Download The Normal Curve

NORMAL DISTRIBUTIONS &
PROBABILITY
DAY 1
Exploring Normal Distributions with
the
EMPIRICAL RULE
Normal Distribution
• A normal distribution is a continuous, symmetrical, bellshaped distribution of a variable.
Characteristics of Normal Distribution
1. A normal distribution curve is bell-shaped.
2. The mean, median, and mode are equal and are
located at the center of the distribution.
3. The curve is unimodal (i.e., it has only one mode)
4. The curve is symmetric about the mean (its
shape is the same on both sides of a vertical line
passing through the center)
5. The curve is continuous. There are no gaps or
holes. For each value of X, there is a
corresponding value of Y.
Characteristics of Normal Distribution
6. The curve never touches the x axis. No matter
how far in either direction the curve extends,
it never meets the x axis – but it gets
increasingly closer.
7. The total area under a normal distribution
curve is equal to 1.00, or 100%.
The area under the part of a normal curve that lies within
1 standard deviation of the mean is approximately 0.68, or
68%; within 2 standard deviations, about 0.95, or 95%; and
within 3 standard deviations, about 0.997, or 99.7%.
The Empirical Rule
(a.k.a. the “68-95-99.7 Rule”)
• In a normal distribution, almost all data will lie
within 3 standard deviations of the mean.
o About 68% of all data lies within 1 standard
deviation of the mean.
o About 95% of all data lies within 2 standard
deviations of the mean.
o About 99.7% of all data lies within 3 standard
deviations of the mean.
Why Do We Need It?
• The Empirical Rule is most often used in statistics
for forecasting or predicting final outcomes.
• After a standard deviation is calculated, and
before exact data can be collected, the Empirical
Rule can be used to estimate impending data.
• The probability based on the Empirical Rule can
be used if gathering appropriate data may be
time consuming, or even impossible to obtain.
Tips for Using the Empirical Rule
• Before applying the Empirical Rule it is a good
idea to identify the data being described and
the value of the mean and standard deviation.
• Sketch a graph summarizing the information
provided by the empirical rule and identify the
percentages for each region of the graph
(+/- 1, +/- 2, +/- 3)
• Remember that data must be normally
distributed for the Empirical Rule to apply.
The Empirical Rule
for a Normal Distribution
The amount of mustard dispensed from
a machine at The Hotdog Emporium
is normally distributed with a mean
of 0.9 ounce and a standard
deviation of 0.1 ounce. If the
machine is used 500 times,
approximately how many times will it
be expected to dispense 1 or more
ounces of mustard.
Choose:
5
16
80
100
The mean is 0.9 and the standard deviation is 0.1. If one standard deviation is
added to the mean, the result is 1.0 ounce. Therefore, dispensing 1 or more
ounces falls into the category above one standard deviation to the right of the
mean. Using the Empirical Rule, 16% of data falls at or above 1 standard
deviation.
16% x 500 = 80 times to dispense one or more ounces of mustard.
A machine is used to fill soda bottles. The amount of soda dispensed
into each bottle varies slightly. Suppose the amount of soda dispensed
into the bottles is normally distributed. If at least 99% of the bottles
must have between 585 and 595 milliliters of soda, find the greatest
standard deviation, to the nearest hundredth, that can be allowed.
The 99% implies a distribution within 3 standard deviations of the mean. The
difference from 585 milliliters to 595 milliliters is 10 milliliters. Symmetrically
divided, there are 5 milliliters used to create 3 standard deviations on one
side of the mean. Dividing 5 by 3, we get the standard deviation to be 1.67
milliliters, to the nearest hundredth.
Battery lifetime is normally distributed for large
samples. The mean lifetime is 500 days and
the standard deviation is 61 days. To the
nearest percent, what percent of batteries have
lifetimes less than 439 days?
Subtracting, we see that 1 s.d. below is 439 days, an exact match to
our question and an indication that the Empirical Rule can be used to
find the answer. The question is asking what percent of a distribution is
beyond one standard deviation to the left of the mean.
Answer: 16%
A shoe manufacturer collected data
regarding men's shoe sizes and found
that the distribution of sizes exactly fits
the normal curve. If the mean shoe size
is 11 and the standard deviation is 1.5,
find:
a. the probability that a man's shoe size
is greater than or equal to 11.
b. the probability that a man's shoe size
is greater than or equal to 14.
a. 50% In a normal distribution, the mean divides the data into two equal
areas. Since 11 is the mean, 50% of the data is above 11 and 50% is below 11.
The probability is 0.5.
b. 14 is exactly two standard deviations above the mean. Using the Empirical Rule
we see that 2.5% will fall above two standard deviations. Probability is 0.025.
Five hundred values are normally distributed with a mean of 125 and a standard
deviation of 10.
a. What percent of the values lies in the interval 115 - 135,
to the nearest percent?
b. What interval about the mean includes 95% of the data?
a. What percent of the values is in the interval 115 - 135?
mean + one standard deviation = 135
mean - one standard deviation = 115
Percent within one standard deviation of the mean = 68%
b. 2 standard deviations about the mean for a total interval size
of 40, with the mean in the center.
mean + 2 standard deviations = 145
mean - 2 standard deviations = 105
Interval: [105,145]
Example:
Estimating with the Empirical Rule
• You have purchased fluorescent light bulbs for
your home. The average bulb life is 500 hours
with a standard deviation of 24. The data is
normally distributed.
• One of your bulbs burns out at 450 hours.
Would you send the bulb back for a refund?
Problem: You have purchased fluorescent light bulbs for your home. The average bulb
life is 500 hours with a standard deviation of 24. The data is normally distributed. One
of your bulbs burns out at 450 hours. Would you send the bulb back for a refund?
Solution:
• According to the Empirical Rule:
o 68% of the light bulbs should last between
500 ± 24 or between 476 to 524 hours.
o 95% of the light bulbs should last between
500 ± 2(24) or between 452 to 548 hours.
o 99.7% of the light bulbs should last between
500 ± 3(24) or between 428 to 572 hours.
• If the light bulb only lasted 450 hours, I would
consider it a defective bulb. Less than 2.5%
should last less than 452 hours.
Your Turn
• The scores for all high school seniors taking the verbal
section of the Scholastic Aptitude Test (SAT) in a
particular year had a mean of 490 and a standard
deviation of 100. The distribution of SAT scores is
normal.
1. What percentage of seniors scored between 390 and
590 on this SAT test?
2. One student scored 795 on this test. How did this
student do compared to the rest of the scores?
3. A rather exclusive university only admits students who
were among the highest 16% of the scores on this test.
What score would a student need on this test to be
qualified for admittance to this university?
1. What percentage of seniors scored between 390
and 590 on this SAT test?
• The mean = 490 and the standard deviation = 100. So, we can
draw the Normal Curve to depict the data.
Calculate the difference
between the mean and
the test scores:
|390-490| = 100  1
s.d.
|590-490| = 100  1
s.d.
So, about 68% of
seniors would score
between 390 and 590
on the SAT test.
2. One student scored 795 on this test. How did this student
do compared to the other scores?
• Calculate the difference between the mean and the student’s
test score:795-390 = 405
• This student’s score is more than 3 std. deviations from the
mean
(3 standard deviations = 3 * 100)
Since about 99.7% of seniors
would score between 190 and
790 on the SAT test, this is an
exceptionally high score.
Only 0.15% of students would
have scores above 790. This
is an example of using the
Empirical Rule to estimate an
answer.
3 . A rather exclusive university only admits students who were among the
highest 16% of the scores on this test. What score would a student need on
this test to be qualified for admittance to this university?
• About 68% of the scores are between 390 and 590, so this
leaves 32% of the scores outside this interval.
• Since a bell-shaped curve is symmetrical, one-half of the
scores (16%) are on each end of the distribution.
• 16% of the students scored above 590 on this SAT test.
So, to qualify for admittance to
this university, a student
would need to score 590 or
above on the SAT test.