Download 3-4

Here is the data for the class heights
in inches:
69 71 70 67 71 61 68 64 63 64 71 63
64 74 63 72 74 64 69 64 140 64 68
65 65 65 67 68 75 78
What can we conclude from…
a) Chebyshev’s Thereom?
b) The Empirical Rule?
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Section 3-4
Measures of Relative
Standing
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Key Concept
This section introduces measures that can be
used to compare values from different data
sets, or to compare values within the same
data set. The most important of these is the
concept of the z score.
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z
Definition
Score (or standardized value)
the number of standard deviations
given value x is above or below
the mean
that a
-Can be used to compare values from different or the same
data sets
-Found by converting a value to a standardized score
-No units
-Measure of position: describes location of a value (in
terms of standard deviations) relative to the mean
EX: z=2.0: A value is 2 standard deviations above the mean
z = -3.0: A value is 3 standard deviations below the
mean
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Measures of Position z score
Sample
x
x
z= s
Population
x
µ
z=

Round z to 2 decimal places
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Interpreting Z Scores
Whenever a value is less than the mean, its
corresponding z score is negative
Ordinary values:
Unusual Values:
z score between –2 and 2
z score < -2 or z score > 2
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Notice: Unusual/usual scores
• In 3-3, we used the range rule of thumb
to conclude that a value is “unusual” if
it is more than ____ standard deviations
away from the mean.
• It follows that unusual values have z
scores less than _____ or greater than
_____.
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Example
Over the past 30 years, heights of
basketball players at Newport
University have a mean of 74.5 in. and a
standard deviation of 2.5 in. The latest
recruit has a height of 79.0 in.
1) Find the z-score.
2) Is the height of 79.0 in. unusual among
the heights of players over the past 30
years? Why or why not?
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Example
Our tallest president was Lyndon Johnson at
75”. The tallest Miami Heat basketball player
is Shaquille O’Neil at 85”. These guys are
from very different populations; find who is
relatively taller among their populations.
Standardize their scores.
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Definition
 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%.
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Quartiles and Percents
• Used for comparing values within
the same data set
• Also measures of position
• Useful for comparing values within
the same data set or between
different sets of data.
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Quartiles
Q1, Q2, Q3
divide ranked scores into four equal parts
25%
(minimum)
25%
25% 25%
Q1 Q2 Q3
(maximum)
(median)
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Percentiles
Just as there are three quartiles
separating data into four parts, there
are 99 percentiles denoted P1, P2, . . .
P99, which partition the data into 100
groups.
Process
-Sort data (ascending order)
-Count total data = n
-Count # of values less than the score
you are looking for)
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Finding the Percentile
of a Given Score
Percentile of value x =
number of values less than x
• 100
total number of values
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Converting from the kth Percentile to
the Corresponding Data Value
Notation
L=
k
100
•n
n
k
L
Pk
total number of values in the data set
percentile being used
locator that gives the position of a value
kth percentile
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Example
• Find the value of the 30th percentile =
P30.
• K = ___ since we are trying to find the
value of the ____th percentile.
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Converting from the
kth Percentile to the
Corresponding Data Value
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Some Other Statistics
 Interquartile Range (or IQR): Q3 - Q1
 Semi-interquartile Range:
Q3 - Q1
2
 Midquartile:
Q3 + Q1
2
 10 - 90 Percentile Range: P90 - P10
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