Transcript z-score - Lyndhurst Schools

Chapter 7:
The Normal Distribution
Important Properties of a Normal
Distribution
• The mean, median, and mode have the same
value, which is located exactly in the center of the
distribution.
• The total area under the Normal curve equals 1.
The area to the left of the mean is .5 and the area
to the right of the mean is .5.
• Data that lie beyond two standard deviations
from the mean are rare, and data that lie beyond
three standard deviations from the mean are very
rare.
• Recall that a Normal distribution is roughly
symmetric, unimodal, and bell-shaped.
• The graphical display of a Normal distribution is a
Normal curve.
• Many variables approximate the Normal distribution.
• The mean and standard deviation are the two most
important parameters for any normal distribution.
• We know approximately what percent of data
fall exactly 1, 2, and 3 standard deviations
from the mean. However, how can we find a
percent if a value does not fall exactly 1, 2, or
3 standard deviations from the mean.
• The first thing we have to do is standardize
our score(s). This is referred to as finding the
z-score.
Z-Scores
• A z-score is a measure of the relative position of a
data item in terms of the number of standard
deviations it is from the mean.
• Z-scores can be negative, positive, or 0.
• A positive z-score would indicate the original
value is above the mean. For example, a z-score
of 1.24 would mean that the score is 1.24
standard deviations above the mean.
• A negative z-score would indicate the original
value is below the mean. For example, a z-score
of -0.27 would mean that the score is 0.27
standard deviations below the mean.
• A z-score of 0 would indicate that the original
value was the same as the mean.
• Question: Are all negative z-scores bad?
Example: A distribution is approximately Normal
with a mean of 26 and a standard deviation of 7.
Calculate and interpret the z-score for a value of 21.
The value 21 is 0.71 standard
deviations below the mean.
Example: A distribution is approximately Normal
with a mean of 100 and a standard deviation of
10. Calculate and interpret the z-score for a
value of 117.
The value 117 is 1.7
standard deviations
above the mean.
Example: Seth recently took two tests in school. On
his history test he scored a 75. The class average on
the test was a 63 and the test had a standard
deviation of 3. On his biology test, Seth scored a 81.
The class average on the test was a 76 and the test
had a standard deviation of 7. Relatively speaking,
on which test did he perform better?
He performed better on his history test as he was more
standard deviations above the class mean.
• In order to find the area under the Normal curve
(percentage) for values that do not fall exactly 1,
2, or 3 standard deviations from the mean, we
need to utilize the standard Normal table.
• Once scores have been standardized, the mean
becomes 0.
Steps to Use the standard Normal table
1) Draw the Normal curve and label the mean of 0
along with 3 standard deviations in each direction.
2) Standardize the score(s) of interest.
3) Plot the z-score(s), draw a vertical line(s), and shade
the area of interest.
4) Look up the z-score on the standard Normal table.
– If the area of interest is shaded to the left, the value in the
table is the desired area.
– If the area of interest is shaded to the right, we need to
subtract the area in the table from 1.
– If the area of interest is shaded between two z-scores, we
need to look up the area for both z-scores and subtract.
Example A: A data set is Normally distributed with a
mean of 259 and a standard deviation of 74. Find
the area under the curve less than a score of 180.
• We have to look up the z-score
of -1.07 on the table.
• Since the score starts out as
“-1.0”, go to the z column and go
down until you reach -1.0.
• Since there is a 7 in the hundredths place (.07), go to the right until
you reach the .07 column. You should now be located in the spot
that has -1.0 to the left, and .07 on the top.
This value should be .1423.
Example B: A data set is Normally distributed
with a mean of 259 and a standard deviation of
74. Find the area under the curve less than a
score of 350.
Since the z-score is 1.23,
look for 1.2 in the z column
and .03 on top. This should
give us an area of .8907
Example C: A data set is Normally distributed with a
mean of 26 and a standard deviation of 2.4. Find
the area under the curve more than a score of 29.
A z-score of 1.25 on the table gives an area of .8944. However, this
is the area to the LEFT of the score. We want the area to the right
of the score. Since the area under the Normal curve totals 1, we
need to subtract .8944 from 1. This gives us our desired area,
which is .1056.
Example D: A data set is Normally distributed with a
mean of 26 and a standard deviation of 2.4. Find
the area under the curve above a score of 21.
1-.0188=.9812
Example E: In the 2008 Wimbledon tennis tournament,
Rafael Nadal averaged 115 miles per hour on his first-serves.
Assuming that the distribution of his first-serve speeds is
Normal with a standard deviation of 6 mph, find what
proportion of his first-serves you would expect to be
between 110 and 125 mph.
Look up both areas and subtract.
.9525-.2033=.7492
Example F: On the 2009 PGA tour, Tiger Woods had an
average driving distance of 298 yards. Assuming that the
distribution of his driving distances is Normal with a
standard deviation of 12 yards, find what proportion of his
drives you would expect to be between 290 and 310 yards.
.8413-.2514=.5899