ap statistics: section 2.1 a. measuring relative standing: z-scores a z-score describes a particular...

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