1. 3.4 - 1 Copyright © 2010, 2007, 2004 Pearson Education, Inc. Lecture Slides Elementary Statistics Eleventh Edition and the Triola Statistics Series by Mario F. Triola

2. 3.4 - 2 Copyright © 2010, 2007, 2004 Pearson Education, Inc. Chapter 3 Statistics for Describing, Exploring, and Comparing Data 3-1 Review and Preview 3-2 Measures of Center 3-3 Measures of Variation 3-4 Measures of Relative Standing and Boxplots

3. 3.4 - 3 Copyright © 2010, 2007, 2004 Pearson Education, Inc. Section 3-4 Measures of Relative Standing and Boxplots

4. 3.4 - 4 Copyright © 2010, 2007, 2004 Pearson Education, Inc. Key Concept This section introduces measures of relative standing, which are numbers showing the location of data values relative to the other values within a data set. They can be used to compare values from different data sets, or to compare values within the same data set. The most important concept is the z score. We will also discuss percentiles and quartiles, as well as a new statistical graph called the boxplot.

5. 3.4 - 5 Copyright © 2010, 2007, 2004 Pearson Education, Inc. Basics of z Scores, Percentiles, Quartiles, and Boxplots Part 1

6. 3.4 - 6 Copyright © 2010, 2007, 2004 Pearson Education, Inc.  z Score (or standardized value) the number of standard deviations that a given value x is above or below the mean Z score

7. 3.4 - 7 Copyright © 2010, 2007, 2004 Pearson Education, Inc. Sample Population x – µ z =  Round z scores to 2 decimal places Measures of Position z Score z = x – x s

8. 3.4 - 8 Copyright © 2010, 2007, 2004 Pearson Education, Inc. Interpreting Z Scores Whenever a value is less than the mean, its corresponding z score is negative Ordinary values: –2 ≤ z score ≤ 2 Unusual Values: z score 2

9. 3.4 - 9 Copyright © 2010, 2007, 2004 Pearson Education, Inc. Percentiles are measures of location. There are 99 percentiles denoted P 1, P 2,... P 99, which divide a set of data into 100 groups with about 1% of the values in each group.

10. 3.4 - 10 Copyright © 2010, 2007, 2004 Pearson Education, Inc. Finding the Percentile of a Data Value Percentile of value x = 100 Do problems 17-28 on page 125 number of values less than x total number of values

11. 3.4 - 11 Copyright © 2010, 2007, 2004 Pearson Education, Inc. n total number of values in the data set k percentile being used L locator that gives the position of a value P k k th percentile L = n k 100 Notation Converting from the kth Percentile to the Corresponding Data Value

12. 3.4 - 12 Copyright © 2010, 2007, 2004 Pearson Education, Inc. Converting from the kth Percentile to the Corresponding Data Value

13. 3.4 - 13 Copyright © 2010, 2007, 2004 Pearson Education, Inc. Quartiles  Q 1 (First Quartile) separates the bottom 25% of sorted values from the top 75%.  Q 2 (Second Quartile) same as the median; separates the bottom 50% of sorted values from the top 50%.  Q 3 (Third Quartile) separates the bottom 75% of sorted values from the top 25%. Are measures of location, denoted Q 1, Q 2, and Q 3, which divide a set of data into four groups with about 25% of the values in each group.

14. 3.4 - 14 Copyright © 2010, 2007, 2004 Pearson Education, Inc. Q 1, Q 2, Q 3 divide ranked scores into four equal parts Quartiles 25% Q3Q3 Q2Q2 Q1Q1 (minimum)(maximum) (median)

15. 3.4 - 15 Copyright © 2010, 2007, 2004 Pearson Education, Inc.  Interquartile Range (or IQR): Q 3 – Q 1  10 - 90 Percentile Range: P 90 – P 10  Semi-interquartile Range: 2 Q 3 – Q 1  Midquartile: 2 Q 3 + Q 1 Some Other Statistics

16. 3.4 - 16 Copyright © 2005 Pearson Education, Inc. Slide 6-16 Why Variation Matters 6-B Big Bank (three line wait times): 4.15.25.66.26.77.2 7.77.78.59.311.0 Best Bank (one line wait times): 6.66.76.76.97.17.2 7.37.47.77.87.8

17. 3.4 - 17 Copyright © 2005 Pearson Education, Inc. Slide 6-17 Five Number Summaries & Box Plots 6-B lower quartile = 5.6 high value (max) = 7.8 upper quartile = 7.7 median = 7.2 lower quartile = 6.7 low value (min) = 6.6 Best BankBig Bank high value (max) = 11.0 low value (min) = 4.1 upper quartile = 8.5 median = 7.2

18. 3.4 - 18 Copyright © 2010, 2007, 2004 Pearson Education, Inc. Boxplots - Normal Distribution Normal Distribution: Heights from a Simple Random Sample of Women

19. 3.4 - 19 Copyright © 2010, 2007, 2004 Pearson Education, Inc. Boxplots - Skewed Distribution Skewed Distribution: Salaries (in thousands of dollars) of NCAA Football Coaches

20. 3.4 - 20 Copyright © 2010, 2007, 2004 Pearson Education, Inc. Outliers and Modified Boxplots Part 2

21. 3.4 - 21 Copyright © 2010, 2007, 2004 Pearson Education, Inc. Outliers  An outlier is a value that lies very far away from the vast majority of the other values in a data set.

22. 3.4 - 22 Copyright © 2010, 2007, 2004 Pearson Education, Inc. Important Principles  An outlier can have a dramatic effect on the mean.  An outlier can have a dramatic effect on the standard deviation.  An outlier can have a dramatic effect on the scale of the histogram so that the true nature of the distribution is totally obscured.

23. 3.4 - 23 Copyright © 2010, 2007, 2004 Pearson Education, Inc. Outliers for Modified Boxplots For purposes of constructing modified boxplots, we can consider outliers to be data values meeting specific criteria. In modified boxplots, a data value is an outlier if it is... Data > Q 3 + 1.5  IQR Data < Q 1 - 1.5  IQR or

24. 3.4 - 24 Copyright © 2010, 2007, 2004 Pearson Education, Inc. Modified Boxplots Boxplots described earlier are called skeletal (or regular) boxplots. Some statistical packages provide modified boxplots which represent outliers as special points.

25. 3.4 - 25 Copyright © 2010, 2007, 2004 Pearson Education, Inc. Modified Boxplot Construction  A special symbol (such as an asterisk) is used to identify outliers.  The solid horizontal line extends only as far as the minimum data value that is not an outlier and the maximum data value that is not an outlier. A modified boxplot is constructed with these specifications:

26. 3.4 - 26 Copyright © 2010, 2007, 2004 Pearson Education, Inc. Modified Boxplots - Example Pulse rates of females listed in Data Set 1 in Appendix B.

27. 3.4 - 27 Copyright © 2010, 2007, 2004 Pearson Education, Inc. Recap In this section we have discussed:  z Scores  z Scores and unusual values  Quartiles  Percentiles  Converting a percentile to corresponding data values  Other statistics  Effects of outliers  5-number summary  Boxplots and modified boxplots

28. 3.4 - 28 Copyright © 2010, 2007, 2004 Pearson Education, Inc. Putting It All Together Always consider certain key factors:  Context of the data  Source of the data  Measures of Center  Sampling Method  Measures of Variation  Outliers  Practical Implications  Changing patterns over time  Conclusions  Distribution