I gave a quiz with 10 questions

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Lets suppose

• I gave a quiz with 10 questions
• There are 36 students in my class
• The students score is the number of questions
  the student answered correctly
• The following is a list of the scores

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Raw Data- Test Scores

• 0
• 1
• 1
• 2
• 2
• 2
• 3
• 3
• 3
• 3

• 4
• 4
• 4
• 4
• 4
• 5
• 5
• 5
• 5
• 5

• 5
• 6
• 6
• 6
• 6
• 6
• 7
• 7
• 7
• 7

• 8
• 8
• 8
• 9
• 9
• 10

3
Frequency Data
Looking at the scores, I see that 1 student
missed every question and got a score of 0
But one student got every question correct and
got a score of 10
4
If we were to graph the scores they would look
like this
5
If we were to graph the scores they would look
like this
This graph shows that 3 students got scores of 2
correct on the quiz
3
2
6
And if we were to connect the dots with a line,
it would start to resemble a bell shaped (aka
normal) curve
Number of students with each score
Test Scores
7
Along the bottom (the x axis), we are measuring
the trait in which we are interested (in this
case, test scores)
On the vertical axis (y) we are measuring the
number of students
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5
Number of students with each score
4
3
2
1
5
0
10
1
2
3
7
6
4
8
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By convention, the horizontal line is called the
x axis and the vertical line is called the y axis
Test Scores
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The formula for the normal distribution curve is
                 
Any curve that fits this formula is a normal
curve 
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9
Many things have a normal distribution when
graphed

• The following slide shows what the normal
  distribution looks like when actual people are
  used to form the picture.

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Average Height (µ) 5 10
Normal distribution of height of males (blue) and
females (red)
Average Height (µ) 5 4
These Photos were downloaded from the following
web site http//acsweb.fmarion.edu/Pryor/bellcurv
e.htm
1
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Both distributions are normal although they have
different shapes. The men are more spread out
while the women are more centered in the middle.
These Photos were downloaded from the following
web site http//acsweb.fmarion.edu/Pryor/bellcurv
e.htm
2
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Class photo at Ben Hill Griffin Stadium ("The
Swamp" at UF, 2001) These are photographs of
students in my biology class at the University of
Florida. I arranged the students according to
height, from shortest to tallest (left-to-right,
within each photo), in 1-inch increments.
Therefore, one column of students equals a group
of students of the same height. The purpose of
this class project was to depict an interesting
biological phenomenon called a bell curve
distribution, or normal distribution (explained
below). These photos were published in a biology
textbook (C. Starr and R. Taggart. 2003. The
Unity and Diversity of Life. 10th Ed. Page 189.)
as a demonstration of this phenomenon. When
arranged as a whole, it is obvious that most
students have a near-average height, while only a
few students are much shorter or taller than the
average. The average height corresponds to the
highest column of students in each picture (the
average height was 5'4" for females, wearing
orange shirts and 5'10" for males, wearing blue
shirts). This creates a symmetrical,
"bell-shaped" pattern that is typical of most
biological measurements. Other examples of
biological measurements that make a bell curve
are body weights, heart rates, and blood pressure
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Two Normal Curves with the Same Mean (µ) (center)
and Different Standard Deviations (s) (spread)
2
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Two Normal Curves with Different Means (µ)
(center) and Equal Standard Deviations (s)
(spread)
2
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Two Normal Curves with Different Means (µ)
(center) and Different Standard Deviations (s)
(spread)
2
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FYI

• Every trait that is normally distributed has its
  own bell shaped curve
• If you know the average score (the mean) and the
  spread of the scores (the standard deviation),
  you can draw the bell shaped curve.

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Facts about a Normal (aka Bell Shaped) Curve

• It has a bell shape
• It is symmetrical around the mean (µ)
• The measured trait is continuous
• The tails dont touch the x axis
• Every different combination of Mean and Standard
  Deviation has its own curve
• Most important All normal curves have the same
  mathematical properties in common

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50 of the area under the curve lies below the
mean (red area)
and 50 of the area under the curve lies above
the mean (white area)
4
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All Normal Curves have these same percentages

• About 2/3 of the cases are within or 1 SD from
  the mean
• About 95 of the cases are within or- 2 SD from
  the mean
• About 99 of all cases are within or- 3 SD from
  the mean

3
20
More precisely

• But for our purposes now, we can use the previous
  rounded percentages

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Notice that even if the spread (the SD/s) changes

• The areas under the curve do not change

22
Notice that even if the average (the mean/µ)
changes
2
6
4
8
12
10
14
? 12
16
? 5

• The areas under the curve do not change

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Since the area under the curve is constant

• As you change the shape from wide to narrow (i.e.
  the SD/s) decreases, the height becomes taller
  (note the change in shape from red to blue)

24
Note that changing the spread (standard
deviation/s) did not change the center (mean/µ)
mean/µ
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Why is it important that the percentages under
the curve are constant?

• Because knowing this allows us to make
  comparisons between different individuals or
  groups and to draw conclusions

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Age Distribution
For Example
Half the people are older than the mean age and
half the people are younger than the mean age
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Age Distribution
But what if we want to know where someone fits on
the curve in relationship to the rest of the
group or population?
?
?
?
?
?
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Age Distribution
Or if we know their correct place under the
curve, how do we calculate how many are above or
below them?
?
?
?
?
?
?
?
?
29
The area under the curve corresponds to the
percentage of people or subjects that have the
trait we are measuring
Important FACT
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The red area under this curve equals all or 100
of the area
The red area under this curve equals half or 50
of the area
The red area under this curve equals ¼ or 25 of
the area
31
Thus, we can see that 50 of the population is
older than Tony and 50 is younger. If we chose
someone from the population at random, there
would be a 50/50 chance that the person would be
older than Tony- and a 50/50 chance that the
person would be younger.
Age Distribution
But how does Nick compare in age to the rest of
the population?
Tony
Nick
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The hatched area below shows the portion of the
population that is younger than Nick
The dotted area below shows the portion of the
population that is older than Nick
o
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o
Q How can we turn these areas into numbers that
we can understand?
Mean (µ)
33
A If we know the mean (µ) and standard deviation
(s) of any curve, we can draw the curve and can
find any position under the curve
s
µ
Remember our formula for the normal curve
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We can also estimate areas under the curve
82?
86?
14?
?
12?

• But calculating a percentage from looking at a
  graph is not very accurate

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But do we really want to do all this work when we
can use the internet or Excel to calculate all
the values for us?
Click on the following link to go to one of the
many web sites that calculates areas of the
normal curvehttp//www.stat.ucla.edu/dinov/course
s_students.dir/Applets.dir/NormalCurveInteractive.
html
36
But what is the process behind comparing
something or someone in one group to something or
someone in another group?
37
How do we compare different things?
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Just as in the US, although there are many
different languages

• English is the standard language that we use to
  communicate

39
The same is true about normal curves

• Although each normal curve has its own mean (µ)
  and standard deviation (s) and its own unique
  shape, we can convert any data to a standard
  normal distribution

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How do we do this?
Z Scores
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Converting a raw score to a Z Score does not
change its position in relation to the other
scores.

• The score will still keep its same place in the
  bell curve

43
When we take the scores from our test..

• 0
• 1
• 1
• 2
• 2
• 2
• 3
• 3
• 3
• 3

• 4
• 4
• 4
• 4
• 4
• 5
• 5
• 5
• 5
• 5

• 5
• 6
• 6
• 6
• 6
• 6
• 7
• 7
• 7
• 7

• 8
• 8
• 8
• 9
• 9
• 10

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And convert all the scores to Z Scores using the
formula
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Substituting in the following

• X is an individual score that we want to convert
  to a Z Score
• µ is the mean of the numbers and 5
• s is the Standard Deviation and 2.449

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For example..A score of 0 is converted to a Z
Score in the following manner

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The Z Scores for the test will look like this

• -2.04
• -1.63
• -1.63
• -1.22
• -1.22
• -1.22
• -.816
• -.816
• -.816
• -.816

• -.408
• -.408
• -.408
• -.408
• -.408
• 0
• 0
• 0
• 0
• 0

• 0
• .408
• .408
• .408
• .408
• .408
• .816
• .816
• .816
• .816

• 1.22
• 1.22
• 1.22
• 1.63
• 1.63
• 2.04

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And the Graph of the Z Scores will look like this
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RAW SCORES
Note that converting the raw scores to Z Scores
did not change the shape of the graph.
Z SCORES
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RAW SCORES
i.e. The New distribution of Z Scores is still
the same as the original distribution of the data
that we started with.
Z SCORES
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RAW SCORES
It is as though we kept the same information but
gave it a different name.
Z SCORES
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The distribution of the scores DID NOT change
RAW SCORES
Z SCORES
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The mean and standard deviation of the original
scores DID change
RAW SCORES
µ 5 ? 2.449
5
µ 0 ? 1
Z SCORES
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BUT...
RAW SCORES
The values of the Z Scores are EQUIVALENT to the
values of the original distribution
Z SCORES
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For the original scores µ 5 And for the Z
Scores µ 0
RAW SCORES
Z SCORES
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For the original scores s 2.449 For the new Z
distribution s 1
RAW SCORES
Z SCORES
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Note that ½ of the Z scores (the ones below the
mean) are negative and ½ are positive (the ones
above the mean)
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Resource List

• http//acsweb.fmarion.edu/Pryor/bellcurve.htm
• http//www.route79.com/journal/archives/000130.htm
  l
• http//www.csun.edu/gk45683/Lecture20620-20Z-s
  cores20and20Normal20Curve.pdf
• http//www.utah.edu/stat/introstats/web-text/Norma
  l_Distribution/Back20to20Menu20Locator20Map

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So that instead of saying that the mean 5, we
say that the m
RAW SCORES
µ changed from 5 to 0
Z SCORES
s changed from 2.449 to 1
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We do this by thinking of the area under the
curve as the probability of occurrence or the
percentage of time that we can expect something
to occur.
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