The Normal Probability Distribution

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The Normal Probability Distribution
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Review relative frequency histogram
Values of a variable, say test scores
1/10 2/10 4/10 2/10 1/10
60 70 80 90
In this example 10 people took a test. The
height of each bar is the relative frequency or
percentage of those in that range of scores. What
of people had test scores between 70 and 80?
40 What of people had scores less than 70?
30 If you add up all the fractions what do you
get? 1
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The Normal Distributions - Basic idea

• The normal distribution is a tool we use to try
  to convey the same information as we get from a
  relative frequency histogram.
• The normal distribution has been used a lot in
  statistics and we will use it later, so we will
  look at some details about it.
• But, first lets look at circles - yes I mean
  circles!

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circles and density

• Imagine you are at the intersection by Dairy
  Queen. Now imagine a large circle is placed on
  the earth such that the center of the circle is
  at the intersection plus enough houses have been
  included so 1,000,000 people live in the circle.
• New York City has a similar intersection and
  circle, except the circle is smaller (WHY?).

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circles and density

• The New York circle is shorter because you travel
  a shorter distance from the center to get an
  equal density of people in New York.
• The smaller circle in a sense has a smaller
  standard deviation(actually it has a smaller
  radius) - the distance is less spread out.
• One thing similar about the two circles is that
  you can divide them both up into quarters. Lets
  do this on the next screen with two circles

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circles and density
A
a
25 of the area is in A on the large circle and
25 of the area of the small circle is in part
a. How can they both be 25? It is 25 of its
own total. There are as many different normal
distributions as there are circles. BUT, normal
distributions are divided up, not into quarters,
but in another way.
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The Normal Distributions - normal dist. and
density

• Normal distributions can roughly be drawn by
  modifying a circle.

flip this part out to left
flip this part out to right
like this
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The Normal Distributions - normal dist. and
density

• Lets label parts of the normal distributions.

This point on the number line is directly below
the inflection point. It turns out that the
point on the number line is one standard
deviation away from the center.

This point is where the bottom part of the circle
flipped. Lets call it the inflection
point. There is one on the other side as well.
number line for the variable- like test score
This is the center of the distribution. It
really is the mean value.
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On the previous screen we see a graph of a normal
distribution. Lets consider an example to
highlight some points. Say a company has
developed a new tire for cars. In testing the
tire it has been determined that the mean tire
mileage is 36,500 miles and the standard
deviation is 5000 miles. Along the horizontal
axis we measure tire mileage. The normal
distribution rises above the axis. Note the
highest point of the curve occurs above the mean
- in our tire example we would be at 36,500.
On the curve we have two inflection points, and
these occur 1 standard deviation away from the
mean. So, mileages 31,500 and 41,500 are 1
standard deviation for the mean and the
inflection points occur above them.
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The Normal Distributions - notation

• In general, now, we will talk about a variable
  having a normal distribution. We will say
  variable X is normally distributed with mean mu
  and standard deviation sigma.
• More simply, we say X is N(mu,sigma).
• Dont let the N(---) part fool you, it means
  N(mean value listed first, then standard
  deviation value listed).

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The Normal Distributions - example with graphical
thinking

• Say we have a variable X is N(3, 1)

Why is this dot, and the one across, above
s 2 and 4?
X is measured on the line
4
3
2
Use the dots as your guide to draw the
normal dist.
3 is the mean
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The Normal Distributions - another example with
graphical thinking

• Say we have a variable X is N(3, 2)

Why is this dot, and the one across, above
s 1 and 5?
X is measured on the line
4
5
3
2
1
Use the dots as your guide to draw the normal
dist.
3 is the mean
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The Normal Distributions - compare the two
examples

• here is what the two examples look like, one on
  top of the other

X is N(3, 1)
X is N(3, 2)
4
5
3
2
1
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The Normal Distributions - compare the two
examples

• Note on the previous screen how the X is N(3, 2)
  had its inflection points wider than on the X is
  N(3, 1).
• Remember how we labeled the quarters of the
  different circles A and a. We said there was 25
  of the circle in both A and a, but based on its
  own total.
• Normal dist.s have a similar rule. 68 of the
  area under the curve is between the two
  inflection points. There is more.

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The Normal Distributions - 68-95-99.7 rule

• On any normal distribution the inflection points
  will be 1 standard deviation on either side of
  the mean. 68 of the area under the curve will
  be within this one standard dev.
• By moving out 2 standard deviations on either
  side of the mean you have about 95 of the area
  under the curve.
• By moving out 3 stand. dev.s you have 99.7 of
  the area under the curve.

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The Normal Distributions - 68-95-99.7 rule

• What is the meaning of the phrase, 1 standard
  dev. on either side of the mean?
• The answer is best seen by an example. X is
  (10, 2.5) means X is normal with mean 10 and
  standard deviation 2.5. Thus 7.5 is 1 stand.
  dev. on the low side of the mean and 12.5 is 1
  stand. dev. on the high side of the mean. Thus
  being anywhere from 7.5 to 12.5 is within 1
  standard deviation of the mean.

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Note about normal distribution 1. There are many
normal distributions, each characterized by a
mean value and a standard deviation. 2. The high
point of the curve is above the mean and for a
normal distribution the mean median mode. 3.
Depending on the variable, the mean can be
negative, zero, or positive. 4. The normal curve
is symmetric. This means each side is a mirror
image of itself. 5. Larger standard deviations
result in a flatter, wider distribution. 6.
Probabilities for the variable are found from
areas under the curve - the 65, 95, 99.7 rule is
an example of this.
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26,500 31,500 36,500 41,500
46,500
miles
-2 -1 0 1
2 z
Remember the concept of a z score from earlier.
z ( a value minus the mean)/standard
deviation. So the value 26,500 has a z (26,500
- 36,500)/5000 -2. This means 26,500 is 2
standard deviations below the mean. You can
check the other values.
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The standard normal distribution Remember how we
said there are many different circles and many
different normal distribution? Sure you do. The
z value translates any normally distributed
variable into what is called the standard normal
variable. Technically the picture I have on the
previous screen is misleading because the zs are
a different scale than the miles, but dont
worry. In the book there is a table with z values
and areas under the curve. Lets see how to use
the table. Here is one place where I want you to
be extra careful when you calculate z. Round z
to 2 decimal places. The z value is broken up
into two parts a.b and .0c. when added we get
a.bc. For example the number 2.13 is broken up
into 2.1 and .03
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Using the standard normal table The z 2.13
means we should go down the table to 2.1 and then
over to .03. The number in the table is .9834.
This means the probability of getting a value
less than z 2.13 is 98.34. In the tire example
if we look at the mean value 36,500, we see the z
(36,500 - 36,500)/5000 0.00 and in the table
we see the value .5000. Thus, there is a 50
chance the tire mileage will be less than
36,500. So the table has the area under the curve
to the left of the value of interest. We may want
other zs and other areas. What do we do?
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Say we want the area to the right of a z that is
greater than 0? The table has the area to the
left. Whatever the z is,
go into the table and get the area and then take
1 minus the area in the table.
The z here would be negative. Say we want area
b. Area a is in the table and b is 1 minus area
a.
b
a
Area b would be found in a similar way to what is
above.
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Back in the old days when I had to walk to school
uphill both ways in three feet of snow, the
standard normal table was all we had to calculate
probabilities for a normal distribution. Now we
have Microsoft Excel to make the
calculations. The NORMSDIST function assumes we
have a z value and we want to find the area the
the left of the z - the area to the left is the
cumulative probability. The function has the
form NORMSDIST(z), where z is the value we have.
z can be negative in Excel. The NORMDIST
function allows us to just work with the variable
without getting the z and we can still have the
cumulative probability. The function has the
form NORMDIST(value, mean, standard deviation,
TRUE). This is an innovation of Excel over the
old days.
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Sometimes we may have an area and want to know
the z. The function NORMSINV asks us to give an
area to the left of a value and the function will
give us the z value. The form of the function is
NORMSINV(cumulative probability). The function
NORMINV does the same, except not in z value
form. It just give the value in the same form as
the variable. The form of the function is
NORMINV(cumulative prob, mean, standard
deviation)