Download Chapter 6 Part 1 Notes - peacock

Chapter 6: The Standard Deviation
as a Ruler and the Normal Model
Key Vocabulary:
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standard deviation
standardized value
rescaling
z-score
normal model
Calculator Skills:
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parameter
statistic
standard Normal model
68-95-99.7 Rule
normal probability plot
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N( 
, )
invNorm(
normal probability plot
normalcdf(
Chapter 6 Part 1 Notes: Standard Deviation and the Normal Model
Adding a constant to all of the values in a set of data adds the same constant to the measures of
_______________________. It does not, however, affect the ________________________.
Example: Add 5 to each value in the given set of data (on the left) to form a new set of data (on
the right). Then find the indicated measures of center and spread.
{5, 5, 10, 35, 45}.
Center:
{_____,
_,
,
,
}.
Center:
x=
x=
M=
M=
Mode =
Mode =
Spread:
Spread:
Range =
Range =
IQR =
IQR =
SD =
SD =
Multiplying a constant to all of the values in a set of data multiplies the same constant to the measures
of _____________________and ___________________.
Stats: Modeling the World – Chapter 6
Example: Multiply each value in the given set of data (on the left) by 2 to form a new set of data
(on the right). Then find the indicated measures of center and spread.
{5, 5, 10, 35, 45}.
Center:
{
,
,
,
,
}.
Center:
x=
x=
M=
M=
Mode =
Mode =
Spread:
Spread:
Range =
Range =
IQR =
IQR =
SD =
SD =
Summary of Linear Transformations:
•
Multiplying each observation by a positive number b multiples both measures of
__________________________________ and measures of
__________________________________ by b.
•
Adding the same number a (either positive or negative) to each observation adds a to
measures of _______________ and to _____________________, but does not change
measures of ______________________.
•
Linear transformations do not change the ______________________ of a distribution.
Your Turn:
Maria measures the lengths of 5 cockroaches that she finds at school. Here are her results (in
inches):
1.4, 2.2, 1.1, 1.6, 1.2
a. Find the mean and standard deviation of Maria’s measurements (use calc).
Stats: Modeling the World – Chapter 6
b. Maria’s science teacher is furious to discover that she has measured the cockroach lengths in
inches rather than centimeters (There are 2.54 cm in 1 inch). She gives Maria two minutes to
report the mean and standard deviation of the 5 cockroaches in centimeters. Find the mean
and standard deviation in centimeters.
Class Problem:
We have a company with employees with the following salaries:
1200 900 1400 2100 1800 1000 1300 700
1700
2300 1200
1. What is the mean and standard deviation of the company salaries?
mean = _________
st dev = _________
2. Suppose we give everyone a $500 raise. What is the new mean and standard deviation?
mean = _________
st dev = _________
3. Suppose we have to cut everyone’s pay by $500 due to the economy. What is the mean and
standard deviation now?
mean = _________
st dev = _________
4. Suppose we give everyone a 30% raise. What is the new mean and standard deviation?
mean = _________
st dev = _________
Stats: Modeling the World – Chapter 6
5. Suppose we cut everyone’s pay by 7%. What is the new mean and standard deviation?
mean = _________
st dev = _________
Standard deviation is a measure of spread, or________________________. The smaller the standard
deviation, the ___________________variability is present in the data. The larger the standard
deviation, the _____________________ variability is present in the data.
Standard deviation can be used as a ruler for measuring how an individual compares to a
_______________________. To measure how far above or below the mean any given data value is,
we find its _________________________________, or________________________________.
z 


To standardize a value, subtract the ____________________ and divide by the__________________.
Measure your height in inches. Calculate the standardized value for your height given that the average
height for women is 64.5 inches with a standard deviation of 2.5 inches and for men is 69 inches with a
standard deviation of 2.5 inches. Are you tall? ______________________________
zheight 
Suppose the average woman’s shoe size is 8.25 with a standard deviation 1.15 and the average male
shoe size is 10 with a standard deviation of 1.5. Do you have big feet? ________________________

zshoe 
Stats: Modeling the World – Chapter 6
Suppose Sharon wears a size 9 shoe and Andrew wears a size 9. Does Sharon have big
feet?_________________________ Does Andrew? ___________________________
zSharon  _____________________
zAndrew 


In order to compare values that are measured using different scales, you must first
________________________the values. The standardized values have no _____________and are
called __________________________. Z-scores represent how far the value is above the
_______________________(if_______) or below the _________________________(if _________).
Example:
z = 1 means the value is ______________ standard deviation ______________ the mean
z = -0.5 means the value is ______________of a standard deviation
___________________ the mean The ________________________the z-score, the more
unusual it is.
Your Turn:
Bob is 64 inches tall. The heights of men are unimodal symmetric with a mean of 69 inches
and standard deviation of 2.5 inches. How does Bob’s height compare to other men.
________________
Problems:
Which one of the following is a FALSE statement about a standardized value (z-score)?
a) It represents how many standard deviations an observation lies from the mean.
b) It represents in which direction an observation lies from the mean.
c) It is measured in the same units as the variable.
Rachael got a 670 on the analytical portion of the Graduate Record Exam (GRE). If GRE scores
are unimodal symmetric and have mean
standardized score?
a)
670  600
 2.33
30
b)
600  670
 2.33
30
= 600 and standard deviation s = 30, what is her
Stats: Modeling the World – Chapter 6
Standardized values, because they have no units, are therefore useful when comparing values that are
measured on different ____________________________, with different
______________________________, or from different ______________________________.
Your Turn:
Timmy gets a 680 on the math of the SAT. The SAT score distribution is Unimodal
symmetric with a mean of 500 and a standard deviation of 100. Little Jimmy scores a 27 on
the math of the ACT. The ACT score distribution is unimodal symmetric with a mean of 18
and a standard deviation of 6.
Who does better? (Hint: standardize both scores then compare z-scores)
___________________________________________________________________________
Your Turn:
Which is better, an ACT score of 28 or a combined SAT score of 2100?
•
ACT:
= 21, s = 5
• SAT: = 1500, s = 325
Assume ACT and SAT scores have unimodal symmetric distributions.
a)
ACT score of 28
b)
SAT score of 2100
c)
I don’t know
Class Problem:
A town’s January high temp averages 36 ̊F with a standard deviation of 10, while in July, the mean
high temp is 74 ̊F with a standard deviation of 8. In which month is it more unusual to have a day
with a high temp of 55 ̊F?
_____________________________________________________________________________
Stats: Modeling the World – Chapter 6
Your Turn:
The distribution of SAT scores has a mean of 500 and a standard deviation of 100. The
distribution of ACT scores has a mean of 18 and a standard deviation of 6. Jill scored a 680
on the math part of the SAT and a 30 on the ACT math test. Jack scored a 740 on the math
SAT and a 27 on the math ACT.
Who had the better combined SAT/ACT math score?
______________________________________________________________________
Smooth Curve:
Sometimes the overall pattern of a histogram is so regular that it can be described by a
_____________________________. This can help describe the location of
_____________________________ within the distribution. The distribution of a histogram
depends on the choice of classes, while with a ___________________________ it does not.
Smooth curve is a ________________________________ of the distribution.
The smooth curve describes what ___________________________ of the observations fall in each
range of values, not the frequency of observations like a histogram.
_________________________________ represents the proportion of observations in an interval.
The total area under the curve is _________________.
Properties of the Normal Distribution:
•
The curve is _______________________.
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The curve is ____________________________.
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The curve is __________________________ about the mean.
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The ___________________, ______________________, and _____________________
are located at the center of the distribution and are ________________to each other.
•
The curve is _______________________ (single mode)
•
The curve never touches the _________________________.
•
The total area under the normal curve is equal to _______________.
Stats: Modeling the World – Chapter 6
Standardizing the Normal Distribution:
By standardizing values, we shift the distribution so that the mean is__________________,
and rescale it so that the standard deviation is ___________________________.
Standardizing does not change the ___________________________________of the
distribution.
The standard Normal model has mean ___________________and standard deviation
___________________.
The Normal model is determined by _____________________and ____________________.
We use the Greek letters sigma and mu because this is a __________________________; it
does not come from actual ______________________________. Sigma and mu are the
________________________________that specify the model.
Stats: Modeling the World – Chapter 6
The larger sigma, the _________________________________spread out the normal model
appears. The inflection points occur a distance of _______________________on either side
of __________________________.
To standardize Normal data, subtract the _________________(________________) and divide
by the ________________________ (__________________).
z =______________________________
Stats: Modeling the World – Chapter 6
Standard Normal Dist. Problems:
Which one of the following is a FALSE statement about the standard normal
distribution?
a) The mean is greater than the median.
b) It is symmetric.
c) It is bell-shaped.
d) It has one peak.
If you knew that the  = 0 and  = 3, which normal curve would match the data?
a) Dataset 1
b) Dataset 2
Which one of the following is a FALSE statement about the standard normal curve?
a) Its standard deviation  can vary with different datasets.
b) It is bell-shaped.
c) It is symmetric around 0.
d) Its mean  = 0.
Suppose the lengths of sport-utility vehicles (SUV) are normally distributed with mean 
= 190 inches and standard deviation  = 5 inches. Marshall just bought a brand-new
SUV that is 194.5 inches long and he is interested in knowing what percentage of SUVs
is longer than his. Using his statistical knowledge, he drew a normal curve and labeled
the appropriate area of interest. Which picture best represents what Marshall drew?
a) Plot A
b) Plot B