64, 95, 99.7

The normal distribution is one of the most
important distributions. The histogram has
this general shape.

When the histogram of the normal
distribution is smoothed to form a curve, the
curve is bell-shaped. This curve is called a
normal curve and is used to model the
normal distribution.

The bell can vary in size but they all have the
same basic properties.
◦ The curve is bell-shaped with the highest point at
the mean µ.
◦ The curve is symmetrical about a vertical line 𝑥 = 𝜇.
◦ The mean, median, and mode are all equal.
◦ 50% of the data values of the distribution are to the
right of the mean µ; 50% of the data values are to
the left of the mean µ.


Approximately 68% of the data values fall between 𝜇 − 𝜎
and 𝜇 + 𝜎; that is, 68% are between one standard
deviation less and one standard deviation more than the
mean.
The normal curve model approaches the horizontal axis,
but never touches or crosses the axis.


Normal curves give us an idea of how
extreme a value is by telling us how likely it is
to find one that far from the mean.
We can find these numbers precisely, but
until then we will use a simple rule that tells
us a lot about the normal curve.



About 68% of the values fall within one
standard deviation ơ of the mean µ.
About 95% of the values fall within two
standard deviations ơ of the mean µ.
About 99.7% (almost all!) of the values fall
within three standard deviations ơ of the
mean µ.

The following shows what the 68-95-99.7
rule looks like on the graph.


Data is considered “normal” if it falls within
two standard deviations ơ of the mean µ, or
within the central 95% of the curve.
Data is considered “unusual” if it falls outside
two standard deviations ơ of the mean µ, or
within the upper 2.5% and the lower 2.5% of
the curve.

pg 866 #3 & 4