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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 ushow many standard deviations a particular score is above or below the mean.
Since a z-score is in standard deviation units, converting a data
to a z-score is called____________standardizing.
If x is an observation from a distribution that has known mean
and known standard deviation , then the standardized value of x is:
xs
xxz
Observations larger than the mean have ________ z-scores, whereas
observations smaller than the mean have ________ z-scores.
positive
negative
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.
14.26.07
80-93z :Kerry
32.16.07
80-72z :Norman
Kerry’s score is 2.14 stand. dev. above the mean
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?
5.14
76-82 :Calculus
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 to fall at the _________ percentile. That is why you never see an ACT score reported
above the 99th percentile.
th100
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.
8888.25
22 Por
9292.25
23P
Example 4: Norman’s score was the 2nd lowest score. Determine his percentile.
808.25
2P
404.25
1or P
There is no simple method to convert a z-score to a percentile. The percentile
depends on the shape of the distribution. In a perfectly symmetrical distribution, the
mean _______ the median. Thus, in a perfectly symmetrical distribution a z-score of 0 will equal the ______ percentile. But,
in a left-skewed distribution, where the mean is ______ than the median, this
observation will be somewhere ________ the 50th percentile.
equals
50th
lessbelow
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
2
11100k
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
)11(100
)1
11(100
2
%75 %89.88 %75.93 %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 z-score. For that, we need more
advanced models known as ______________.density curves