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Daniel S. Yates
The Practice of Statistics
Third Edition
Chapter 2:
Describing Location in a Distribution
Section 2.1 Measures of Relative
Standing and Density Curves
Copyright © 2008 by W. H. Freeman & Company
Objectives
• What are measures of relative standing?
• What is a Standardize Value?
• How do you compute a z-score of an
observation given the mean and standard
deviation of a distribution?
• What does the z-score measure?
• How do you find the pth percentile of an
observation in a data set?
• What is a Mathematical Model?
• What is a Density Curve?
Measures of Relative Standing
• Suppose we have a data set of grades for
Algebra 2AB Chapter Test :
( 94, 61, 40, 72, 73, 88, 68, 62, 73, 57, 35, 82,
48, 66, 65, 79, 45, 91, 66, 71, 63, 11, 69, 64, 38,
59, 70, 70, 79, 77, 39, 55)
• We can discuss a particular student’s grade
relative standing in the class in two ways:
– Relative to the median (percentile)
– Relative to the mean (how far way from the mean)
pth Percentile
• Pth percentile of a distribution – the value
with p percent of the observations less
than or equal to the observation in
question.
• For example we are interested in the
percentile for a test grade of 45.
• Data sorted:
(11,35, 38, 39, 40, 45, 48, 55, 57, 59, 61,
62, 63, 64, 65, 66, 66, 68, 69, 70, 70, 71,
72, 73, 73, 77, 79, 79, 82, 88, 91, 94 )
• 6/32 x 100 = 18.75%
Your turn!
• Data sorted:
(11,35, 38, 39, 40, 45, 48, 55, 57, 59, 61,
62, 63, 64, 65, 66, 66, 68, 69, 70, 70, 71,
72, 73, 73, 77, 79, 79, 82, 88, 91, 94 )
• What is the percentile for a grade of 66?
• 17/32 x 100 = 53.1%
• What is the 50% percentile?
• Grade of 66.
Relative to the Means of the Data
Set
• We standardize each data by:
• The standard value (z-score) is a measure of
how many standard deviations a data value is
from the means of the data set.
Back to Our Example
• Data sorted:
(11,35, 38, 39, 40, 45, 48, 55, 57, 59, 61, 62, 63, 64, 65,
66, 66, 68, 69, 70, 70, 71, 72, 73, 73, 77, 79, 79, 82, 88,
91, 94 )
• What is the standard value for a test grade of 45?
• First we need to find the mean and standard deviation.
• Mean = 63.4375 and Sx = 17.8234
x  mean
45  63.4375
z

 1.034
standard deviation
17.8234
• What is the z-score for a grade of 66?
• z = 0.144
Calculator Exercise
• We will convert all the grades to z-score.
• Data sorted:
(11,35, 38, 39, 40, 45, 48, 55, 57, 59, 61,
62, 63, 64, 65, 66, 66, 68, 69, 70, 70, 71,
72, 73, 73, 77, 79, 79, 82, 88, 91, 94 )
Impact on the Distribution When we
Standardize
We need to look at the mean and standard deviation, to
find out what is the impact.
Recall:
• Linear Transformation: xnew = a + bx
• When we add (or subtract) a constant from each
data we move the distribution by that amount but we
do not change the spread.
• When we multiply (or divide) each data by a
constant we change the spread. We can quickly
compute the new standard deviation by multiplying
the old standard deviation by dividing it by the
constant.
Impact Continued
So looking at the formula to convert data to a standard
value
x  mean
z
standard deviation
we can see we are moving the distribution by a
constant and by dividing the standard deviation we
are changing the spread.
If x = mean, then mean – mean = 0. The new
mean is 0.
By dividing by the standard deviation we are changing the
standard deviation to 1 since standard deviation
standard deviation
1
Data Analysis Toolbox (p123)
When describing a distribution –
1. Always plot the data.
2. Look for overall pattern (shape, center,
spread) and striking deviations such as
outliers.
3. Calculate a numerical summary to
describe center and spread.
4. For large data sets, can we fit a smooth
curve to the distribution.
For Example
The smooth curve are an idealized description (mathematical
model) for the distribution.
Smooth curves are easier to work with than histograms
Density Curves
When we adjust the scale so that the
area underneath the curve is one we
have density curve.
Definition :
Density Curves
Density Curves come in different shapes , but they all have
the same area of 1.
Density Curve Parameters
Since density curves are idealized descriptions of
the data distribution we use different symbols to
represent:
Sample
Density curve
Mean
x
µ
Standard Deviation
Sx
σ
Algebra 2 Grades
The Median and Mean of a Density
Curve
Problem 2.12 page 128
(a) Mean C, Median B
(b) Mean A, Median A
(c) Mean A, Median B