Download Measures of Position

Organizing Data
Measures of Position &
Exploratory Data Analysis
Essentials: Measures of Position
(Better understanding distribution shapes.)
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Know the types of measures used to look at specific positions within a
data distribution.
Be able to calculate the inter-quartile range, three quartiles, Pearson’s
Index of Skewness, z-score, Coefficient of Variation.
Be familiar with symmetry vs. skewness and distribution shapes.
Be able to build both traditional and modified box plots (aka: box-andwhiskers plot).
Measures of Position
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Measures of position are points within
the data that are used to describe
characteristics of the data. Percentiles,
Deciles, Quartiles, Minimum and
Maximum are among these points.
We will focus on 5 specific values...
The Five-Number-Summary
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Five numbers are frequently used to
indicate positions within a data set and
much more...
These points are:
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Minimum - Q1 - Median - Q3 - Maximum
Components of the
Five-Number Summary
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The Minimum and
Maximum values
represent the extremes
in the data set.
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Obtaining these points is
a simple matter of
looking for them.
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Q1, Median (Q2), and
Q3 represent points
within the data. They
are the 25th, 50th, and
75th percentiles,
respectively.
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Formulas are used to
locate these positions.
The Quartiles
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The Quartiles are obtained using the
following three formulas.
Q1(25th percentile) = (n+1)/4
Median (Q2, 50th percentile) = (n+1)/2
Q3 (75th percentile) = [3(n+1)]/4
(Where n = number of observations. Note that these formulas
identify POSITIONS within the data, not values; the data must
be placed in numeric order)
The Interquartile Range
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The Interquartile Range (IQR)
describes the middle 50% of the data.
It is obtained by the formula:
 IQR = Q3 - Q1
The Interquartile Range is used as a
measure of variation around the Median
of a set of data.
Quartiles divide a data set that has been
ordered from smallest to largest into four
sections, each containing 25% of the data values.
A term more familiar to you might be percentile.
For example, the 25th percentile,
50th percentile, or 75th percentile.
The Minimum value, First Quartile (Q1), Median
(Q2), Third Quartile (Q3) and Maximum value
comprise what is referred to as the Five Number
Summary. Each component of this summary
represents a measure of position within a set of data.
Together, these five values are referred to as the
Five Number Summary.
Anatomy of Measures of Position
440 481 482 483 483 514 514 554 554 554 562 612 623 631 638 664 671 677 690 707
Recall that the median in this data set is found
using the formula (n+1)/2 to obtain the POSITION of the median
(here the POSITION is (20 + 1)/2 = 10.5). Determining the value half way
between the 10th and 11th values yields a Median of 558: [(554 + 562)/2 = 558].
Another name for the median, is the second quartile (Q2) . It is the value
in the data set such that 50% of the values are lower than it,
The third quartile (Q3) is the value
and 50% of the values are higher than it.
such that 75% of the values are lower
than it, and 25% of the values are higher than it.
To find this value apply the formula [3(n+1)/4] to obtain
The first quartile (Q1) is the value
the POSITION of the third quartile. Here the formula yields
such that 25% of the values are lower
POSITION 15.75: [(3(20+1))/4 = 15.75]. Determine the value ¾
than it, and 75% of the values are higher than it.
of the way between the 15th value (638) and the 16th value (664)
To find this value apply the formula (n+1)/4 to obtain
by determining the difference between these two
the POSITION of the first quartile. Here the formula yields
numbers (664 - 638 = 26); multiplying the
POSITION 5.25: [(20+1)/4 = 5.25]. Determine the value ¼
difference
by .75 (26 * .75 = 19.5); and adding
of the way between the 5th value (483) and the 6th value (514)
this value to the smaller number
by determining the difference between these two
(638 + 19.5 = 657.5 = Q3)
numbers (514 – 483 = 31); multiplying the
The Interquartile Range is the
difference by .25 (31 * .25 = 7.75); and adding
difference between the third quartile,
this value to the smaller number
and the first quartile. Here, the
(483 + 7.75 = 490.75 = Q1)
interquartile range is
657.5 - 490.75 = 166.75
Exploring and Comparing Data
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Exploratory Data Analysis (EDA): EDA is
the process of using statistical tools (graphs,
measures of center, variation, and position)
to investigate data sets in order to
understand their important characteristics.
Box-and-Whiskers Plot
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a.k.a. Box plot
A Box plot is used to display the fivenumber-summary. One can also
examine the shape of the distribution
with a Box plot.
Data Presentation:
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Box plot: Traditional vs. Modified Box plot
Display of Outliers and Adjacent Values
Anatomy of a Traditional Box plot
DUDLEY’S DOUGHNUTS
(flour in pounds, used on 20 consecutive days during the
month of Dec. 1999)
This is Q3, the
third quartile. Here,
Q3 is 657.5.
Title
800
This is the Upper
Whisker. It is the
maximum value in the
data set. Here, the
maximum value is 707.
700
600
This is the
Median (Q2). Here, the
median is 558.
500
400
N=20
20
Flour (in lbs.) Used
This is the
Lower Whisker. It is the
minimum value in the data
set. Here, the minimum
value is 440.
This is Q1, the first
quartile. Here, Q1 is 490.75.
Flour (in lbs.) Used
440, 481, 482, 483, 483, 514, 514, 554, 554, 554,
562, 612, 623, 631, 638, 664, 671, 677, 690, 707
Historical Note
Who invented this useful tool for quick data
analysis?
John Tukey
Statistician
Outliers, Limits, and Adjacent Points
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An Outlier is a data point found at one of the
extremes of the data, and is well outside the general
pattern of the data.
Upper and Lower Limits: Used as a tool to identify
observations that may be outliers.
Lower Limit = Q1 - 1.5(IQR)
 Upper Limit = Q3 + 1.5(IQR)
Adjacent Points: The last data value that occurs before (or at)
the Upper or Lower Limit. In modified Box plots the whisker
would stop at this data value rather than being drawn out to
1.5(IQR).
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Mean Price of a Movie Ticket for a Sample
of 12 U.S. Cities
8
US
4
5
6
7
8
Example of a Modified Box Plot
9
Grouped Box plots
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A grouped box plot is a good way to
visualize differences/similarities between
groups.
Symmetry
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Symmetry – a distribution is symmetric if the left half of the
distribution is roughly a mirror image of its right half.
Skewness – a distribution is skewed if it is not symmetric and
if it extends more to one side than the other
Mode
=
Mean
=
Median
SYMMETRIC
Mean
Median
Mode
SKEWED LEFT
(negatively)
Mode
Median
Mean
SKEWED RIGHT
(positively)
Pearson’s Index of Skewness
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Skewness can be measured using
Pearson’s Index of Skewness.
I
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3(x  median )
s
Example: Given a set of date whose statistics include: mean =
40, median = 41, S.D = 4, determine if this distribution id
skewed.
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I = (3(40 - 41))/4 = -.75
Given that the value is within the range from -1 to 1, inclusive, this
distribution would not be considered to be significantly skewed.
Pearson’s Index of Skewness
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When symmetric, I = 0.
Values usually range from –3 to +3.
A distribution is considered symmetric if the index
value is between -1 and +1
If the index value (I) is less than -1 the data are
negatively (left) skewed.
If the index value is greater than +1 The data are
positively (right) skewed.
Pearson’s Index of Skewness:
Example
Use Pearson’s Index of Skewness to
determine if the distribution of 406
automobile
weights is approximately normally
distributed or does it display a degree of
skewness?
Mean Wt: 2969.56 lb.
Median Wt: 2811.00 lb.
Standard Deviation: 849.83 lb.
Measures of Position
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Standard Scores: a standard score, or
z-score is the number of standard
deviations that a given value x is above or
below the mean. To find a z-score
For populations:
z  x 

For samples:
z  xx
s
(Always round z to two decimal places.)
Standard Scores (z-score)
z  xx
s
Recall, a z-score is a measure of a value’s distance
away from a distribution’s mean as measured in
standard deviations.
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Example: Ozzie just took two tests. Given his scores, the mean for the
tests, and the standard deviations, on which test did Ozzie perform
better relative to the other students?
Calculus Exam: Grade = 65, class mean = 50, S.D. = 10
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History exam: Grade = 30, class mean = 25, S.D. = 5
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z = (65 - 50)/10 = 1.5 (or 1.5 standard deviations above the mean)
z = (30 - 25)/5 = 1 (or 1 standard deviation above the mean)
Since the z-score for the calculus exam is larger, Ozzie’s relative
position is higher in the calculus class than it is in the history class.
Coefficient of Variation
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Allows us to compare standard deviations.
The result is expressed as a percentage.
CVar 
s
_
100
x
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Example:Trinity’s test statistics included: Anthropology test - mean of
50 and S.D. of 10; Music test - mean of 40 and S.D. of 5. Which test
showed greater variation in test scores?
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Anthropology: (10/50)*100 = 20%
History: (5/40)*100 = 12.5%
Thus, there was greater variation in test scores for the Anthropology test.
Coefficient of Variation: Example
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The heights and weights and ages of the starting members of the 2008
World Champion Boston Celtics are noted below. Determine the
coefficient of determination for these variables to determine which has
the greatest variation.
Ray Allen:
Rajon Rondo:
Paul Pierce:
Kevin Garnett:
Kendrick Perkins:
77
73
79
83
82
in.,
in.,
in.,
in.,
in.,
205
171
235
253
280
lb.,
lb.,
lb.,
lb.,
lb.,
33
21
30
32
23
yrs.
yrs.
yrs.
yrs.
yrs.