Download Sec 6C

6C – The Normal Distribution
The
distribution with a
, and
is a
, _______________
peak. Its peak corresponds to the
,
of the distribution.
Both of these are normal distributions with a mean of 50; they differ in their
The second would have a larger
.
.
What’s the big deal about the Normal Distribution?
It is so
. Most
are normally
distributed. Any quantity that is the result of
factors is likely to follow
a normal distribution, as are scores on standardized tests, sports statistics (many people
with many different levels of skill).
Conditions for a Normal Distribution
A data set that satisfies the following 4 criteria is likely to have a nearly normal
distribution.
1. Most data values are
near the
, giving the
distribution a well-defined
.
2. Data values are spread
around the mean, making the
distribution
.
3. Larger deviations from the mean become increasingly rare, producing the
_______________________________ of the distribution.
4. Individual data values result from a combination of many different factors, such as
genetic and environmental factors.
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The Standard Deviation in Normal Distributions
We can characterize the
of any normal distribution with just
number: the
. The standard
deviation gives a precise description of the variation base on the
rule.
(Also known as the Empirical Rule.)
Note: These percentages are actually 68.3% within 1 standard deviation of the mean;
95.4% within 2 standard deviations of the mean.
Applying the 68-95-99.7 Rule
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Example: The resting heart rates for a sample of individuals are normally distributed
with a mean of 70 and a standard deviation of 15. Use the 68-95-99.7 Rule to find the
percentage of heart rates in each of the following categories.
a. less than 55
b. greater than 90
c. less than 70
d. greater than 40
e. between 55 and 85
f. between 40 and 85
The 68-95-99.7 Rule applies to data values that are exactly 1, 2, or 3 standard deviations
from the mean. What do we do in other cases?
Standard Scores (or z-scores)
The number of standard deviations a data value lies above or below the mean is called
the
(or z-score), defined by:
z = s tan dard score =
data value " mean
s tan dard deviation
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The standard score is
and
for values
for values
the mean,
the mean.
• The standard score of the mean is z = 0, because it is 0 standard deviations from the
mean.
• The standard score of a data value 1.5 standard deviations above the mean is z = 1.5.
• The standard score of a data value 2.4 standard deviations below the mean is z =
−2.4.
Example: The Stanford-Binet IQ test is designed so that scores are normally distributed
with a mean of 100 and a standard deviation of 16. Find the standard scores for IQ
scores of 95 and 125.
Now we can use the properties of the normal Distribution to find the percentage of
scores
.
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For the standard scores given, the table give the
distribution
of values in the
that value.
For example, we see that about 29% of the data has a standard score less than or equal
to z = −0.55. We say that this score lies in the 29th percentile.
Examples: The Stanford-Binet IQ test is designed so that scores are normally
distributed with a mean of 100 and a standard deviation of 16.
a. About what percentage of the scores were above 123?
b. About what percentage of the scores were below 89?
c. Scores that are 2 or more standard deviations below the mean are flagged for
investigation. What was this cutoff? Approximately what percent of the scores were in
the range?
d. Suppose 1000 people were tested. About how many scored above 130?
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e. What score corresponds to the 90th percentile?
f. What score corresponds to the 40th percentile?
g. What percent of the scores falls between 95 and 105?
h. What percent of the scores falls between 68 and 100?
i. John Doe scores 120. What is the percentile for his score?
j. In what percentile is a data value that is 0.95 standard deviations above the mean?
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Cholesterol levels in men 18 to 4 years of age are normally distributed with a mean of
178 and a standard deviation of 41. An a group of 1000 men, about how many have a
cholesterol level of 219 or higher?
Suppose that in a company with 5000 employees the annual salaries are normally
distributed with a mean of $35,000 and a standard deviation of $3850. About how
many of the employees earn more than $40,000?
Be sure you can/know:
• What is meant by a normal distribution
• The 68-95-99.7 Rule and be able to use it to solve problems
• How to calculate a z-score, use a z-score table, calculate percentile of a particular
data value
• Solve problems on data sets that are normally distributed
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