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Math 160 - Cooley
Intro To Statistics
OCC
Section 6.1 – Introducing Normally Distributed Variables
Density Curve – A smooth curve that represents the distribution of a continuous variable
Basic Properties of Density Curves
Property 1: A density curve is always on or above the horizontal axis.
Property 2: The total area under a density curve (and above the horizontal axis) equals 1.
Variables and Their Density Curves
For a variable with a density curve, the percentage of all possible observations of the variable that lie within any
specified range equals (at least approximately) the corresponding area under the density curve, expressed as a
percentage.
Normally Distributed Variable
A variable is said to be a normally distributed variable or to have a normal distribution if its distribution has the
shape of a normal curve.
Normally Distributed Variables and Normal-Curve Areas
For a normally distributed variable, the percentage of all possible observations that lie within any specified
range equals the corresponding area under its associated normal curve, expressed as a percentage. This result
holds approximately for a variable that is approximately normally distributed.
Standard Normally Distribution; Standard Normal Curve
A normally distributed variable having mean 0 and standard deviation 1 is said to have the standard normal
distribution. Its associated normal curve is called the standard normal curve, which is shown below.
Standardized Normally Distributed Variable
The standardized version of a normally distributed variable x,
z
x

has the standard normal distribution.
 Exercises:
1)
The area under the density curve that lies to the left of 10 is 0.654.
a)
What percentage of all possible observations of the variable are less than 10?
b)
What percentage of all possible observations of the variable are at least 10?
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Math 160 - Cooley
Intro To Statistics
OCC
Section 6.1 – Introducing Normally Distributed Variables
 Exercises:
2)
Sketch the normal distribution with mean 32 and standard deviation of 6.
3)
Sketch the normal distribution with µ = –3 and σ = 2.
4)
From the paper “Effects of Chronic Nitrate Exposure on Gonad Growth in Green Sea Urchin
Strongylocentrotus Droebachiensis” by S. Siikavuopio et al., we found that weights of adult green sea
urchins are normally distributed with mean 52.0 g and standard deviation 17.2 g. Let x denote weight of
adult green sea urchins.
a) Sketch the distribution of the variable x.
b) Obtain the standardized version, z, of x.
c) Identify and sketch the distribution of z.
d) The percentage of adult green sea urchins with weights between 50 g and 60 g is equal to the area
under the standard normal curve between ________ and ________.
e) The percentage of adult green sea urchins with weights above 40 g is equal to the area under the
standard normal curve above ________.
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