Download Lesson 21 Sep PS

Prob and Stats, Sep 21
The Empirical Rule and the Z-Score
Book Sections: 7.1 & 7.2
Essential Questions: How do I compute and use statistical values? What
is the Empirical Rule? What is a Z-Score and how can I compute
and use it? How are Z-Scores tied to the Empirical Rule?
Standards: PS.SPID.1, .2, .3
The Empirical Rule
• A symmetric data set (mean and median about equal) has a
consistent distribution that relates the mean, standard
deviation, and a percent of data interval for all such
distributions.
• The Empirical Rule and Z-Scores only apply to symmetric
distributions.
TheWords
Empirical
Rule
and Graph Together
• 68% of the data lies within
one standard deviation of the
mean
• 95% of the data lies between
two standard deviations of the
mean
• 99.7% of the data lies
between three standard
deviations of the mean
Standard Deviation Thresholds
• 68% - 1 x Standard deviation
• Range ( x - s, x + s)
• 95% - 2 x Standard deviations
• Range ( x - 2s, x + 2s)
• 99.7% - 3 x Standard deviations
• Range ( x - 3s, x + 3s)
Example
• In a symmetrically distributed data set with a mean of 12
and a standard deviation of 3, what is the range of the
middle 99.7% of the data?
Empirical Rule Problems Backwards
• Based on deviations (1, 2, or 3), you are trying to list a
range containing a percent.
• Given a range, determine a deviation (1, 2, or 3) that
corresponds to a percent. Solve this equation for d to
determine the deviation:
x  sd  upper _ bound
If d = 1, ans = 68%; d = 2, ans = 95%; d = 3, ans = 99.7%.
Examples
• In a symmetrically distributed data set with a mean of 40
and a standard deviation of 8, what percent of that data lies
x  sd  upper _ bound
between 16 and 64?
• What is the range of the middle 95% of the data in a symmetrically
distributed data set with a mean of 25 and a standard deviation of 2.5?
Example
• In a symmetrically distributed data set with a mean of 100
and a standard deviation of 15, what percent of that data lies
x  sd  upper _ bound
between 70 and 130?
Example
• In a symmetrically distributed data set with a mean of 77
and a standard deviation of 12, what is the range of the
middle 68% of the data?
The Standard Score
• The standard score, or z-score (z) is a number that
is representative of the number of standard
deviations a given value of x falls from the mean.
__
z=
(Value – Mean)
Standard Deviation
or
(x  x )
z
s
In this scheme, you need a mean and standard deviation. The x is
the value you want the z-score of. Now its plug and chug.
Properties of the Z-Score
• A z-score can be positive, negative, or zero
• Positive – the x-value was greater than the mean
• Negative – the x-value was less than the mean
• Zero – the x-value was equal to the mean
• A z-score
can be used to identify an unusual value
of a data set that is symmetrically distributed
The Purpose of the Z-Score
• A z-score tells you exactly how many standard
deviations a value is above or below the mean in a
symmetrically distributed data set.
• Positive is above the mean, negative is below the
mean.
The z-score transforms the mean to 0 and the standard deviation to 1.
Finding Z-Scores
• The weight of polar bears in or near the town of Barry,
Canada is symmetrically distributed with a mean weight of
840 pounds with a standard deviation of 67 pounds.
Compute the z-score for Slim, a young male bear captured
and tagged, who weighed 725 pounds.
Example
• A symmetrically distributed data set has a mean of 69 and
a standard deviation of 7. Compute the z-score for a value
of 60.
Example
• Cars driving along a stretch of I-85 have a mean speed of
71 mph and a standard deviation of 3 mph. What is the zscore of a car clocked here going 81 mph?
The Empirical Rule with Z-Scores
99.7% of Data
95% of Data
68% of
Data
In the world of z-scores, mean is always 0 and standard deviation is 1.
Interpreting Z-Scores
• It is rare to have a Z-Score smaller than -3 or larger than 3.
• It is nearly impossible to have a Z-Score outside the range of [-6, 6].
 That would be 6 standard deviations away from the mean.
Class work: CW 9/21/16, 1-15
Homework: None