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Lesson 2.3 Density Curves and the The empirical rule provides quick estimates of the per-
Normal Distribution centage of obs ervations in certain intervals for normal
distributions.
Data distributions that display a recognizable pattern can Because normal distributions have special properties,
be modeled with a density curve. All density curves must
it’s important to assess whether a distribution is approxi-
• lie on or above the horizontal axis, and mately normal or not. A basic plan to assess the normality
• have an area of exactly 1 between the curve and the of a distribution is to examine a graph of the data and
horizontal axis. check the empirical rule. A distribution can be considered
approximately normal if it has
The area under the curve and above an interval will be
approximately equal to the proportion of observations within • a graph that looks approximately symmetric and bell-
that interval. The shapes of density curves can be described shaped, and
with the same terminology as the shapes of distributions of • about 68%, 95%, and 99.7% of its values fall
data. When a density curve is skewed left, the mean is less within 1, 2, and 3 standard deviations of the mean,
than the median and when it is skewed right, the mean is respectively.
greater than the median. The mean and median will be equal
for density curves that are symmetric.
One special family of density curves is normal density Lesson 2.5 Normal Distributions: Finding
curves. Normal density curves are symmetric, bell-shaped, Areas from Values
and single-peaked. The mean µ and median of a normal Although all normal distributions have their own mean and
distribution are equal and are located directly beneath the standard deviation, they can be standardized by converting
peak of the density curve. The standard deviation σ of a x − µ
z
normal distribution measures the variability of the distri- each value into a standardized score (z score): = .
σ
bution. In a normal distribution, the inflection points of Transforming the values results in a new normal distribu-
the density curve are ocated one standard deviation to the tion called the standard normal distribution. The standard
l
left and right of the mean.
normal distribution has a mean of 0 and a standard devi-
ation of 1. The areas under normal density curves do not
Lesson 2.4 The Empirical Rule and conform to any standard geometric shapes. So, a table of
standard normal probabilities (Table A) or technology can
Assessing Normality
be used to find the areas. The table only reports the left-tail
The empirical rule (the 68–95–99.7 rule) states that in a area, so some simple calculations can be used to find right-
normal distribution: tail areas and areas between two z-scores.It is also possible
to use technology to find these areas without standardizing.
• about 68% of the observations fall within one standard The area above an interval and under a normal curve rep-
deviation of the mean (between µ − σ and µ + ),
σ
resents the approximate percentage of values that fall in that
• about 95% of the observations fall within two standard interval. Only three basic types of areas exist: left-tail areas,
σ
σ
deviations of the mean (between µ − 2and µ + 2 ), and right-tail areas, and “between” areas (areas between two
• about 99.7% of the observations fall within three standard boundary values).
σ
σ
deviations of the mean (between µ − 3 and µ + 3 ).

CHAPTER 2 • Modeling One-Variable Quantitative Data 2-5

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