 4 Are these the same? and
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Numerical Measures of Central Tendency
Central Tendency – tendency of data to center about certain numerical values 3 commonly used measures of Central Tendency: Mean Median Mode
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Numerical Measures of Central Tendency
The Mean Arithmetic average of the elements of the data set Sample mean denoted by Population mean denoted by Calculated as and
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Numerical Measures of Central Tendency
The Median Middle number when observations are arranged in order Median denoted by m Identified as the observation if n is odd, and the mean of the and observations if n is even
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Numerical Measures of Central Tendency
The Mode The most frequently occurring value in the data set Data set can be multi-modal – have more than one mode Data displayed in a histogram will have a modal class – the class with the largest frequency
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Numerical Measures of Central Tendency
The Data set Mean Median is the or 5th observation, 8 Mode is 8
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Numerical Measures of Variability
Variability – the spread of the data across possible values 3 commonly used measures of Variability: Range Variance Standard Deviation
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Numerical Measures of Variability
The Range Largest measurement minus the smallest measurement Loses sensitivity when data sets are large These 2 distributions have the same range. How much does the range tell you about the data variability?
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Numerical Measures of Variability
The Sample Variance (s2) The sum of the squared deviations from the mean divided by (n-1). Expressed as units squared Why square the deviations? The sum of the deviations from the mean is zero
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Numerical Measures of Variability
The Sample Standard Deviation (s) The positive square root of the sample variance Expressed in the original units of measurement
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Numerical Measures of Variability
Samples and Populations - Notation Sample Population Variance s2 Standard Deviation s
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Numerical Measures of Relative Standing
Descriptive measures of relationship of a measurement to the rest of the data Common measures: percentile ranking z-score
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Numerical Measures of Relative Standing
Percentile rankings make use of the pth percentile The median is an example of percentiles. Median is the 50th percentile – 50 % of observations lie above it, and 50% lie below it For any p, the pth percentile has p% of the measures lying below it, and (100-p)% above it
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Numerical Measures of Relative Standing
z-score – the distance between a measurement x and the mean, expressed in standard units Use of standard units allows comparison across data sets
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Numerical Measures of Relative Standing
More on z-scores Z-scores follow the empirical rule for mounded distributions
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Methods for Detecting Outliers
Outlier – an observation that is unusually large or small relative to the data values being described Causes: Invalid measurement Misclassified measurement A rare (chance) event 2 detection methods: Box Plots z-scores
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Methods for Detecting Outliers
Box Plots based on quartiles, values that divide the dataset into 4 groups Lower Quartile QL – 25th percentile Middle Quartile - median Upper Quartile QU – 75th percentile Interquartile Range (IQR) = QU - QL
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Methods for Detecting Outliers
Box Plots Not on plot – inner and outer fences, which determine potential outliers QU (hinge) QL (hinge) Median Potential Outlier Whiskers
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Methods for Detecting Outliers
Rules of thumb Box Plots measurements between inner and outer fences are suspect measurements beyond outer fences are highly suspect Z-scores Scores of 3 in mounded distributions (2 in highly skewed distributions) are considered outliers
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