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Normal Distribution: Empirical Rule
The Normal Distribution is a continuous distribution. This is probably
the most important probability distribution.
Graphs of Normal Probability Distribution
Properties of a normal curve
1. The curve is bell-shaped
with highest point over
the mean µ.
2. The curve is
symmetrical about a
vertical line through µ.
3. The curve approaches
the horizontal axis but
never touches or
crosses it.
4. The inflection
(transition) points
between cupping up
and down occur at µ - σ
and µ + σ.
5. The total area under the
curve is 1.
The mean µ and standard deviation
σ control the shape of the normal
curve. The curves on the right show
examples of normal curves of
different height.
The larger the value for σ, the
fatter the curve will be. A normal
curve with a very small standard
deviation (σ) will appear to be very
narrow.
Applying Chebyshev’s Theorem to the normal distribution we get the
Empirical Rule. For a distribution that is symmetrical and bell-shaped
1. Approximately 68% of the data values will lie within one
standard deviation on each side of the mean.
2. Approximately 95% of the data values will lie within two
standard deviation on each side of the mean.
3. Approximately 99.7% of the data values will lie within three
standard deviation on each side of the mean.
Examples showing the empirical rule:
1. The mean value of equipment
from an inventory sample is
µ = $1500 with a standard
deviation σ = $200. The data
set is a normal distribution.
Estimate the percent of the
equipment whose values are
between $1300 and $1700.
The interval is 1500 – 200
and 1500 + 200 which
translates to µ - 1σ and µ +
1σ. Using the empirical rule,
68% of the equipment falls
in this interval.
2. Using µ = $1500 with a
standard deviation σ =
$200, between what 2
values does 95 % of the
values fall.
From the empirical rule this
would be µ - 2σ and µ + 2σ.
The calculations would be
1500 – 2(200) and 1500 +
2(200) or $1100 and $1900.
3. Using µ = $1500 with a
standard deviation σ =
$200, between what 2
values does 99.7% of the
values fall.
From the empirical rule this
would be µ - 3σ and µ + 3σ.
The calculations would be
1500 – 3(200) and 1500 +
3(200) or $900 and $2100.