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Chapter 2
– Class 15
Numerical
Summary of
Quantitative Data
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Class Work
What kind of numerical summary have you
learned so far?
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5 number summary
Min Q1 Median Q3 Max
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Example 2.14 Fastest Speeds for Men
Ordered Data (in rows of 10 values) for the 87 males:
• Median = (87+1)/2 = 44th value in the list = 110 mph
• Q1 = median of the 43 values below the median =
(43+1)/2 = 22nd value from the start of the list = 95 mph
• Q3 = median of the 43 values above the median =
(43+1)/2 = 22nd value from the end of the list = 120 mph
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Numerical Summaries
of Quantitative Data
Notation for Raw Data:
n = number of individuals in a data set
x1, x2 , x3,…, xn represent individual raw data values
Example: A data set consists of handspan
values in centimeters for six females;
the values are 21, 19, 20, 20, 22, and 19.
Then, n = 6
x1= 21, x2 = 19, x3 = 20, x4 = 20, x5 = 22, and x6 = 19
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Notation and Finding the Quartiles
Split the ordered values into the half
that is below the median and the half
that is above the median.
Q1 = lower quartile
= median of data values
that are below the median
Q3 = upper quartile
= median of data values
that are above the median
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Percentiles
The kth percentile is a number that has
k% of the data values at or below it and
(100 – k)% of the data values at or above it.
• Lower quartile = 25th percentile
• Median = 50th percentile
• Upper quartile = 75th percentile
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Describing the Location
of a Data Set
• Mean: the numerical average
• Median: the middle value (if n odd)
or the average of the middle two
values (n even)
Symmetric: mean = median
Skewed Left: mean < median
Skewed Right: mean > median
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Determining the Mean and Median
x

x
i
The Mean
where
x
i
n
means “add together all the values”
The Median
If n is odd: M = middle of ordered values.
Count (n + 1)/2 down from top of ordered list.
If n is even: M = average of middle two ordered values.
Average values that are (n/2) and (n/2) + 1
down from top of ordered list.
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Example 2.12 Will “Normal” Rainfall
Get Rid of Those Odors?
Data: Average rainfall (inches)
for Davis, California for 47 years
Mean = 18.69 inches
Median = 16.72 inches
In 1997-98, a company
with odor problem blamed
it on excessive rain.
That year rainfall was
29.69 inches. More rain
occurred in 4 other years.
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Mean VS Median
Kobe Bryant
25.2 million
Derek Fisher
3.4 million
Pau Gasol
18.7 million
Matt Barnes
1.9 million
Andrew
Bynum
15.2 million
Troy Murphy
1.4 million
Lamar Odom
8.9 million
Jason Kapono
1.2 million
Metta World
Peace
6.8 million
Derrick
Caracter
0.8 million
Luke Walton
5.7 million
Devin Ebanks
0.8 million
Steve Blake
4.0 million
• 2011-2012: Salaries of Los Angeles Lakers
• Find the five number salary
• Find the mean
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Choose a Summary
• Skewed Distribution
– Use 5 number summary
• Reasonably symmetric distribution – free of
outliers
– Mean and standard deviation (Since they are
strongly affected by outliers)
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The Influence of Outliers
on the Mean and Median
Larger influence on mean than median.
High outliers will increase the mean.
Low outliers will decrease the mean.
If ages at death are: 76, 78, 80, 82, and 84
then mean = median = 80 years.
If ages at death are: 46, 78, 80, 82, and 84
then median = 80 but mean = 74 years.
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Describing Spread: Range
and Interquartile Range
• Range = high value – low value
• Interquartile Range (IQR) =
upper quartile – lower quartile
• Standard Deviation
(covered later )
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Example 2.13 Fastest Speeds Ever Driven
Five-Number
Summary
for 87 males
•
•
•
Median = 110 mph measures the center of the data
Two extremes describe spread over 100% of data
Range = 150 – 55 = 95 mph
Two quartiles describe spread over middle 50% of data
Interquartile Range = 120 – 95 = 25 mph
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How to Handle Outliers
Outlier: a data point that is not
consistent with the bulk of the data.
• Look for them via graphs.
• Can have big influence on conclusions.
• Can cause complications in some
statistical analyses.
• Cannot discard without justification.
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Possible Reasons for Outliers
and Reasonable Actions
• Outlier is legitimate data value and represents natural
variability for the group and variable(s) measured.
Values may not be discarded — they provide important
information about location and spread.
• Mistake made while taking measurement or entering it
into computer. If verified, should be discarded/corrected.
• Individual in question belongs to a different group
than bulk of individuals measured. Values may be
discarded if summary is desired and reported for the
majority group only.
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Example 2.16 Tiny Boatsmen
Weights (in pounds) of 18 men on crew team:
Cambridge:188.5, 183.0, 194.5, 185.0, 214.0,
203.5, 186.0, 178.5, 109.0
Oxford:
186.0, 184.5, 204.0, 184.5, 195.5,
202.5, 174.0, 183.0, 109.5
Note: last weight in each list is unusually small.
They are the coxswains for their teams,
while others are rowers.
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Homework
•
•
•
•
•
Assignment:
Chapter 2 – Exercise 2.43 and 2.44
Chapter 2 – Exercise 2.74 and 2.81
Reading:
Chapter 2 – p. 37-46
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