1. Measures of Position Section 2-6
M A R I O F. T R I O L A
Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman

2. Measure of Position

3. Measure of Position Z Score (or standard score)
the number of standard deviations a score is above or below the mean

4. Measure of Position
z-score
Sample
x – x
z = s

5. Measure of Position
z-score
Sample
Population
x – µ
x – x
z =
z = s s

6. round to 2 decimal places
Measure of Position
z-score
Sample
Population
x – µ
x – x
z =
z = s s
round to 2 decimal places

7. Interpreting Z Scores Z FIGURE Unusual Values Ordinary Values Unusual
– 3
– 2
– 1 1 2 3 Z

8. Example

9. Quartiles, Deciles, Percentiles
Measures of Position
Quartiles, Deciles,
Percentiles

10. Quartiles

11. Quartiles
Q1, Q2, Q3

12. divides ranked scores into four equal parts
Quartiles
Q1, Q2, Q3
divides ranked scores into four equal parts

13. divides ranked scores into four equal parts
Quartiles
Q1, Q2, Q3
divides ranked scores into four equal parts
25%
25%
25%
25% Q1 Q2 Q3

14. divides ranked data into ten equal parts
Deciles
D1, D2, D3, D4, D5, D6, D7, D8, D9
divides ranked data into ten equal parts

15. divides ranked data into ten equal parts
Deciles
D1, D2, D3, D4, D5, D6, D7, D8, D9
divides ranked data into ten equal parts
10%
D D D D D D D D D9

16. k th Percentiles (for a given number k)

17. Quartiles, Deciles, Percentiles
Fractiles

18. Quartiles, Deciles, Percentiles Fractiles
partitions data into approximately equal parts

19. Finding the Percentile of a Given Score

20. Finding the Percentile of a Given Score
number of scores less than x
Percentile of score x = • 100
total number of scores

21. Finding the Score
Given a Percentile

22. Finding the Score Given a Percentile k L = • n
n number of scores in the data set
k percentile being used
L locator that gives the position of a score
Pk kth percentile k
L = • n
100

23. Finding the Value of the kth Percentile
Start
Rank the data.
(Arrange the data in
order of lowest to
highest.)
Finding the Value of the kth Percentile
Compute
L = n where
n = number of scores
k = percentile in question ) ( k
100
The value of the kth percentile
is midway between the Lth score
and the next higher score in the
original set of data. Find Pk by
adding the L th score and the
next higher score and dividing the
total by 2. Is
L a whole
number ?
Yes No
Change L by rounding
it up to the next
larger whole number.
The value of Pk is the
Lth score, counting from the lowest

24. Quartiles
Q1 = P25
Q2 = P50
Q3 = P75

25. Quartiles
Deciles
Q1 = P25
Q2 = P50
Q3 = P75
D1 = P10
D2 = P20
D3 = P30 •
D9 = P90

26. Interquartile Range: Q3 – Q1 Semi-interquartile Range: Midquartile:
Percentile Range: P90 - P10
Q3- Q1 2
Q1+ Q3 2

27. Interquartile Range: Q3 – Q1 Semi-interquartile Range: Midquartile:
Percentile Range: P90 - P10
Q3 – Q1 2
Q1+ Q3 2

28. Interquartile Range: Q3 – Q1 Semi-interquartile Range: Midquartile:
Percentile Range: P90 - P10
Q3 – Q1 2
Q1 + Q3 2

29. Interquartile Range: Q3 – Q1 Semi-interquartile Range: Midquartile:
10–90 Percentile Range: P90 – P10
Q3 – Q1 2
Q1 + Q3 2

30. Exploratory Data Analysis Section 2-7
M A R I O F. T R I O L A
Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman

31. Exploratory Data Analysis
Used to explore data at a preliminary level
Few or no assumptions are made about the data
Tends to involve relatively simple calculations and graphs

32. Exploratory Data Analysis Traditional Statistics
Used to explore data at a preliminary level
Few or no assumptions are made about the data
Tends to involve relatively simple calculations and graphs
Traditional Statistics
Used to confirm final conclusions about data
Typically requires some very important assumptions about the data
Calculations are often complex, and graphs are often unnecessary

33. Boxplots Box-and-Whisker Diagram
5 - number summary
Minimum
first quartile Q1
Median
third quartile Q3
Maximum

34. Boxplots Box-and-Whisker Diagram60
68.5 78 52 90 50 55 60 65 70 75 80 85 90
Boxplot of Pulse Rates (Beats per minute) of Smokers

35. Figure Boxplots
Normal

36. Figure Boxplots
Normal
Uniform

37. Figure Boxplots
Normal
Uniform
Skewed

38. Values that are very far away from most of the data
Outliers
Values that are very far away from most of the data

39. When comparing two or more boxplots, it is necessary to use the same scale.

40. When comparing two or more boxplots, it is necessary to use the same scale.40 50 60 70 80 90
100
PULSE 1 2
(yes) SMOKE (No)

41. Modified Boxplots Box-and-Whisker Diagram
See page 103, Problem 11.