Download Summarizing Data: Theoretical models A density curve is a curve

Chapter 3
1
Summarizing Data: Theoretical models
A density curve is a curve that models the shape of
the histogram of a data distribution.
The area between the curve, the horizontal axis, and
any two vertical lines at x = a and x = b measures the
(theoretical) proportion of the data that lies between
x = a and x = b. The total area underneath the curve
measures 1 = 100%.
The mean of a distribution is the point at which the
curve would balance on its baseline if it were a physical
object.
The median of a distribution is the point which
separates the area under the curve into two equal
halves.
The mean and median of a symmetric density curve are
equal. The mean of a skewed density curve lies further
out in the tail of the distribution than the median.
The theoretical mean associated with a density curve is
labeled µ (Greek letter mu). The theoretical standard
deviation is denoted σ (Greek letter sigma).
Chapter 3
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Summarizing Data: The Normal Distribution
The most common and important type of density curve
is a normal curve.
• It faithfully describes many real data sets.
• It approximates important types of chance
outcomes.
• It is the basis for most statistical inference
techniques.
A normal curve is symmetric and bell-shaped. Its mean
µ lies at the position of the central peak. The points on
either side of the mean at which the curve changes
concavity are a distance σ from the mean. We denote
the normal distribution with N(µ, σ).
[TI-83: If a quantity x satisfies a normal distribution
N(µ, σ), then the proportion of values of x which lie
between two particular values a and b ( a ≤ x ≤ b ) is
computed as
DISTR normalcdf( a , b , µ , σ )
If no upper bound b is given, it is understood that b = ∞;
on the calculator, use 1E99 for ∞; if no lower bound is
given, it is understood that a = –∞; on the calculator,
use -1E99 for –∞.]
Chapter 3
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The 68–95–99.7 Rule
For the normal distribution,
• 68% of the data will lie within one standard
deviation of the mean
(between µ – σ and µ + σ);
• 95% of the data will lie within two standard
deviations of the mean
(between µ – 2σ and µ + 2σ);
• 99.7% of the data will lie within three standard
deviations of the mean
(between µ – 3σ and µ + 3σ).
Chapter 3
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Describing Data: Measuring Relative Standing
• The z-score, or standardized score, of an
observation x in a normal distribution is its
distance from the mean,
−µ
z = xσ
measured in terms of standard deviations; that is,
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x = µ + zσ . Positive z-scores lie above the mean,
negative z-scores below the mean.
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• The standard normal distribution N(0, 1) is the
one with mean 0 and standard deviation 1. If x is a
variable with normal distribution N(µ, σ), then the
corresponding standardized variable z has the
standard normal distribution.
[TI-83: DISTR normalcdf( a , b )]
• extreme z-scores
values z* of z which satisfy certain probability
conditions of the form P(z ≤ z*) = P for a given
proportion value P.
[TI-83: DISTR invNorm( P ) ]