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Chapter 2

Chapter 2. Methods for Describing Sets of Data. Objectives. Describe Data using Graphs Describe Data using Charts. Describing Qualitative Data. Qualitative data are nonnumeric in nature Best described by using Classes 2 descriptive measures

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Chapter 2

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  1. Chapter 2 Methods for Describing Sets of Data

  2. Objectives • Describe Data using Graphs • Describe Data using Charts

  3. 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

  4. Describing Qualitative Data – Displaying Descriptive Measures • Summary Table Class Frequency Class percentage – class relative frequency x 100

  5. Describing Qualitative Data – Qualitative Data Displays • Bar Graph

  6. Describing Qualitative Data – Qualitative Data Displays • Pie chart

  7. Describing Qualitative Data – Qualitative Data Displays • Pareto Diagram

  8. Graphical Methods for Describing Quantitative Data • The Data

  9. 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

  10. Graphical Methods for Describing Quantitative Data • Dot Plot

  11. Graphical Methods for Describing Quantitative Data • Stem-and-Leaf Display

  12. Graphical Methods for Describing Quantitative Data • Histogram

  13. Graphical Methods for Describing Quantitative Data • More on Histograms

  14. 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

  15. Summation Notation • Data set of 1, 2, 3, 4 • Are these the same? and

  16. 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

  17. 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

  18. 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

  19. 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

  20. Numerical Measures of Central Tendency • The Data set 1 3 5 6 8 8 9 11 12 • Mean • Median is the or 5th observation, 8 • Mode is 8

  21. Numerical Measures of Variability • Variability – the spread of the data across possible values • 3 commonly used measures of Variability: • Range • Variance • Standard Deviation

  22. Numerical Measures of Variability • The Range • Largest measurement minus the smallest measurement • Loses sensitivity when data sets are large • These 2 distributionshave the same range. • How much does therange tell you about the data variability?

  23. 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

  24. Numerical Measures of Variability • The Sample Standard Deviation (s) • The positive square root of the sample variance • Expressed in the original units of measurement

  25. Numerical Measures of Variability • Samples and Populations - Notation

  26. Numerical Measures of Relative Standing • Descriptive measures of relationship of a measurement to the rest of the data • Common measures: • percentile ranking • z-score

  27. 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

  28. 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

  29. Numerical Measures of Relative Standing • More on z-scores • Z-scores follow the empirical rule for mounded distributions

  30. 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

  31. 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

  32. Potential Outlier QU (hinge) Whiskers Median QL (hinge) Methods for Detecting Outliers • Box Plots • Not on plot – inner and outer fences, which determine potential outliers

  33. 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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