Transcript Chapter 6: The Standard Deviation as a Ruler and

Chapter 6: The Standard Deviation
as a Ruler and the Normal Model
AP Statistics
Shifting and Rescaling Data
Suppose your class was given a test . The mean
score was 85%, the median was 80%, the
standard deviation was 5% and the IQR was
7%. What if I made a mistake on the test and
need to add 3 points to everyone’s grade?
What would happen to those summary
statistics? What if, instead, I decided to
multiply everyone’s grade by 1.5? Then what
would happen to those summary statistics?
Shifting and Rescaling Data
• Adding (or subtracting) a constant to every
data value adds (or subtracts) the same
constant to measures of position (mean,
median, percentiles, min, max) , but will leave
measures of spread (range, IQR, standard
deviation) unchanged.
• This represents a “shift” of the data
• The shape of the distribution will remain the
same.
Shifting and Rescaling Data
• Multiplying (or dividing) all the data values by
any constant, all measures of position (mean,
median, percentiles, min, max) and all
measures of spread (range, IQR, standard
deviation) are multiplied (or divided) by that
same constant.
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
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
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?
Normal Curve
• When we take a sample and get real discrete
data, we display it as a histogram.
• This histogram also displays a good estimation
of the population parameter and the shape of
the true distribution.
• For example, if we take a sample of 1000
people and their IQ, the sample will have a
mean of 100, a standard deviation of 15, and
will be normally distributed. SO WILL THE
POPULATION (mostly)
Normal curve
• Here is a histogram of
IQ scores. The y-axis
scale is in hundreds.
• This sample can also be
used to model the
entire population.
Normal curve
• We could then model
the IQ scores of all
adults with the density
curve to the right.
• This is called a standard
normal model
• The highest point in the
graph represents the
mean.
Normal Curve
• The standard normal curve is a model—a
model of reality –not reality itself
• Therefore, the mean and standard deviation
of the model are not summary statistics. They
are values we use to help us specify the
model.
• Therefore, when we create a model for real
world data we always define it by: N   ,  
Standard Normal Curve
• Typically, we
standardize the data
first (convert into zscores)—therefore
creating the standard
normal curve model:
N 0 ,1 
Normal Curve
Normal Curve and Standard Deviation
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
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 68-95-99.7 Rule
• One of the main goals of statistics is to find
out how extreme certain values are. For
example, is an IQ of 110 extremely high?
What about 125?
• The way we do this is by determining how
likely it is to find a value that far from the
mean
• We can find them precisely (soon) or we can
find them by using the 68-95-99.7 Rule
The 68-95-99.7 Rule
The 68-95-99.7 Rule
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 400 and 600 on the verbal part
of the exam? GO THROUGH COMPLETE
ANSWER!!!! See pg 110 for example
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 above 700?
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?
This cannot be done by using 68-95-99.7 Rule
Need to use the properties of normal curve and
technology
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?
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:
area 350  y  620   area   1 . 5  z  1 . 2   . 818
6. Interpret you results in context
Example
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?
Example
(need to find cutoff)
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?
Example
(need to find cutoff)
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?
Example
WORK
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?