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Guided Notes for Lesson: What is Normality
Preliminaries
Slide 1:
Discussion Question: How do we describe data in Statistics?
In this course, the way we describe data is by using the C.U.S.S.
Method. We look at these characteristics:
C. C
U. U
S. S
S. S
Slide 2
• Recall that a
is a standard way of displaying the
distribution of data graphically using the
.
• EX 1: Make a box plot using the following distribution of numbers
in your calculator.
• 2, 3, 5, 5, 6, 7, 7, 7, 8, 9, 9, 9 10, 10, 10, 10, 10, 10, 11, 11, 12, 14,
14, 14, 15, 16, 18, 18, 18, 18, 19, 19, 20, 20, 22, 22, 24, 24, 25, 26,
26, 27, 28, 28, 28, 33, 45, 50, 55, 66
• After you obtain the graph in your calculator draw it in your notes,
write down the five number summary, and then use C.U.S.S to
describe the distribution.
Slide 3: Calculator Use
1. Go to
button and push
2. Copy the data into list
3. Push
.
to get to the
4. Select
.
page.
.
5. Select Type:
(bottom left)
6. Select Xlist:
.
7. Select
.
Slide 5
Recall that a
is a
distribution using different
has a similar shape to a
a
of the data of a
and
. The distribution
. Below is a visual representation of
.
Video Questions
1. How do label the axes of a Histogram?
2. Are the bars of a Histogram connected?
3. What is the range of a Histogram?
Slide 7:
• As more
is added to a Histogram the
more and more similar to a
becomes
. You can see the
resemblance in the examples we covered and in the following
picture.
Slide 8:
• To assume that a set of data follows a
of the most important
is one
in
.
• When we think about the concept of
of numbers that has
,
consider a list
, and
If you have LOTS of these types of numbers then they MIGHT
follow a “
”
• Some examples of Normally distributed data are:
.
• Can you think of some examples based on the description given
above? Write them in your notes.
Slide 9:
The Normality Trap
• Not all data follows a
• Data with
.
or
may not be
distributed
•
will be closer to a
distribution than
samples
• Real life data is almost
.
Characteristics of a Normal Distribution
• The
,
, and
• The curve is
crosses the
all have the same value
and
.
about the line that
• The curve approaches, but never touches the
away from the
• The
.
under the curve is equal to
• Almost all of the
as you move
.
under the curve exists within
standard deviations of the mean.
• When data follows a normal distribution it is denoted
where 𝜇
is the mean and 𝜎 is the standard deviation of the distribution.
How to Draw a Normal Curve
Ex: In your notes draw the following curve. The data is N(10, 5).
The Empirical Rule
• Recall the characteristics of the Normal distribution said that
almost all of the area under the curve exists within three standard
deviations of the mean
•
How do we know what the almost means? For this we have the
A.K.A The
• The
Rule
states that;
•
of the data is within 1
of the mean
•
of the data is within 2
of the mean
•
of the data is within 3
of the mean
• Discussion Question: Where do you think this rule originated
from?
Estimating Areas
• Complete this example in your notes with a partner.
• Suppose the scores from your AP Stats exam are N(75, 5).
Answer the following questions.
1. Sketch the distribution
2. What percentage of the scores is within 60 and 90?
3. What percentage is the 50th percentile?
4. What score would be the 16th percentile?
Z-Scores
• A
is how many
away from the
a data point is
.
• Z-scores are used to find the
under the
by
standardizing it.
• The Standard Normal Curve is a Normal curve with a mean of
and standard deviation
.
• The formula for the z-score is as follows:
• Now to interpret this result we would say: A score of 82 in the test
is
.
• If your z-score is negative then the data is
deviations away from the mean.
standard
Using Z-scores
• Suppose we said that the show size for males in the U.S. is N(9.5,
1.25).
• How would you calculate the percentage of shoe sizes that are
below 10?
• We could use the
to estimate this, but an quicker
way is to use
.
• Let’s work through this example with our calculators
• Using the z-score formula we get: 𝑧 =
10−9.5
1.5
= .3333
• Now on the calculator press 2nd vars, normalcdf(
,
) and
you get the answer.
• If it were to ask you for the percentage above 10, you would press
2nd vars, normalcdf(
,
).
Now You Try
• Using the previous scenario answer the following questions in your
notes with a partner.
• Find the percentage of shoe sizes above 8.5
• Find the percentage of shoe sizes between 8.5 and 9.5
• Find the percentage of show sizes below 9.25
• Draw and shade the Normal Curves representing these scenarios
Slide 19:
• Discussion Question: How could we use the Empirical Rule to
check if the data follows a Normal Distribution?
The steps to checking the data for Normality are as follows
1. Check the
(68-95-99.7 percent of the data lie within
this boundary)
2. Check the
(The closer the
is to the mean
suggest that the data might be Normally distributed)
3. Draw a
or
in your calculator and check
C.U.S.S. The more symmetrical the graph the more evidence that
suggest that the data is Normally Distributed
Great White Sharks Activity
Below are the length of a sample of 44 Great white Sharks in feet. In
your notes use the steps previously stated to check if the length of this
sample of Great White Sharks is Normally Distributed? Write a
Paragraph describing your findings. Discuss your findings with your
partner.