### Powerpoint 2.1 A

```AP Statistics:
Section 2.1 A
Measuring Relative Standing: z-scores
A z-score describes a particular
data value’s position in relation to
the rest of the data.
In particular, a z-score tells us
how many standard deviations a
particular score is above or below
the mean.
Since a z-score is in standard
deviation units, converting a data
standardizing.
to a z-score is called____________
If x is an observation from a
distribution that has known mean
and known standard deviation ,
then the standardized value of x is:
xx
z
sx
Observations larger than the mean
have ________
positive z-scores, whereas
observations smaller than the
negative z-scores.
mean have ________
Example 1: Kerry earned a 93 on the Chapter 1
test and Norman earned a 72. The median and
mean for the class were both 80 and the
standard deviation was 6.07. Determine, and
interpret their respective z-scores.
93 - 80
Kerry : z 
 2.14
6.07
Kerry’s score is 2.14
stand. dev. above the
mean
72 - 80
Norman : z 
 1.32
6.07
Norman’s score is
1.31 stand. dev.
below the mean.
Example 2: Kerry earned a scored of 82 in his
calculus class. The class had a mean of 76 and a
standard deviation of 4. Did he do worse on this
test than on his stats test?
82 - 76
Calculus :
 1.5
4
Compared to the class, Kerry did worse on his
calculus test, because he is only 1.5 stand. dev.
above the mean.
Measuring Relative Standing:
Percentiles
In Chapter 1, we defined the pth percentile
of a distribution as the value that has p
percent of the observations fall at or below
it. Some people define the pth percentile
of a distribution as the value with p
percent of observations below it. Using this
definition, it is impossible for an individual
th
to fall at the _________
100 percentile. That is
why you never see an ACT score reported
above the 99th percentile.
As long as you use a definition for
percentile that is common use, you
will use receive full credit on the
AP exam.
Example 3: Jenny’s score on her stats test was
the 3rd highest score in the class. If there are 25
students in the class, determine her percentile.
23
 .92  P92
25
22
or
 .88  P88
25
Example 4: Norman’s score was the 2nd
lowest score. Determine his percentile.
2
 .08  P8
25
1
or
 .04  P4
25
There is no simple method to convert a zscore to a percentile. The percentile
depends on the shape of the distribution.
In a perfectly symmetrical distribution, the
mean _______
equals the median. Thus, in a
perfectly symmetrical distribution a z-score
of 0 will equal the ______
50th percentile. But,
in a left-skewed distribution, where the
mean is ______
less than the median, this
observation will be somewhere ________
below
the 50th percentile.
There is a theorem that describes
the percent of observations in any
distribution that falls within a
specified number of standard
deviations of the mean. It is know
as Chebyshev’s Inequality.
In any distribution,
the percent of observations falling
within k standard deviations of the
mean is
1

 1001  2 
 k 
Example 5: Complete the following table to
determine the percent of observations in any
distribution that must fall within k standard
deviations of the mean.
0%
1
100(1  2 )
1
100(1  1)
75 %
88.89% 93.75%
96 %
Chebyshev’s Inequality gives us some
insight into how observations are
distributed within distributions. It
does not help us determine the
percentile corresponding to a given zscore. For that, we need more