Chapter 3 – Data Description
section 3.3 –Measures of Position
Measures of Position
Z-Scores



Different data sets can have vastly different
characteristics. (apples and oranges)
Z-Scores allow us to compare them.
a z-score (or standard score) for a value is obtained by
subtracting the mean from the value and dividing the result
by the standard deviation.
z
x

value  mean
z
standard deviation
xx
z
s
Measures of Position
Z-Scores
Example:
 a student scored 65 on a calculus test that had a mean of
50 and a standard deviation of 10.
 She scored 30 on a history test with a mean of 25 and a
standard deviation of 5.
 Compare her relative positions for the two tests
z
x

value  mean
z
standard deviation
xx
z
s
Measures of Position
Percentiles


Percentiles divide the data set into 100 equal parts.
SAT Score Reports usually give percentiles scores


a percentile score of 73 means that you did better than 73 percent
of everyone who took the test at the same time as you.
Percentile graphs are similar to Ogives

take a look at the graph on page 133…
Measures of Position
Percentiles
The percentile corresponding to a given value X is computed
by using the following formula:
(number of values below X)  0.5
percentile 
100%
total number of values
Measures of Position
Percentiles
(number of values below X)  0.5
percentile 
100%
total number of values
Example
 A teacher give a 20-point quiz to 10 students. Here are
the scores: 18, 15, 12, 6, 8, 2, 3, 5, 20, 10
 Find the percentile rank for a score of 12.
 So the kid who scored 12 did better than 65% of the class.

find the percentile rank for a score of 6.
Measures of Position
Percentiles
Example
 A teacher give a 20-point quiz to 10 students. Here are the
scores: 18, 15, 12, 6, 8, 2, 3, 5, 20, 10


Using the same data, find the score corresponding
to the 25th percentile.
what about the 60th percentile?
Measures of Position
Percentiles
Going backwards is a little different:
 Arrange the data from lowest to highest
 substitute into the formula c = np/100 where:




n = total number of values
p = percentile
If c is not a whole number, round up to the next whole
number. Count up to that number.
If c is a whole number, average the cth and (c+1)th values.
Measures of Position
Percentiles
Example
 A teacher give a 20-point quiz to 10 students. Here are the
scores: 18, 15, 12, 6, 8, 2, 3, 5, 20, 10


Using the same data, find the score corresponding to the 25th
percentile.
what about the 60th percentile?
Measures of Position
Quartiles

Quartiles divide the data into quarters.

We call the quartiles Q1, Q2, Q3

First find the median


Next find the “median” of the data that falls below Q2


this is Q2
this is Q1
Finally, find the “median” of the data that falls above Q2

this is Q3
Measures of Position
Quartiles



The Inter-quartile range (IQR) is another useful measure
of variability.
IQR = Q3 – Q1
This is the range of the middle 50% of the data.



we discussed resistant measures.
is the IQR resistant?
calculate the IQR for the quiz data.
Measures of Position
Outliers

an outlier is an extremely high or low data value when
compared to the rest of the data values.

Outliers are:



Data values smaller than Q1 – 1.5(IQR)
Data values larger than Q3 + 1.5(IQR)
Check this set for outliers:

5, 6, 12, 13, 15, 18, 22, 50
Practice

page 141 #1 – 29 odd!