Transcript Standard Normal Distribution

Think about this….
If Jenny gets an 86% on her first statistics test, should
she be satisfied or disappointed? Could the scores of
the other students in the class affect her feelings?
In this chapter, we will investigate the individual
observations (data values) in comparison to the
entire distribution.
Empirical Rule:
For data with a symmetric, bell shaped distribution, the standard
deviation has the following characteristics:
The distribution of heights of
adult American males is
approximately normal with a
mean of 69 inches and a
standard deviation of 2.5
inches.
1) What percent of men are
taller than 74 inches?
2) Between what heights do the middle 95% of men fall?
3) What percent of men are shorter than 66.5 inches?
4) A height of 71.5 inches corresponds to what percentile of adult
American heights?
If a distribution is fairly symmetric and can be “idealized” by
a normal distribution curve, the distribution can be
symbolized by the function, N(m,s).
Standard Scores (z-scores) are based upon the Standard
Normal Distribution of N(0,1) where the mean is 0 and a
standard deviation of 1.
This scale
shows the zscore values
(# of standard
deviations
from the
mean)
Properties of a Density Curve
•Always on or above the horizontal axis
•Has an area of exactly 1 under the curve
•Describes the overall pattern of a distribution
•Because a density curve is an “idealized” description of the
distribution, the density curve can be used to generalize from a
sample to a population
•The mean of a density curve is symbolized m and
standard deviation is symbolized s
the
Here is the list of the 25 students in Jenny’s class:
79
81
80
77
73
83
74
93
78
80
75
67
73
77
83
86
90
79
85
83
89
84
82
77
72
If we wanted to compare Jenny’s score (86) to the
class, what could we do?
xx
z
s
observation (value)
mean
standard deviation
What if the value is not EXACTLY at a whole standard
deviation?
Putting the z-scores on a standard normal distribution together with
the fact that the area under the curve represents the percent of data
that falls within a particular interval… we can use z-scores to
calculate exact percentiles of an observation.
This shaded
area would
equal the
percentile of
that z-score
To find these
percentiles, we
can use a table.
This is near the
front cover of
your textbook.
What would be the percentile for a z-score of 1.15?
This means that an
observation with a z-score
of 1.15, would fall in the
87.49th percentile.
Finding Percentiles of Normal Distributions
with a Calculator
2nd VARS (Distr), #2 (normalcdf)
normalcdf(lowest x value, highest x-value, m, s)
Use -9999 for -
 and 9999 for + 
If you given a z-score instead of x values, use m = 0 & s = 1.
For Jenny’s stat test, you would enter:
Normalcdf (0, 86, 80, 6.07)
Using a standard normal distribution, we can relate an observations
standard score with its percentile.
This shaded
area would
equal the
percentile of
that z-score
If you know the percentile, how
would you find the value of the
observation?
What would be the z-score for an observation that had a
percentile value of approximately 92% ?
Find the
percentile that
is CLOSEST to
the desired
percent
Once you know the z-score, you can use the formula to find
the actual observation value…but you must know the original
data’s mean and standard deviation
Example: The mean score on a test was 76 with a standard
deviation of 10.4. If a teacher wishes to give awards to those
students in approximately the top 8%, what is cutoff score for those
students?
top 8% = 92% to the left of the cutoff score
xx
z
s
x  76
1.41 
10.4
x  90.66
Finding a Value given a Percentile for a Normal
Distribution with a Calculator
2nd VARS (Distr), #3 (invnorm)
Invnorm(percentile or area to the left, m, s)
For the previous example you would enter:
InvNorm( .08, 76, 10.4)