Download Real-Life and The 68-95

16 Mathematics of Normal Distributions
16.1 Approximately Normal Distributions
of Data
16.2 Normal Curves and Normal
Distributions
16.3 Standardizing Normal Data
16.4 The 68-95-99.7 Rule
16.5 Normal Curves as Models of RealLife Data Sets
16.6 Distribution of Random Events
16.7 Statistical Inference
Copyright © 2010 Pearson Education, Inc.
Excursions in Modern Mathematics, 7e: 16.5 - 2
Real-Life and The 68-95-99.7 Rule
The reason we like to idealize a real-life,
approximately normal data set by means of
a normal distribution is that we can use
many of the properties we just learned
about normal distributions to draw useful
conclusions about our data. For example,
the 68-95-99.7 rule for normal curves can
be reinterpreted in the context of an
approximately normal data set as follows.
Copyright © 2010 Pearson Education, Inc.
Excursions in Modern Mathematics, 7e: 16.5 - 3
THE 68-95-99.7 RULE FOR
APPROXIMATELY NORMAL DATA
1. In an approximately normal data set,
about 68% of the data values fall
within (plus or minus) one standard
deviation of the mean.
2. In an approximately normal data set,
about 95% of the data values fall
within (plus or minus) two standard
deviations of the mean.
Copyright © 2010 Pearson Education, Inc.
Excursions in Modern Mathematics, 7e: 16.5 - 4
THE 68-95-99.7 RULE FOR
APPROXIMATELY NORMAL DATA
3. In an approximately normal data set,
about 99.7%, or practically 100%, of
the data values fall within (plus or
minus) three standard deviations of
the mean.
Copyright © 2010 Pearson Education, Inc.
Excursions in Modern Mathematics, 7e: 16.5 - 5
Example 16.7 2007 SAT Math Scores:
Part 2
We are now going to use what we learned so
far to analyze the 2007 SAT mathematics
scores. As you may recall, there were N =
1,494,531 scores, distributed in a nice,
approximately normal distribution. The two
new pieces of information that we are going
to use now are that the mean score was
 = 515 points and the standard deviation
was  = 114 points.
Copyright © 2010 Pearson Education, Inc.
Excursions in Modern Mathematics, 7e: 16.5 - 6
Example 16.7 2007 SAT Math Scores:
Part 2
Just knowing the mean and the standard
deviation (and that the distribution of test
scores is approximately normal) allows us to
draw a lot of useful conclusions:
Copyright © 2010 Pearson Education, Inc.
Excursions in Modern Mathematics, 7e: 16.5 - 7
Example 16.7 2007 SAT Math Scores:
Part 2
Median. In an approximately normal
distribution, the mean and the median should
be about the same. Given that the mean
score was  = 515, we can expect the median
score to be close to 515. Moreover, the
median has to be an actual test score when N
is odd (which it is in this example), and SAT
scores come in multiples of 10, so a
reasonable guess for the median would be
either 510 or 520 points.
Copyright © 2010 Pearson Education, Inc.
Excursions in Modern Mathematics, 7e: 16.5 - 8
Example 16.7 2007 SAT Math Scores:
Part 2
First Quartile. Recall that the first quartile is
located 0.675 standard deviation below the
mean. This means that in this example the
first quartile should be close to
515 – 0.675  114 ≈ 438 points. But again,
the first quartile has to be an actual test score
(the only time this is not the case is when N is
divisible by 4), so a reasonable guess is that
the first quartile of the test scores is either
430 or 440 points.
Copyright © 2010 Pearson Education, Inc.
Excursions in Modern Mathematics, 7e: 16.5 - 9
Example 16.7 2007 SAT Math Scores:
Part 2
Third Quartile. We know that the third
quartile is located 0.675 standard deviation
above the mean. In this case this gives
515 + 0.675  114 ≈ 592 points.
The most reasonable guess is that the third
quartile of the test scores is either 590 or 600
points.
Copyright © 2010 Pearson Education, Inc.
Excursions in Modern Mathematics, 7e: 16.5 - 10
Example 16.7 2007 SAT Math Scores:
Part 2
In all three cases our guesses were very
good–as reported by the College Board, the
2007 SAT math scores had median M = 510,
first quartile Q1 = 430, and third quartile
Q3 = 590.
We are now going to go analyze the 2007
SAT scores in a little more depth–using the
68-95-99.7 rule.
Copyright © 2010 Pearson Education, Inc.
Excursions in Modern Mathematics, 7e: 16.5 - 11
Example 16.7 2007 SAT Math Scores:
Part 2
The Middle 68.
Approximately 68% of the scores should have
fallen within plus or minus one standard
deviation from the mean. In this case, this
range of scores goes from 515 – 114 = 401 to
515 + 114 = 629 points. Since SAT scores
can only come in multiples of 10, we can
estimate that a little over two-thirds of
students had scores between 400 and 630
points.
Copyright © 2010 Pearson Education, Inc.
Excursions in Modern Mathematics, 7e: 16.5 - 12
Example 16.7 2007 SAT Math Scores:
Part 2
The Middle 68.
The remaining third were equally divided
between those scoring 630 points or more
(about 16% of test takers) and those scoring
400 points or less (the other 16%).
Copyright © 2010 Pearson Education, Inc.
Excursions in Modern Mathematics, 7e: 16.5 - 13
Example 16.7 2007 SAT Math Scores:
Part 2
The Middle 95.
Approximately 95% of the scores should have
fallen within plus or minus two standard
deviations from the mean, that is, between
515 – 228 = 287 and 515 – 228 = 743 points.
In practice this means SAT scores between
290 and 740 points. The remaining 5% of the
scores were 740 points or above (about
2.5%) and 290 points or below (the other
2.5%).
Copyright © 2010 Pearson Education, Inc.
Excursions in Modern Mathematics, 7e: 16.5 - 14
Example 16.7 2007 SAT Math Scores:
Part 2
Everyone.
The 99.7 part of the 68-95-99.7 rule is not
much help in this example. Essentially, it says
that all test scores fell between
515 – 342 = 173 points and 515 + 342 = 857
points.
Duh! SAT mathematics scores are always
between 200 and 800 points, so what else is
new?
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Excursions in Modern Mathematics, 7e: 16.5 - 15