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Using the Empirical
Rule
Remember: Normal Distributions

These are special density curves.

They have the same overall shape
 Symmetric
 Single-Peaked
 Bell-Shaped

They are completely described by giving its
mean () and its standard deviation ().

We abbreviate it N(,)
Normal Curves….
•Changing the mean without changing the standard
deviation simply moves the curve horizontally.
•The Standard deviation controls the spread of a Normal
Curve.
Standard Deviation

It’s the natural measure of spread for Normal
distributions.

It can be located by eye on a Normal curve.
 It’s
the point at which the curve changes from concave
down to concave up.
Empirical Rule (68-95-99.7 Rule)

In the Normal distribution with mean ()
and standard deviation ():
1 of  ≈ 68% of the observations
 Within 2 of  ≈ 95% of the observations
 Within 3 of  ≈ 99.7% of the observations
 Within
Scores on the Wechsler adult Intelligence Scale (a
standard IQ test) for the 20 to 34 age group are
approximately Normally distributed. N(110, 25).

What percent are between 85 and 135?

What percent are below 185?

What percent are below 100?
Standard Normal Distribution

It is the Normal distribution with mean 0 and
standard deviation 1.

If a variable x has any Normal distribution N(,
), then the standardized variable
z
x

has the standard Normal distribution.
A standard Normal table give the area
under the curve to the left of z. Find the
area to the left of z = 0.21
Using the chart

Find P(z<1.23)

Find P(z > 2.01)
More examples

Find P(z< -0.13)

Find P(z > -1.72)
More examples

Find P(-1.56 < z < 1.01)

Find P(-2.23 < z < -0.27)
Try the following:

P(z < 1.39)

P(z > -2.15)

P(-0.56 < z < 1.81)
Find the z-score that correlates with the
20th percentile.
For what z-score are 45% of all
observations greater than z?
Homework

Correction to Assignment Sheet
 Page
131 (43-52)