Download Normal Curve and z scores AP Stats

Normal Curve, Standard Deviation as
Ruler, and Estimations about
Populations
AP Statistics
Normal Model
• When we take a sample
and get real data, we
display it as a
histogram.
• For example, the
histogram to the right
displays the distribution
of 1000 IQs. The mean
of the sample is about
100 and the standard
deviation is
approximately 15.
Normal Model
• All this distribution tells
us is information about
this single sample.
• In statistics, we want to
gain information about
a population.
• The information about
our sample can be
(under the right
circumstances) used to
gain information about
the population
Normal Model
• As long as the sample was
well obtained the
histogram provides a
good estimation of the
shape, center and spread
of the distribution.
• This estimation of the
population is illustrated
with a MODEL.
• If the sample is
approximately symmetric
the model we use is
called the NORMAL
MODEL
Normal Model
• The graph to the left is
the Normal Model that
represents the IQ scores
of all adults.
• It has a mean of 100
(highest point) and a
standard deviation of
15.
• This is called the
standard normal model
Normal Model
• The standard normal curve is a model—a
model of reality –not reality itself.
• It is not perfect—but will give us a good
general idea of how the population looks
Histogram
Real Data
Sample
Statistic
Histogram
Mean: y
Standard Deviation: s x
Normal Model
Model of Population
Population
Parameter
Normal Curve (or other)
Mean: 
Standard Deviation:

Histogram—real data
Normal model—model
of population
Normal Model
• Is always defined by the mean and standard
deviation N  ,  
Normal Model Summary
• Perfectly symmetric and unimodal
• Continues infinitely in both directions. It is asymptotic
to the horizontal axis as it approaches  
• It is defined by only two parameters, mean and
standard deviation
• Centered at mean
• The “points of inflection” where the curve changes
from curving downward to curving upward, occur at
exactly  1
• The total area under the curve equals 1.
Nearly Normal Condition
• When we use the normal model, we are
assuming that the data is normal (symmetric).
• Therefore, before you use a normal model to help
in your analysis, you need to check to make sure
the data is basically normal—remember, real
world data is not perfectly normal
• Use: Nearly Normal Condition: The shape of
the data’s distribution is unimodal and
symmetric
Nearly Normal Condition
Two ways to check the Nearly Normal Condition
1. Make a histogram
 Also, look at mean and median
2. Make a Normal Probability plot
If the distribution of the data is roughly
Normal, the plot is roughly a diagonal
straight line.
Normal Probability Plot
Normal
Not Normal
Normal Probability Plot—calculator
• We are plotting the
data along the y-axis in
this example
The Standard Deviation as a Ruler
John recently scored a 113 on Test A. The scores on the test are
distributed with a mean of 100 and a standard deviation of
10. Mary took a different test, Test B, and scored 263. The
scores on her test are distributed with a mean of 250 and a
standard deviation of 25. Which student did relatively better
on his particular test?
(A) John did better on his test
(B) Mary did better on her test
(C) They both performed equally well
(D) It is impossible to tell since they did not take the same test
(E) It is impossible to tell since the number of students taking
the test is unknown.
The Standard Deviation as a Ruler
• We need to have a level
playing field—we need a way
to make the numbers mean
the same thing.
• We do that by using a z-score
• A z-score measures how many
standard deviations a point is
from the mean
• NO UNITS for a z-score

y  y
z
s
Standard Deviation as Ruler
The Standard Deviation as a Ruler
• Standardizing data into z-scores does not
change the shape of the data.
• Standardizing data into z-scores does change
the center by making the mean 0.
• Standardizing data into z-scores does change
the spread by making the standard deviation 1
• Positive z-score –data point above mean
• Negative z-score—data point below mean
Standard Deviation as a Ruler:
Examples
• The SATs have a
distribution that has a
mean of 1500 and a
standard deviation of
250. Suppose you score
a 1850. How many
standard deviations
away from the mean is
your score?
• Suppose your friend
took the ACT, which
scores are distributed
with a mean of 20.8 and
a standard deviation of
4.8. What score would
your friend need to get
in order to have done as
well as you did on the
SATs?
Estimations About Populations
• Many times, we use the Normal Model to
make estimations or predictions about a
population.
• Later in the course, we will use them to
determine just how “unusual” something is
Estimations About Populations
What percent of the population of adults would
you expect to have an IQ between 70 and
130?
Estimations About Populations
Estimations About Populations
Estimations About Populations
The verbal section of the
SAT test is
approximately normally
distributed with a mean
of 500 and a standard
deviation of 100.
Approximately what
percent of students will
score between 400 and
600 on the verbal part
of the exam?
The verbal section of the
SAT test is
approximately normally
distributed with a mean
of 500 and a standard
deviation of 100.
Approximately what
percent of students will
score above 700?
Estimations About Populations
The verbal section of the SAT test is
approximately normally distributed with a
mean of 500 and a standard deviation of 100.
Approximately what percent of students will
score between 350 and 620?
Estimations About
Populations:Example
The verbal section of the
SAT test is
approximately normally
distributed with a mean
of 500 and a standard
deviation of 100.
Approximately what
percent of students will
score between 350 and
620?
Estimating About
Populations
Estimations About Populations
The results of a placement test for an exclusive
private school is normal, with a mean of 56
and a standard deviation of 12.
Approximately what percent of students who
take the test will score below a 40?
Estimations About Populations
The verbal section of the SAT test is
approximately normally distributed with a
mean of 500 and a standard deviation of 100.
What is the lowest score someone could
receive to be in the top 10% of all scores?
Estimations about the Population
Example
The verbal section of
the SAT test is
approximately
normally distributed
with a mean of 500
and a standard
deviation of 100.
What is the lowest
score someone
could receive to be
in the top 10% of all
scores?
Estimations About Populations
The results of a placement test for an exclusive
private school is normal, with a mean of 56
and a standard deviation of 12. The school
will only accept students who score in the top
15% of all scores. What is the minimum score
a student would need in order to be
accepted?
Estimations About Populations
The verbal section of the SAT test is
approximately normally distributed with a
mean of 500 and a standard deviation of 100.
What is the range of the middle 50% of data?
Estimation About Populations
What do you need to show?
1. Check Nearly Normal Condition
2. Draw normal curve model with proper notation
(use parameter notation)
3. Find values you are looking for in model and shade
in appropriate region
4. Convert to z-score
5. Find the area in the shaded region:
P  Value  p350  y  620  p1.5  z  1.2  .818
6. Interpret you results in context