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The 68-95-99.7 Rule
• In any normal distribution:
– 68 % of the individuals fall within 1s of m.
– 95 % of the individuals fall within 2s of m.
– 99.7 % of the individuals fall within 3s of m.
How can we make a valid comparison
of observations from two distributions?
• By standardizing the values of the
observations with respect to the distributions
from which they come.
• If x is an observation from a distribution with
mean m and standard deviation s, then
xm
z
s
is the standardized value (a.k.a. z-score) of x.
Standard Normal Distribution
The standard normal distribution is the normal
distribution N(0,1) with mean 0 and standard
deviation 1.
If a variable X has a N(m,s) distribution, the
standardized variable
X m
Z
s
has the standard normal distribution.
Standard Normal Table
(Table A, inside cover of book or from website)
Examples
1. Find the area under the standard normal curve to the
left of -1.4.
2. Find the area under the N(0,1) curve between 0.76
and 1.4.
3. Find the value z of the N(0,1) which has area 0.25
to its right.
4. Suppose X~N(275, 43). What proportion of the
population is greater than 200? What proportion of
the population is between 200 and 375?
5. Suppose verbal SAT scores follow the N(430, 100)
distribution. How high must a student score in order
to place in the top 5%?
Testing for Normality
• The normal distribution provides a good
model for many real data distributions.
• Furthermore, the normal distribution is a
nice model to work with mathematically.
• However, we need to be cautious when
assuming normality of data. Are we sure our
data are normally distributed?
The Normal Quantile Plot:
A Visual Test of Normality
1. Arrange the observed data values from
smallest to largest and record the percentile of
the data each observation occupies.
2. Compute the z-scores of the same percentiles.
3. Plot each data point against the corresponding
z-score.
If the points on a normal quantile plot lie close to a
straight line then the plot indicates the data are normal.
FIG. 1.34: Normal quantile plot of
IQ scores for 78 seventh-graders