1. Measures of Relative Standing and Boxplots
Section 3.4
Measures of Relative Standing and Boxplots

2. Basics of z Scores, Percentiles, Quartiles, and Boxplots
Part 1
Basics of z Scores, Percentiles, Quartiles, and Boxplots

3. Z Score
the number of standard deviations that a given value x is above or below the mean - Also known as a standardized value

4. Population Sample x – x x – µ z = z = s  Measures of Position z Score
*Round z scores to 2 decimal places if necessary.

5. 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 or z score > 2

6. Example 1: Helen Mirren was 61 when she earned her Oscar-winning Best Actress award. The Oscar-winning Best Actresses have a mean age of 35.8 years and a standard deviation of 11.3 years.
What is the difference between Helen Mirren’s age and the mean age?
b) How many standard deviations is that?
c) Convert Helen Mirren’s age to a z score.
d) If we consider “usual” ages to be those that convert to z scores between –2 and 2, is Helen Mirren’s age usual or unusual?

7. Example 2: Human body temperatures have a mean of 98
Example 2: Human body temperatures have a mean of 98.20°F and a standard deviation of 0.62°F (based on Data Set 2 in Appendix B). Convert each given temperature to a z score and determine whether it is usual or unusual.
a) °F b) 96.90°F c) 96.98°F

8. Example 3: Scores on the SAT test have a mean of 1518 and a standard deviation of 325. Scores on the ACT test have a mean of 21.1 and a standard deviation of 4.8. Which is relatively better: a score of 1840 on the SAT test or a score of 26.0 on the ACT test? Why?

9. Percentiles
measures of location. There are 99 percentiles denoted P1, P2, P99, which divide a set of data into 100 groups with about 1% of the values in each group.

10. Finding the Percentile
of a Data Value
Percentile of value x = • 100
number of values less than x
total number of values

11. Example 4: Use the given sorted values (the numbers of points scored in the Super Bowl for a recent period of 24 years). Find the percentile corresponding to the given number of points.
a) 47 points b) 54 points

12. Converting from the kth Percentile to the Corresponding Data Value
Notation
n total number of values in the data set
k percentile being used
L locator that gives the position of a value
Pk kth percentile
L = • n k
100

13. Corresponding Data Value
Converting from the
kth Percentile to the
Corresponding Data Value

14. Example 5: Use the given sorted values (the numbers of points scored in the Super Bowl for a recent period of 24 years). Find the indicated percentile.
a) P b) P22

15. Quartiles
measures of location, denoted Q1, Q2, and Q3, which divide a set of data into four groups with about 25% of the values in each group.
Q1 (First Quartile) separates the bottom 25% of sorted values from the top 75%.
Q2 (Second Quartile) same as the median; separates the bottom 50% of sorted values from the top 50%.
Q3 (Third Quartile) separates the bottom 75% of sorted values from the top 25%.

16. Quartiles
Q1, Q2, Q3 divide ranked scores into four equal parts
25% Q3 Q2 Q1
(minimum)
(maximum)
(median)

17. Some Other Statistics Interquartile Range (or IQR): Q3 – Q1 Q3 – Q1
Semi-interquartile Range: 2
Q3 – Q1
Midquartile: 2
Q3 + Q1
Percentile Range: P90 – P10

18. Example 6: Use the given sorted values (the numbers of points scored in the Super Bowl for a recent period of 24 years). Find the indicated percentile or quartile.
a) P b) Q3

19. 5-Number Summary
For a set of data, the 5-number summary consists of the minimum value; the first quartile Q1; the median (or second quartile Q2); the third quartile, Q3; and the maximum value.

20. Boxplot
A boxplot (or box-and-whisker-diagram) is a graph of a data set that consists of a line extending from the minimum value to the maximum value, and a box with lines drawn at the first quartile, Q1; the median; and the third quartile, Q3.

21. Boxplot
Boxplot of Movie Budget Amounts

22. Boxplots - Normal Distribution
Normal Distribution: Heights from a Simple Random Sample of Women

23. Boxplots - Skewed Distribution
Skewed Distribution: Salaries (in thousands of dollars) of NCAA Football Coaches

24. Example 7: Use the given sorted values (the numbers of points scored in the Super Bowl for a recent period of 24 years). Construct a boxplot and include the values of the 5 – number summary.

25. Outliers and Modified Boxplots
Part 2
Outliers and Modified Boxplots

26. Outliers
An outlier is a value that lies very far away from the vast majority of the other values in a data set.

27. 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.

28. 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 . . .
above Q3 by an amount greater than 1.5  IQR or
below Q1 by an amount greater than 1.5  IQR

29. Modified Boxplots
Boxplots described earlier are called skeletal (or regular) boxplots.
Some statistical packages provide modified boxplots which represent outliers as special points.

30. Modified Boxplot Construction
A modified boxplot is constructed with these specifications:
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.

31. Modified Boxplots - Example
Pulse rates of females listed in Data Set 1 in Appendix B.

32. Example 8: Use the 40 upper leg lengths (cm) listed for females from Data Set 1 in Appendix B. Construct a modified boxplot. Identify any outliers.
41.6
42.8 39
40.2
36.2
43.2
38.7 41
43.8
37.3
42.3
39.1
40.3
48.6
33.2
43.4
41.5 40
38.2
38.1 38 36
32.1
31.1
39.4
39.2
36.6 27
38.5
39.9
37.5
39.7
33.8

33. Deciles are the 10th, 20th, 30th, 40th, 50th, 60th, 70th, 80th, and 90th percentiles. The are denoted using the following notation: D1, D2, …, D9.
Quintiles are the 20th, 40th, 60th, and 80th percentiles.

34. Example 9: Using the following data, find the deciles D2, D5, and D9.
Sorted Movie Budget Amounts (in millions of dollars)
4.5 5
6.5 7 20 29 30 35 40 41 50 52 60 65 68 70 72 74 75 80
100
113
116
120
125
132
150
160
200
225