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In this chapter, we will look at using the
standard deviation as a measuring stick
and some properties of data sets that are
normally distributed.
The number of standard deviations a particular
observation in a data set is away from the mean of that
set is called it
, and is denoted zx.
It is calculated as follows:
zx =
x-x
x -m
or zx =
s
s
If zx > 0, then the observation is above average
If zx < 0, then the observation is below average
z-scores have no units; when observations are changed
into their z-scores it is called
Once standardized, observations from different data sets
having different centers and spreads can be compared.
In the year 2009, cars sold in the United States had an
average of 135 horsepower with a standard deviation of 40
horsepower.
(a) Find the z-score for a car with 120 horsepower.
(b) Find the z-score for a car with 200 horsepower.
(c) A certain car has z-score of 1.2. How much horsepower does it have?
(d) A certain car has z-score of 0.6. How much horsepower does it have?
Adult Dalmatians have an average weight of 52 pounds
with a standard deviation of 6 pounds. Adult German
Shepherds have an average weight of 77 pounds with a
standard deviation of 3.6 pounds. Which is more unusual,
an adult Dalmatian weighing 63 pounds or an adult
German Shepherd weighing only 68 pounds?
If the histogram of a data set is unimodal and fairly
symmetric, then its distribution is called
l.
If such a distribution has mean m and standard
deviation , it can be modeled with a normal model, the
notation of which
, and the shape is below:
N ( m, sis)
 - 3
 - 2
-

+
 + 2
 + 3
The standard normal curve is N ( 0, 1) .
This is commonly called the z–curve.
It has shape below:
-3
-2
-1
0
1
2
3
For a normal distribution:
≈ 68% of values fall within one standard deviation of the mean
≈ 95% of values fall within two standard deviations of the mean
≈ 99.7% of values fall within three standard deviations of the mean