Download Power Point: Working with Normal distributions

How do you determine your
score percentile on the
math section of the SAT?
I just got my score report for the
SAT. If I made a 630 on the Math
section, what percent of students did
I score as well or better than?
In this lesson, you will learn
how to find percentiles by
using a normal curve to
represent the distribution of
univariate data.
Let’s
Review
Let’s
Review
Remember that normal distributions
are symmetric, bell-shaped curves
with no significant gaps or outliers
that approximately follow the
empirical rule:
Let’s
Review
Let’s
Review
The empirical rule states
that for a normal
distribution:
•about 68% of the data lie within one
standard deviation of the mean
•about 95% of the data lie within two
standard deviations of the mean
•about 99.7% of the data lie within three
standard deviations of the mean.
Let’s
Review
Let’s
Review
The distribution on the following dotplot is
approximately normal. It is symmetric
about the mean, bell-shaped and
approximately follows the empirical rule.
Collection 7
0
2
4
6
8
10
x
12
14
16
18
A Common
Let’s
Review Mistake
If your SAT math score is one
standard deviation to the right of the
mean, it does not mean that your
score is at the 68th percentile.
Percentiles represent the total percent
that score at or below your score.
CoreReview
Lesson
Let’s
Suppose that the distribution of math
SAT scores in your state is
approximately normal with a mean of
520 and a standard deviation of 110.
How can we graph the distribution
without knowing every individual
score?
CoreReview
Lesson
Let’s
Well, statisticians often approximate
normal distributions of data to a
smooth curve that displays the shape
of the data without all the lumpiness.
CoreReview
Lesson
Let’s
We know the normal curve for the
distribution of SAT math scores in
your state would look something
like this:
CoreReview
Lesson
Let’s
Distribution Plot
Normal, Mean=520, StDev=110
0.004
Density
0.003
0.002
0.001
0.000
190
300
410
520
X
630
740
850
CoreReview
Lesson
Let’s
Now we can use this normal curve
to determine the percent of SAT
math scores that are at or below
your score of 630. Since 630 is
one standard deviation to the right
of the mean, the area of interest
would look like this:
CoreReview
Lesson
Let’s
CoreReview
Lesson
Let’s
We know 50% of your state’s SAT
math scores are at or below 520.
We determine that a score of 630 is
one standard deviation above the
mean, so we must add this
additional area in order to
determine your score percentile.
CoreReview
Lesson
Let’s
We also know 68% of the data on a
normal curve is between one
standard deviation left and right of
the mean, so we just need to add
half of that area to 50% in order to
have our area of interest:
50% + 1/2(68%)=84%
CoreReview
Lesson
Let’s
By using the normal curve for this
distribution of data, we have now
been able to determine that your
score on the math SAT is in the 84th
percentile of all test takers in your
state.
CoreReview
Lesson
Let’s
We now see that although 68
percent of the data was between
one standard deviation of the
mean, that is not equivalent to the
percent of data that is located to
the left. Percentiles represent the
total area left of a variable of
interest.
In this lesson you have
learned how to find
percentiles by using a normal
curve to represent the
distribution of univariate
data.
Guided
Practice
Let’s
Review
Now, how would you determine the
percentile of a student who scores a
410 on the SAT math section in your
state?
Guided
Practice
Let’s
Review
Since a score of 410 would be one
standard deviation to the left of the
mean, you would need to subtract
the area from 50%, so the area of
interest is 50%-1/2(68%)=16%.
Guided
Practice
Let’s
Review
So a student who scores a 410 on the
SAT math section is in the 16th
percentile of all test takers in the
state.
Guided
Practice
Let’s
Review
Finally, determine the percentile of a
student who scores a 740 on the SAT
math section in your state.
Guided
Practice
Let’s
Review
A score of 740 would be located two
standard deviations above the mean:
50%+1/2(95%)=97.5%.
A score of 740 on the SAT math
section in your state is in the 97.5th
percentile of all test takers.
Extension
Let’s
ReviewActivities
The distribution of exam scores in the
math department of a large university is
approximately normal with a mean of 81
and a standard deviation of 7. Sketch and
label a normal curve to represent this
distribution. Next, shade the region under
the curve that represents the percent of
exam scores at or below 67. What
percent of the data would be in this
shaded region?
Extension
Let’s
ReviewActivities
The distribution of heights of adult
American men is approximately normal
with a mean of 69.2 inches and a standard
deviation of 3.1 inches. Determine the
height percentile for a male who is 75.4
inches tall.
Extension
Let’s
ReviewActivities
The distribution of exam scores in the
math department of a large university is
approximately normal with a mean of 81
and a standard deviation of 7. What
percent of students score higher than a 95
on the exam?
Quick Quiz
Let’s
Review
Use the information provided here to
answer the questions on the next
slide:
In 2009, the mean ACT score was 18
and the standard deviation was 6.
The distribution of ACT scores is
approximately normal.
Quick Quiz
Let’s
Review
A score of 24 on the ACT in 2009
would place a student in what
percentile?
About what percent of the 2009
ACT test-takers scored at or
below a 6?
Lesson Slides Rubric
Use this rubric to ensure your
lesson plan is great!
Math Rubric
Criteria for Success
Storyline or Arc of
the Lesson


Hook Slide




Objective Slide


Let’s Review



There is a clear arc to the lesson. One slide leads
naturally to the next so that there is a flow and a building
of meaning
The teacher poses a simple question that illicits the
response, “yeah, I do wonder how that works…”
The question is short
A relevant example is included when it is short and
further pulls the learner in
The question mirrors what the student will learn, then
need to do later in the guided practice
The objective follows the form (you will learn X by doing
Y)
Is concise and follows the form provided in the examples
Reminds the student of how this lesson fits with other
lessons (the lesson, however, should still be able to stand
on its own)
Reminds the student of important vocabulary
Is as concise as possible
Uses visuals whenever possible
Things to avoid











All the components of the lesson are there but they seem
disconnected, as if the author wrote each without thinking
about how they fit into the whole.
The question seems formulaic, inauthentic, or overly “schoolish” (message: you have to learn this because you’re in school
rather than, this is genuinely interesting)
The hook is overly-complicated and potentially confusing
The question does not parallel the guided practice questions
Does not follow the form
Is overly vague in describing either the X or the Y
Is too long
Is written for teachers but not students
Is either too detailed or not detailed enough in connecting the
lesson to other lessons
Leaves out important touch points
Makes the lesson overly dependent on the other lessons
(student will be confused or feel like they’ve made a mistake,
if they watch this lesson alone)
Common Mistake





Modeling a Way of
Looking at It





Objective Review

Guided Practice


Points out a common mistake that students make
Concisely explains the thought process that leads to that
mistake
Clearly models a way to look at the standard
Uses visuals as often as possible to show how the way of
looking works
Is in “think aloud” format. The teacher is opening up
his/her thought process to the student
Takes advantage of every opportunity to explain why the
math works the way it works
Engages the learner by asking questions along the way to
build suspense
Uses an an example to show the way in action
Explains how this way of looking at it shows why the
common mistake (see above) is a mistake
Reviews the objective in a way which conveys, “we’ve
come full cicle and now you see this objective with new
eyes.”
Serves as a “let’s pull this all together” moment that
helps organize the lesson in the learner’s mind
Is at the same difficulty level modeled in the lesson
Is connected to the initial hook question


Isn’t actually a mistake students make (too simple)
Is confusing or vague

Focuses on the algorhythm (or trick) instead of on showing a
way of looking at the math
Fails to use visuals to show a way
Fails to explain his/her thinking along the way. The teacher
effortlessly runs through the steps as if it’s all obvious and
easy
Does not ask any questions along the way to pull the learner in
Misses opportunities to explain the why behind the math
Fails to explain why this way of looking at the math addresses
the common mistake






Creates abrupt feeling between the lesson and the reviewing
(subtext: “we’re done with this lesson, let’s quickly bring it to
a close.”)


Seem unrelated to the hook question
Is at a different difficulty level than that modeled in the lesson


Extension Activity
Suggestions





Aesthetics




Includes a suggestion for a struggling student who needs
more opportunities for practice
Includes a suggestion for students who seem to get it
but need more practice
Includes a suggestion for students who get it and are
ready to be challenged further
Suggestions should clearly build from the approach in
the core lesson
The slides use the correct colors (blue, green, red) in the
correct sequence.
The slides use the correct fonts
The slides use handwriting and the handwriting appears
as written in the right places
The slides only use the headers/titles provided
The slides use the provided visuals or include visuals
created by the author or LearnZillion
The slides use animation, highlighting, and circling to
scaffold the learning, keeping the eye focused on what
the teacher is introducing/explaining
The slides clean and uncluttered. The visuals and text do
not exceed the maximum amount (see tutorial for
example of maximum)




Does not include differentiation
Does not thoughtfully connect or flow from the lesson
Does not clearly build from the approach in the core lesson
Does not give a range of activities


The slides use other colors or vary the order of the colors
The slides add new headers/titles that aren’t part of the
template
The slides use clip art
The slides are cluttered
Animation is distracting and feels more like sizzle than part of
the steak



Graphic and Image Templates
Copy and Paste items from these slides to
make your presentation look great!
You can copy and paste
these items into any slide
Green text box that appears letter by letter
Green text box that fades in
Blue text box that appears letter by letter
Blue text box that fades in
Red text box that appears letter by letter
Red text box that fades in
You can copy and paste these items into any
slide—make sure you copy both the bubble
and the text!
Do I feel
strongly
about it?
Do I have a
lot to say?
Do I feel
strongly about
it?
Do I have a lot
to say?
Do I have a lot
to say?
You can copy and paste these items
into any slide. You can resize them as
needed!
Use black text
Use black text
when you
write in me
please! Also,
keep the text
left-justified
rather than
centered!
when you
write in me
please! Also,
keep my text
left-justified
rather than
centered!
All arrows can be recolored by changing the
“shape fill.” You can also resize them or
rotate them!
You can use these when discussing
main ideas or steps in a process…
1
[Write first step here…]
2
[Write second step here…]
3
[Write third step here…]
You can resize any of these boxes and
use them to highlight text or ideas.
Let’s Review
Let’s
Review
A Common
Let’s
ReviewMistake
Core Lesson
Guided
Practice
Let’s
Review
Extension
Activities
Let’s
Review
Quick
Assessment
Let’s
Review