Download Standard Deviation as a Ruler • The z

Chapter 6
1
Standard Deviation as a Ruler
• The z-score, or standardized value, of an
observation y is its distance from the mean
measured in units of standard deviations;

y− y
z=
.
s
Positive z-scores lie above the mean, negative zscores below the mean. A z-score is unchanged
even after €
shifting or rescaling the data
Chapter 6
2
The Normal Distribution
The most commonly occurring distributions are
symmetric and “bell-shaped”. Mathematicians
have devised a theoretical model for such
distributions, the normal model. It faithfully
describes many real data sets and is the basis for
most statistical inference techniques.
A normal curve is symmetric and bell-shaped, and
is characterized by the position of its mean and the
size of its standard deviation. Its mean, always
denoted µ (mu, the Greek letter m), lies at the
position of the central peak. The points on either
side of the mean at which the curve changes
concavity are located one standard deviation away
from the mean. We denote the standard deviation
of a normal model by σ (sigma, the Greek letter s).
The normal distribution with mean and standard
deviation is denoted N(µ, σ).
Chapter 6
3
The 68–95–99.7 Rule
For the normal model,
• 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).
In situations where the normal model is applied to
a given situation, sketch a graph of the model and
identify the appropriate scale by indicating on the
horizontal axis the seven values
µ – 3σ, µ – 2σ, µ – σ, µ, µ + σ, µ + 2σ, µ + 3σ.
Chapter 6
4
Working with the Normal Model
• More general percentages associated with the
normal model N(µ, σ) can be found with your
calculator (or other statistical software): the
percentage of values lying between two particular
values a and b ( a ≤ y ≤ b ) is computed as
DISTR normalcdf( a , b , µ , σ )
(If no upper bound b is given, it is understood that
b = ∞ — use 1E99 for ∞; if no lower bound is given,
it is understood that a = –∞ — use -1E99 for –∞.)
• The standard normal distribution N(0, 1) has
mean 0 and standard deviation 1. If a normal
model N(µ, σ) applies to a data set, then the
corresponding standardized values z will obey the
standard normal distribution N(0, 1). Percentages
associated with the standard normal model N(0, 1)
can also be found with your calculator (or other
statistical software): the percentage of values lying
between two particular values a and b ( a ≤ y ≤ b )
is computed as
DISTR normalcdf( a , b )
[TI-83: DISTR normalcdf]
Chapter 6
5
• Technology can also be used to work with the
inverse problem: to determine data values or zscores that correspond to particular percentages
[TI-83: DISTR invNorm]
Checking for normality
• The Nearly Normal Condition
If a data set has unimodal and symmetric shape, we
may be able to apply the normal model to its study;
check by consulting a histogram or a normal
probability plot
[TI-83: STATPLOT Type]