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Math II
UNIT QUESTION: Can real world
data be modeled by algebraic
functions?
Standard: MM2D1, D2
Today’s Question:
How is a normal distribution used to
curve test scores?
Standard: MM2D1d
The Normal Distribution
Section 7.4
The normal distribution and standard
deviations
Suppose we measured the right foot length of 30
teachers and graphed the results.
Assume the first person had a 10 inch foot. We could
create a bar graph and plot that person on the graph.
Number of People with
that Shoe Size
If our second subject had a 9 inch foot, we would add
her to the graph.
As we continued to plot foot lengths, a
8
pattern would begin to emerge.
7
6
5
4
3
2
1
4
5
6
7
8
9 10 11 12 13 14
Length of Right Foot
Number of People with
that Shoe Size
Notice how there are more people (n=6) with a 10 inch right foot
than any other length. Notice also how as the length becomes
larger or smaller, there are fewer and fewer people with that
measurement. This is a characteristics of many variables that
we measure. There is a tendency to have most measurements
in the middle, and fewer as we approach the high and low
extremes.
If we were to connect the top of each bar, we
8
would create a frequency polygon.
7
6
5
4
3
2
1
4
5
6
7
8
9 10 11 12 13 14
Length of Right Foot
Number of People with
that Shoe Size
You will notice that if we smooth the lines, our data almost
creates a bell shaped curve.
8
7
6
5
4
3
2
1
4
5
6
7
8
9 10 11 12 13 14
Length of Right Foot
You will notice that if we smooth the lines, our data almost
creates a bell shaped curve.
Number of People with
that Shoe Size
This bell shaped curve is known as the “Bell Curve” or the
“Normal Curve.”
8
7
6
5
4
3
2
1
4
5
6
7
8
9 10 11 12 13 14
Length of Right Foot
Number of Students
Whenever you see a normal curve, you should imagine the
bar graph within it.
9
8
7
6
5
4
3
2
1
12 13 14
15 16 17 18 19 20 21 22
Points on a Quiz
The
Nowmean,
lets look
mode,
at quiz
andscores
median
forwill
51all
students.
fall on the same
value in a normal distribution.
12
13 13
12+13+13+14+14+14+14+15+15+15+15+15+15+16+16+16+16+16+16+16+16+
17+17+17+17+17+17+17+17+17+18+18+18+18+18+18+18+18+19+19+19+19+
14 14 14 14
19+ 19+20+20+20+20+ 21+21+22 = 867
15 15 15 15 15 15
12, 13, 13, 14, 14, 14, 14, 15, 15, 15, 15, 15, 15, 16, 16, 16, 16, 16, 16, 16, 16, 17, 17, 17, 17, 17, 17, 17, 17, 17, 18, 18, 18, 18, 18, 18, 18, 18, 19, 19, 19, 19, 19, 19, 20, 20, 20, 20, 21, 21, 22
16 16 16 16 16 16 16 16
867 / 51 = 17
17 17 17 17 17 17 17 17 17
18 18 18 18 18 18 18 18
19 19 19 19 19 19
Number of Students
20 20 20 20
21 21
9
8
7
6
5
4
3
2
1
22
12 13 14
15 16 17 18 19 20 21 22
Points on a Quiz
Normal distributions (bell
shaped) are a family of
distributions that have the
same general shape. They
are symmetric (the left side is
an exact mirror of the right
side) with scores more
concentrated in the middle
than in the tails. Examples of
normal distributions are
shown to the right. Notice
that they differ in how spread
out they are. The area under
each curve is the same.
The normal distribution and standard
deviations
34%
2.35%
34%
13.5%
In a normal distribution:
The total area under the curve is 1.
13.5%
2.35%
The normal distribution and standard
deviations
In a normal distribution:
Approximately 68% of scores will fall within one
standard deviation of the mean
The normal distribution and standard
deviations
In a normal distribution:
Approximately 95% of scores will fall within two
standard deviations of the mean
The normal distribution and standard
deviations
In a normal distribution:
Approximately 99.7% of scores will fall within three
standard deviations of the mean
When you have a subject’s raw score, you can use the mean and
standard deviation to calculate his or her standardized score if the
distribution of scores is normal. Standardized scores are useful
when comparing a student’s performance across different tests, or
when comparing students with each other.
2.35% 13.5% 34%
34% 13.5% 2.35%
z-score
-3
-2
-1
0
1
2
3
T-score
20
30
40
50
60
70
80
IQ-score
65
70
85
100
115
130
145
200
300
400
500
600
700
800
SAT-score
The number of points that one standard deviations equals
varies from distribution to distribution. On one math test, a
standard deviation may be 7 points. If the mean were 45, then
we would know that 68% of the students scored from 38 to 52.
2.35%
24
31
13.5%
38
On another test, a
standard deviation may
equal 5 points. If the mean
were 45, then 68% of the
students would score from
40 to 50 points.
34%
34%
13.5%
2.35%
45
52
59
Points on Math Test
2.35%
30
35
13.5%
66
34%
34%
13.5% 2.35%
40
45
50
55
Points on a Different Test
60
Using standard deviation units to
describe individual scores
Here is a distribution with a mean of 100 and standard deviation of 10:
80
-2 sd
90
-1 sd
100
What score is one sd below the mean?
What score is two sd above the mean?
110
1 sd
90
120
120
2 sd
Using standard deviation units to
describe individual scores
Here is a distribution with a mean of 100 and standard deviation of 10:
80
-2 sd
90
-1 sd
100
110
1 sd
120
2 sd
How many standard deviations below the mean is a score of 90?
1
How many standard deviations above the mean is a score of 120?
2
Using standard deviation units to
describe individual scores
Here is a distribution with a mean of 100 and standard deviation of 10:
80
-2 sd
90
-1 sd
100
110
1 sd
120
2 sd
What percent of your data points are < 80?
2.50%
What percent of your data points are > 90?
84%
Class Work
In class:
Workbook Page 277 #1-21
Homework
Monday in class: Finish the Workbook
page 277 #1-21 and
then do page 266 #1-11
