Section 2.5
Measures of Position
Larson/Farber 4th ed.
Section 2.5 Objectives
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•
•
•
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Determine the quartiles of a data set
Determine the interquartile range of a data set
Create a box-and-whisker plot
Interpret other fractiles such as percentiles
Determine and interpret the standard score (z-score)
Larson/Farber 4th ed.
Quartiles
• Fractiles are numbers that partition (divide) an
ordered data set into equal parts.
• Quartiles approximately divide an ordered data set
into four equal parts.
 First quartile, Q1: About one quarter of the data
fall on or below Q1.
 Second quartile, Q2: About one half of the data
fall on or below Q2 (median).
 Third quartile, Q3: About three quarters of the
data fall on or below Q3.
Larson/Farber 4th ed.
Example: Finding Quartiles
The number of vacation days used by a sample of 20
employees in recent years. Find the first, second, and
third quartiles of the test scores.
3 9 2 1 7 5 3 2 2 6 4 0 10 0 3 5 7 8 6 5
Solution:
• Q2 divides the data set into two halves.
Lower half
Upper half
0 0 1 2 2 2 3 3 3 4 5 5 5 6 6 7 7 8 9 10
Q2 = Median = (4 + 5) ÷ 2 = 4.5
Larson/Farber 4th ed.
Solution: Finding Quartiles
• The first and third quartiles are the medians of the
lower and upper halves of the data set.
Lower half
Upper half
0 0 1 2 2 2 3 3 3 4 5 5 5 6 6 7 7 8 9 10
Q1
Q2
Q3
About one fourth of the employees used 2 or less days
of vacation, about one half used 4 or less; and about
three fourths scored 6 or less.
Larson/Farber 4th ed.
Finding Quartiles
Solution: Finding Quartiles
• To calculate the Quartiles for employees vacations:
Interquartile Range
Interquartile Range (IQR)
• The difference between the third and first quartiles.
• IQR = Q3 – Q1
Larson/Farber 4th ed.
Example: Finding the Interquartile Range
Find the interquartile range of the test scores.
Recall Q1 = 2, Q2 = 4.5, and Q3 = 6.5
Solution:
• IQR = Q3 – Q1 = 6.5 – 2 = 4.5
The days of vacation taken in the middle portion of
the data set vary by at most 4.5 days.
Larson/Farber 4th ed.
Box-and-Whisker Plot
Box-and-whisker plot
• Exploratory data analysis tool.
• Highlights important features of a data set.
• Requires (five-number summary):
 Minimum entry
 First quartile Q1
 Median Q2
 Third quartile Q3
 Maximum entry
Larson/Farber 4th ed.
139
Drawing a Box-and-Whisker Plot
1. Find the five-number summary of the data set: Min, Q1,
Q2,Q3, Max.
2. Construct a horizontal scale that spans the range of the data.
3. Plot the five numbers above the horizontal scale.
4. Draw a box above the horizontal scale from Q1 to Q3 and
draw a vertical line in the box at Q2.
5. Draw whiskers from the box to the minimum and maximum
entries.
Box
Whisker
Minimum
entry
Larson/Farber 4th ed.
Whisker
Q1
Median, Q2
Q3
Maximum
entry
Example: Drawing a Box-and-Whisker
Plot
Draw a box-and-whisker plot that represents the 20
employees taking vacation days.
Recall Min = 0 Q1 = 2 Q2 = 4.5 Q3 = 6.5 Max =
10
Solution:
10
0
4.5
2
Larson/Farber 4th ed.
6.5
Percentiles and Other Fractiles
Fractiles
Summary
Symbols
Quartiles
Divides data into 4 equal
parts
Q1, Q2, Q3
Deciles
Divides data into 10 equal
parts
D1, D2, D3,…, D9
Percentiles
Divides data into 100 equal
parts
P1, P2, P3,…, P99
Larson/Farber 4th ed.
Example: Interpreting Percentiles
The ogive represents the
cumulative frequency
distribution for SAT test
scores of college-bound
students in a recent year. What
test score represents the 72nd
percentile? How should you
interpret this? (Source: College
Board Online)
Larson/Farber 4th ed.
Solution: Interpreting Percentiles
The 72nd percentile
corresponds to a test score
of 1700.
This means that 72% of the
students had an SAT score
of 1700 or less.
Larson/Farber 4th ed.
The Standard Score
Standard Score (z-score)
• Represents the number of standard deviations a given
value x falls from the mean μ.
Larson/Farber 4th ed.
Example: Comparing z-Scores from
Different Data Sets
In 2007, Forest Whitaker won the Best Actor Oscar at
age 45 for his role in the movie The Last King of
Scotland. Helen Mirren won the Best Actress Oscar at
age 61 for her role in The Queen. The mean age of all
best actor winners is 43.7, with a standard deviation of
8.8. The mean age of all best actress winners is 36, with
a standard deviation of 11.5. Find the z-score that
corresponds to the age for each actor or actress. Then
compare your results.
Larson/Farber 4th ed.
Solution: Comparing z-Scores from
Different Data Sets
• Forest Whitaker
0.15 standard
deviations above
the mean
• Helen Mirren
2.17 standard
deviations above
the mean
Larson/Farber 4th ed.
Solution: Comparing z-Scores from
Different Data Sets
z = 0.15
z = 2.17
The z-score corresponding to the age of Helen Mirren
is more than two standard deviations from the mean, so
it is considered unusual. Compared to other Best
Actress winners, she is relatively older, whereas the age
of Forest Whitaker is only slightly higher than the
average age of other Best Actor winners.
Larson/Farber 4th ed.
Section 2.5 Summary
•
•
•
•
•
Determined the quartiles of a data set
Determined the interquartile range of a data set
Created a box-and-whisker plot
Interpreted other fractiles such as percentiles
Determined and interpreted the standard score
(z-score)
Larson/Farber 4th ed.