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A normal distribution is a very important statistical
data distribution pattern occurring in many natural phenomena,
such as height, blood pressure, lengths of objects produced by
machines, etc.
Certain data, when graphed as a histogram (data on the
horizontal axis, amount of data on the vertical axis), creates a
bell-shaped curve known as a normal curve,
or normal distribution.
Normal distributions are symmetrical
with a single central peak at the mean (average)
of the data. The shape of the curve is described as
bell-shaped with the graph falling off
evenly on either side of the mean. Fifty percent of the
distribution lies to the left of the mean and fifty percent lies to the
right of the mean.
The spread of a normal distribution is
controlled by the standard deviation,
 .
The smaller the standard deviation the more concentrated the data.
The mean
and the median are the same in a normal
distribution.
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Chart prepared by the NY State
Education Department |
Reading from the chart, we see that
approximately 19.1% of normally distributed data is located between the mean (the
peak) and 0.5 standard deviations to the right (or left) of the mean.
(The percentages are represented by the area under the curve.)
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Understand that this chart shows only
percentages that correspond to subdivisions up to one-half
of one standard deviation.
Percentages for other subdivisions require a statistical
mathematical table or a graphing calculator.
(See example 4) |
If you add percentages, you will see
that approximately:
• 68% of the distribution
lies within one standard deviation of the mean.
• 95% of the distribution lies within two
standard deviations of the mean.
• 99.8% of the distribution lies within three
standard deviations of the mean.
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These three
statements constitute what is referred to as the "empirical
rule". |
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s.d. in callout boxes = standard deviation |
It is also true that:
• 50% of the distribution lies within 0.67448
standard deviations of the mean.
If you are asked for the interval about the mean
containing 50% of the data, you are actually being asked for the
interquartile range, IQR.
The IQR (the width of an interval which contains the middle 50% of the
data set) is normally computed by subtracting the first quartile from
the third quartile. In a normal distribution (with mean 0 and standard
deviation 1), the first and third quartiles are located at -0.67448 and
+0.67448 respectively. Thus the IQR for a normal distribution is:
Interquartile range = 1.34896 x standard
deviation
(this will be the population IQR)
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Percentiles
and the Normal Curve |
The mean (at the center peak of the curve) is
the 50% percentile.
The term "percentile rank" refers to the area
(probability) to the left of the value.
Adding the given percentages from the chart will let you find
certain percentiles along the curve. |
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Examples: Look for the words "normally
distributed" in a question before referring to the Normal
Distribution Standard Deviation chart seen on this page. When
using the chart, your information should fall on the increments of
one-half of one standard deviation as shown in the chart.
| 1.
Find the percentage of the normally
distributed data that lies within 2 standard
deviations of the mean. |
Solution: Read the
percentages from the chart at the top of this page from -2 to +2 standard deviations.
4.4% + 9.2% + 15.0% + 19.1% + 19.1% + 15.0% + 9.2% + 4.4%
= 95.4%
| 2.
At the New Age Information Corporation, the ages of all
new employees hired during the last 5 years are
normally distributed.
Within this curve, 95.4% of the ages, centered about the
mean, are between 24.6 and 37.4 years. Find the mean
age and the standard deviation of the data. |
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| Solution:
As was seen in Example 1, 95.4% implies a span of
2 standard deviations from the mean.
The mean age is symmetrically located between -2 standard
deviations (24.6) and +2 standard deviations (37.4).
The mean age is
 years of
age.
From 31 to 37.4 (a distance of 6.4 years) is 2
standard deviations. Therefore, 1 standard deviation is
(6.4)/2 = 3.2 years. |
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| 3.
The amount of time that Carlos
plays video games in any given week is
normally distributed.
If Carlos plays video games an average of 15 hours per week,
with a standard deviation of 3 hours, what is the
probability of Carlos playing video games between 15 and 18
hours a week? |
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Solution: The
average (mean) is 15 hours. If the standard deviation is 3,
the interval between 15 and 18 hours is one standard deviation above
the mean, which gives a probability of 34.1% or 0.341, as seen in
the chart at the top of this page. |
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| 4.
The lifetime of a battery is normally distributed with
a mean life of 40 hours and a standard deviation of
1.2 hours. Find the probability that a
randomly selected battery lasts longer than 42
hours. |
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The most accurate answer to a problem such as this cannot be
obtained by
using the chart at the top of this page. One standard
deviation above the mean would be located at 41.2 hours, 2
standard deviations would be at 42.4, and one and one-half
standard deviations would be at 41.8 standard deviations.
None of these locations corresponds exactly to the needed 42 hours.
We need more power than we have in the chart to find the
most accurate answer.
Calculator to the rescue!! |
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Solution:
Graph the normal curve. We
see from the location of 42 on the graph that the answer is
going to be quite small. |
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Now, determine the probability of a value falling
to the right of 42 hours (between 42 hours and infinity).
Answer: 4.779%
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Please Read!
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Disclaimer: It may be the case (as
with past Regents Examinations) that you are expected to
secure your answers solely from working with the chart at the top of
this page. Should this be the case, a question such as
Example 4 would most likely offer a list of possible answers
from which to choose. In such a case, the less
accurate
answer secured from the chart should be sufficient. Be
sure to read directions carefully and follow instructions
given by your teacher. See question #2 under
Practice with Normal
Distributions and Standard Deviations for a
comparison of answers done with the chart and with
the calculator. |
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