Download Chapter 6 Part 1 Powerpoint - peacock

AP Statistics
The Standard Deviation as a Ruler and the
Normal Model
Chapter 6
Learning Goals
1. Understand how adding (subtracting) a constant or
multiplying (dividing) by a constant changes the
center and/or spread of a variable.
2. Understand that standardizing uses the standard
deviation as a ruler.
3. Know how to calculate the z-score of an
observation and what it means.
4. Know how to compare values of two different
variables using their z-scores.
5. Recognize when a Normal model is appropriate.
Learning Goals
6. Recognize when standardization can be used to
compare values.
7. Be able to use Normal models and the 68-95-99.7
Rule to estimate the percentage of observations
falling within 1, 2, or 3 standard deviations of the
mean.
8. Know how to find the percentage of observations
falling below any value in a Normal model using a
Normal table or appropriate technology.
9. Know how to check whether a variable satisfies the
Nearly Normal Condition by making a Normal
Probability plot or histogram.
Learning Goal 1
Understand how adding (subtracting) a constant or multiplying
(dividing) by a constant changes the center and/or spread of a variable.
Learning Goal 1:
Linear Transformation of Data
• Linear transformation
– Shifting (moving left or right) the data
or Rescaling (making the size larger or
smaller) the data.
– Changes the original variable x into the
new variable xnew given by
xnew = a + bx
• Adding the constant a shifts all values of x
upward (right) or downward (left) by the
same amount.
• Multiplying by the positive constant b
changes the size of the values or rescales
the data.
Learning Goal 1:
Shifting Data
• Shifting data:
– Adding (or subtracting) a
constant amount to each value
just adds (or subtracts) the same
constant to (from) the mean. This
is true for the median and other
measures of position too.
– In general, adding a constant to
every data value adds the same
constant to measures of center
and percentiles, but leaves
measures of spread unchanged.
Learning Goal 1:
Shifting Data Example: Adding a Constant
• Given the data: 2, 4, 6, 8, 10
– Center: mean = 6, median = 6
– Spread: s = 3.2, IQR = 6
• Add a constant 5 to each value, new data
7, 9, 11, 13, 15
– New center: mean = 11, median = 11
– New spread: s = 3.2, IQR = 6
• Effects of adding a constant to each data
value
– Center increases by the constant 5
– Spread does not change
– Shape of the distribution does not
change
Learning Goal 1: Example - Subtracting a Constant
• The following histograms show a shift from men’s actual
weights to kilograms above recommended weight.
• Weights of 80 men age
19 to 24 of average height
(5'8" to 5'10") 𝑥 = 82.36 kg.
• NIH recommends maximum healthy
weight of 74 kg. To compare their
weights to the recommended
maximum, subtract 74 kg from each
weight; 𝑥 new = 𝑥 – 74.
So, 𝑥 new = 𝑥 – 74 = 8.36 kg.
1.
No change in shape
2.
No change in spread
3.
Shift by 74
Shift Down
Learning Goal 1:
Rescaling Data
• Rescaling data:
– When we divide or multiply all
the data values by any constant
value, all measures of position
(such as the mean, median and
percentiles) and measures of
spread (such as the range, IQR,
and standard deviation) are
divided and multiplied by that
same constant value.
Learning Goal 1:
Rescaling Example: Multiplying by a Constant
• Given the data: 2, 4, 6, 8, 10
– Center: mean = 6, median = 6
– Spread: s = 3.2, IQR = 6
• Multiple a constant 3 to each value, new data:
6, 12, 18, 24, 30
– New center: mean = 18, median = 18
– New spread: s = 9.6, IQR = 18
• Effects of multiplying each value by a constant
– Center increases by a factor of the
constant (times 3)
– Spread increases by a factor of the
constant (times 3)
– Shape of the distribution does not change
Learning Goal 1: Example - Rescaling Data
• The men’s weight data set measured weights in kilograms. If
we want to think about these weights in pounds, we would
• Change from kilograms to pounds
rescale the data:
•
•
•
•
•
•
Weights of 80 men age 19 to 24, of
average height (5'8" to 5'10")
𝑥 = 82.36 kg
min=54.30 kg
max=161.50 kg
range=107.20 kg
s = 18.35 kg
•
•
•
•
•
(multiple each observation by 2.2 lb/kg):
𝑥 new = 2.2(82.36)=181.19 lb
minnew = 2.2(54.30)=119.46 lb
maxnew = 2.2(161.50)=355.3 lb
rangenew= 2.2(107.20)=235.84 lb
snew = 2.2(18.35) = 40.37 lb
1.
No change in shape
2.
Increase in spread
3.
Shift by to the right
Shifts up
Size (spread) increases
Learning Goal 1:
Summary of Linear Transformations
• Multiplying each observation by a
positive number b multiples both
measures of center (mean and median)
and measures of spread (IQR and
standard deviation) by b.
• Adding the same number a (either
positive or negative) to each
observation adds a to measures of
center and to quartiles, but does not
change measures of spread.
• Linear transformations do not change
the shape of a distribution.
Learning Goal 1:
Summary of Linear Transformations
• Linear transformations do not affect the
shape of the distribution of the data.
-for example, if the original data is rightskewed, the transformed data is rightskewed.
• Example, changing the units of data from
minutes to seconds (multiplying by 60
sec/min).
Assembly Time (seconds)
Assembly Time (minutes)
30
20
10
0
Shape remains
the same
25
Frequency
Frequency
30
20
15
10
5
0
Learning Goal 1: Example
Los Angeles Laker’s Salaries (2000)
𝑥 = $4.14 mil, Med. = $2.6 mil, s = $4.76 mil, IQR = $3.6 mil
(a)
Suppose that each member of the team receives a
$100,000 bonus for winning the NBA championship.
How will this affect the center and spread?
$100,000 = $0.1 mil
Learning Goal 1: Example - Solution
a) The mean and median will all increase
by $0.1 mil and the standard deviation
and IQR will not change.
Original: 𝑥 = $4.14 mil, Med. = $2.6 mil,
s = $4.76 mil, IQR = $3.6 mil
𝑥 new = 0.1 + 4.14 = $4.24 mil
Med.new = 0.1 + 2.6 = $2.7 mil
snew = no change = $4.76 mil
IQRnew = no change = $3.6 mil
Learning Goal 1: Example
(b)
Each player is offered a 10%
increase base salary. How will
this affect the center and
spread?
𝑥 = $4.14 mil, Med. = $2.6 mil,
s = $4.76 mil, IQR = $3.6 mil
Learning Goal 1: Example - Solution
(b)
The mean, median, IQR and
standard deviation will all increase
by a factor of 1.1 (100% + 10% = 110%,
as decimal 1.1).
Original: 𝑥 = $4.14 mil, Med. = $2.6 mil,
s = $4.76 mil, IOR = $3.6 mil
𝑥 new = (1.1)(4.14) = $4.55 mil
Med.new = (1.1)(2.6) = $2.86 mil
snew = (1.1)(4.76) = $5.24 mil
IQRnew = (1.1)(3.6) = $3.96 mil
Learning Goal 1: Your Turn
•
Maria measures the lengths of 5
cockroaches that she finds at school. Here
are her results (in inches):
1.4
a.
b.
2.2
1.1
1.6
1.2
Find the mean and standard deviation of
Maria’s measurements (use calc).
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.
Learning Goal 1: 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 = _________
Learning Goal 1: Class Problem (continued)
4. Suppose we give everyone a 30% raise. What is the new
mean and standard deviation?
mean = _________
st dev = _________
5. Suppose we cut everyone’s pay by 7%. What is the new
mean and standard deviation?
mean = _________
st dev = _________
Learning Goal 2
Understand that standardizing uses the standard deviation as a
ruler.
Learning Goal 2:
Comparing Apples to Oranges
• In order to compare data values
with different units (apples and
oranges), we need to make sure we
are using the same scale.
• The trick is to look at how the values
deviate from the mean. Look at
whether the data point is above or
below the mean, and by how much.
• Standard deviation measures that,
the deviation of the data values
from the mean.
Learning Goal 2:
The Standard Deviation as a Ruler
• As the most common measure of
variation, the standard deviation
plays a crucial role in how we look
at data.
• The trick in comparing very
different-looking values is to use
standard deviations as our ruler.
• The standard deviation tells us how
the whole collection of values
varies, so it’s a natural ruler for
comparing an individual to a group.
Learning Goal 2:
The Standard Deviation as a Ruler
• We compare individual data values
to their mean, relative to their
standard deviation using the
following formula:
x  x

z
s
• We call the resulting values
standardized values, denoted as z.
They can also be called z-scores.
Learning Goal 3
Know how to calculate the z-score of an observation
and what it means.
Learning Goal 3: Standardizing with z-scores
A z-score measures the number of standard deviations that a data
value 𝑥 is from the mean 𝑥.
When 𝑥 is 1 standard deviation larger
than the mean, then z = 1.
(x  x)
z
s
for x  x  s, z 
( x  s)  x s
 1
s
s
When 𝑥 is 2 standard deviations larger
than the mean, then z = 2.
for x  x  2 s,
z
( x  2s)  x 2s

2
s
s
When x is larger than the mean, z is positive.
When x is smaller than the mean, z is negative.
Learning Goal 3:
Standardizing with z-scores
• A z-score puts values on a common
scale. Standardized values have no
units.
• z-scores measure the distance of each
data value is from the mean in
standard deviations.
• z-scores farther from 0 are more
extreme. z-scores beyond -2 or 2 are
considered unusual.
• A negative z-score tells us that the data
value is below the mean, while a
positive z-score tells us that the data
value is above the mean.
Learning Goal 3: Standardizing with z-scores
• Gives a common scale.
2.15 SD
Z=-2.15
This Z-Score tells
us it is 2.15
Standard
Deviations from
the mean
– We can compare two
different distributions with
different means and standard
deviations.
• Z-Score tells us how many
standard deviations the
observation falls away from
the mean.
• Observations greater than
the mean are positive when
standardized and
observations less than the
mean are negative.
Learning Goal 3:
Standardizing with z-scores - Example
•
If 𝑥 is from a unimodal symmetric
distribution with mean of 100 and
standard deviation of 50, the z-score
for 𝑥 = 200 is
x  x 200  100
z

 2.0
s
50
•
This says that 𝑥 = 200 is two
standard deviations (2 increments of
50 units) above the mean of 100.
Learning Goal 3:
Standardizing with z-scores – 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.
Learning Goal 3:
Problem
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.
Learning Goal 3:
Problem
Rachael got a 670 on the analytical
portion of the Graduate Record Exam
(GRE). If GRE scores are unimodal
symmetric and have mean 𝑥 = 600 and
standard deviation s = 30, what is her
standardized score?
a)
670  600
 2.33
30
b)
600  670
 2.33
30
Learning Goal 4
Know how to compare values of two different variables
using their z-scores.
Learning Goal 4:
Benefits of Standardizing
• Standardized values have been
converted from their original units
to the standard statistical unit of
standard deviations from the mean
(z-score).
• Thus, we can compare values that
are measured on different scales,
with different units, or from
different populations.
Learning Goal 4:
Standardizing – Comparing Distributions
• The men’s combined skiing event in the in
the winter Olympics consists of two races:
a downhill and a slalom. In the 2006
Winter Olympics, the mean slalom time
was 94.2714 seconds with a standard
deviation of 5.2844 seconds. The mean
downhill time was 101.807 seconds with a
standard deviation of 1.8356 seconds. Ted
Ligety of the U.S., who won the gold
medal with a combined time of 189.35
seconds, skied the slalom in 87.93 seconds
and the downhill in 101.42 seconds.
• On which race did he do better compared
with the competition?
Learning Goal 4:
Standardizing – Comparing Distributions
•
Slalom time (𝑥): 87.93 sec.
Slalom mean 𝑥 : 94.2714 sec.
Slalom standard deviation (s): 5.2844 sec.
x  x

z
s
•
zSlalom 
87.93  94.2714
 1.2
5.2844
Downhill time (𝑥): 101.42 sec.
Downhill mean 𝑥 : 101.807 sec.
Downhill standard deviation (s): 1.8356 sec.
zDownhill 
•
101.42  101.807
 0.21
1.8356
The z-scores show that Ligety’s time in the slalom is
farther below the mean than his time in the downhill.
Therefore, his performance in the slalom was better.
Learning Goal 4:
Standardizing – 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)
Learning Goal 4: Standardizing – Your Turn
Which is better, an ACT score of 28 or a combined
SAT score of 2100?
• ACT: 𝑥 = 21, s = 5
• SAT: 𝑥 = 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
Learning Goal 4: Standardizing – 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?
Learning Goal 4:
Standardizing – Combining z-scores
Because z-scores are standardized
values, measure the distance of each
data value from the mean in standard
deviations and have no units, we can
also combine z-scores of different
variables.
Learning Goal 4:
Standardizing – Example: Combining z-scores
• In the 2006 Winter Olympics men’s
combined event, Ted Ligety of the
U.S. won the gold medal with a
combined time of 189.35 seconds.
Ivica Kostelic of Croatia skied the
slalom in 89.44 seconds and the
downhill in 100.44 seconds, for a
combined time of 189.88 seconds.
• Considered in terms of combined zscores, who should have won the
gold medal?
Learning Goal 4: Solution
• Ted Ligety:
zSlalom 
zDownhill
• Combined z-score: -1.41
• Ivica Kostelic:
87.93  94.2714
 1.2
5.2844
101.42  101.807

 0.21
1.8356
zSlalom 
89.44  94.2714
 0.91
5.2844
zDownhill 
100.44  101.807
 0.74
1.8356
• Combined z-score: -1.65
• Using standardized scores, overall Kostelic did better
and should have won the gold.
Learning Goal 4:
Combining z-scores - 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?
Learning Goal 5
Recognize when a Normal model is appropriate.
Learning Goal 5:
Smooth Curve
• Sometimes the overall pattern of a
histogram is so regular that it can be
described by a Smooth Curve.
• This can help describe the location
of individual observations within the
distribution.
Learning Goal 5:
Smooth Curve
• The distribution of a histogram depends
on the choice of classes, while with a
smooth curve it does not.
• Smooth curve is a mathematical model of
the distribution.
– How?
• The smooth curve describes what
proportion of the observations fall in each
range of values, not the frequency of
observations like a histogram.
• Area under the curve represents the
proportion of observations in an interval.
• The total area under the curve is 1.
Learning Goal 5: Mathematical Model
A density curve is a mathematical model of a distribution.
The total area under the curve, by definition, is equal to 1, or 100%.
The area under the curve for a range of values is the proportion of all observations for
that range.
A mathematical model more represents a population then a sample like a histogram.
Therefore, when calculating z-scores for a mathematical model we use 𝜇 and 𝜎 , the
population mean and standard deviation, instead of 𝑥 and s, sample mean and
standard deviation.
Histogram of a sample with the
smoothed, density curve
describing theoretically the
population.
z
x

Learning Goal 5:
The Normal Model
• There is no universal standard for zscores, but there is a model that
shows up over and over in Statistics.
• This model is called the Normal
Model (You may have heard of “bellshaped curves.”).
• Normal models are appropriate for
distributions whose shapes are
unimodal and roughly symmetric.
• These distributions provide a
measure of how extreme a z-score
is.
Learning Goal 5:
The Normal Model
• Normal Model: One Particular class
of distributions or model.
1. Symmetric
2. Single Peaked
3. Bell Shaped
• All have the same overall shape.
Learning Goal 5:
The Normal Model or Normal Distribution
• The normal distribution is
considered the most important
distribution in all of statistics.
• It is used to describe the
distribution of many natural
phenomena, such as the height of a
person, IQ scores, weight, blood
pressure etc.
9-50
Learning Goal 5:
The Normal Distribution
• The mathematical equation for the
normal distribution is given below:
 x    /2 2
2
y
e
Not required
to know.
 2
where e  2.718,   3.14,  = population mean, and  =
population standard deviation.
Learning Goal 5:
Properties of the Normal Distribution
• When this equation is graphed for a
given  and , a continuous, bellshaped, symmetric graph will result.
• Thus, we can display an infinite
number of graphs for this equation,
depending on the value of  and .
• In such a case, we say we have a
family of normal curves.
• Some representations of the
normal curve are displayed in the
following slides.
9-52
Learning Goal 5:Properties of the Normal Dist.
Here, means are the same ( = 15)
while standard deviations are
different ( = 2, 4, and 6).
0
2
4
6
8
10
12
14
16
18
20
22
24
26
28
30
Here, means are different
( = 10, 15, and 20) while standard
deviations are the same ( = 3)
0
2
4
6
8
10
12
14
16
18
20
22
24
26
28
30
Learning Goal 5:
Properties of the Normal Distribution
Normal distributions with the same mean but with
different standard deviations.
9-54
Learning Goal 5:
Properties of the Normal Distribution
Normal distributions with different means but with
the same standard deviation.
9-55
Learning Goal 5:
Properties of the Normal Distribution
Normal distributions with different means and
different standard deviations.
9-56
Learning Goal 5: Describing a Normal Dist.
The exact curve for a particular normal distribution is
described by its Mean (μ) and Standard Deviation (σ).
μ located at the center of
the symmetrical curve
σ controls
the spread
Normal Distribution
Notation: N(μ,σ)
Learning Goal 5:
Properties of the Normal Distribution
• These normal curves have similar
shapes, but are located at different
points along the x-axis.
• Also, the larger the standard
deviation, the more spread out the
distribution, and the curves are
symmetrical about the mean.
• A normal distribution is a
continuous, symmetrical, bellshaped distribution.
9-58
Learning Goal 5:
Properties of the Normal Distribution
• Summary of the Properties of the
normal Distribution:
• The curve is continuous.
• The curve is bell-shaped.
• The curve is symmetrical about the
mean.
• The mean, median, and mode are
located at the center of the
distribution and are equal to each
other.
• The curve is unimodal (single mode)
• The curve never touches the x-axis.
• The total area under the normal curve
is equal to 1.
9-59
Learning Goal 5: Not Normal Curves
• Why
a)
b)
c)
d)
Normal curve gets closer and closer to the horizontal axis, but
never touches it.
Normal curve is symmetrical.
Normal curve has a single peak.
Normal curve tails do not curve away from the horizontal axis.
Learning Goal 5:
More Normal Distribution
• For a normal distribution the Mean
(μ) is located at the center of the
single peak and controls location of
the curve on the horizontal axis.
• The standard deviation (σ) is located
at the inflection points of the curve
and controls the spread of the
curve.
Learning Goal 5: Inflection Points
• The point on the curve where the curve changes from
falling more steeply to falling less steeply (change in
curvature – concave down to concave up).
Inflection point
Inflection point
• Located one standard deviation (σ) from the mean (μ).
• Allows us to visualize on any normal curve the width of one
standard deviation.
Learning Goal 5:
More Normal Model
• There is a Normal model for every
possible combination of mean and
standard deviation.
– We write N(μ,σ) to represent a Normal
model with a mean of μ and a standard
deviation of σ.
• We use Greek letters because this mean
and standard deviation are not numerical
summaries of the data. They are part of
the model and a model is more like a
population. They don’t come from the
data. They are numbers that we choose to
help specify the model.
• Such numbers are called parameters of
the model.
Learning Goal 5:
More Normal Model
• Summaries of data, like the sample
mean and standard deviation, are
written with Latin letters. Such
summaries of data are called
statistics.
• When we standardize Normal data,
we still call the standardized value a
z-score also, and we write
z
x

Learning Goal 5:
Standardizing the Normal Distribution
• All normal distributions are the
same general shape and share many
common properties.
• Normal distribution notation:
N(μ,σ).
• We can make all normal
distributions the same by measuring
them in units of standard deviation
(σ) about the mean (μ), z-scores.
• This is called standardizing and gives
us the Standard Normal Curve.
Learning Goal 5:
Standardizing the Normal Distribution
• How do linear transformations apply to zscores?
• When we convert a data value to a zscores, we are shifting it by the mean (to
set the scale at 0) and then rescaling by
the standard deviation (to reset the
standard deviation to 1).
– Standardizing into z-scores does not
change the shape of the distribution.
– Standardizing into z-scores changes the
center by making the mean 0.
– Standardizing into z-scores changes the
spread by making the standard
deviation 1.
Learning Goal 5: Standardizing the Normal Dist.
• We can standardize a variable that has a normal
distribution to a new variable that has the standard
normal distribution using the z-score formula:
Substitute your
variable as x
z
BAM! Pops out your
z-score
x

Then divide by your
Standard Deviation
Subtract the mean
from your variable
Learning Goal 5: Standardizing the Normal Dist.
x
x

Learning Goal 5: Standardizing the Normal Dist.
Standardizing Data into z-scores
The Standard Normal Distribution
Learning Goal 5: Standardizing the Normal Dist.
Distribution – any 𝝁 and 𝝈


X
Learning Goal 5: Standardizing the Normal Dist.
z 
Distribution – any 𝝁 and 𝝈
x

Standard Normal Distribution
𝝁 = 𝟎 𝒂𝒏𝒅 𝝈 = 𝟏

= 1

X
= 0
Results in a Standardized Normal Distribution (curve)
One Distribution → One set of areas under the curve → One Table
Z
Learning Goal 5:
Standardizing the Normal Distribution
• Subtracting Mu from each value X
just moves the curve around, so
values are centered on 0 instead of
on Mu.
• Once the curve is centered, dividing
each value by sigma>1 moves all
values toward 0, smushing the
curve.
Learning Goal 5: Standard Normal Dist. - Example
Learning Goal 5: Standard Normal Dist. - Example
Distribution 𝝁 = 𝟓, 𝝈 = 𝟏𝟎 𝒂𝒏𝒅 𝒙 = 𝟔. 𝟐
 = 10
= 5 6.2 X
Learning Goal 5: Standard Normal Dist. - Example
x   6.2  5
z

 .12

10
Distribution 𝝁 = 𝟓, 𝝈 = 𝟏𝟎 𝒂𝒏𝒅 𝒙 = 𝟔. 𝟐
 = 10
= 5 6.2 X
Standard Normal Distribution
𝝁 = 𝟎, 𝝈 = 𝟏 𝒂𝒏𝒅 𝒛 =. 𝟏𝟐
=1
= 0 .12
Z
The area under the original curve and the standard normal curve are the same.
Learning Goal 5:
The Standard Normal Curve
• Let x be a normally distributed variable with
mean μ and standard deviation σ, and let a
and b be real numbers with a < b. The
percentage of all possible observations of x
that lie between a and b is the same as the
percentage of all possible observations of z
that lie between (a −μ)/σ and (b−μ)/σ. This
latter percentage equals the area under the
standard normal curve between (a −μ)/σ and
(b−μ)/σ.
Learning Goal 5:
Standard Normal Curve and z-scores
• Same as with any Normal
Distribution.
• A z-score gives us an indication of
how unusual a value is because it tells
us how far it is from the mean.
• A data value that sits right at the
mean, has a z-score equal to 0.
• A z-score of 1 means the data value is
1 standard deviation above the mean.
• A z-score of –1 means the data value
is 1 standard deviation below the
mean.
Learning Goal 5:
Standard Normal Curve and z-scores
• How far from 0 does a z-score have
to be to be interesting or unusual?
• z-scores beyond -2 or 2 are
considered unusual.
• Remember that a negative z-score
tells us that the data value is below
the mean, while a positive z-score
tells us that the data value is above
the mean.
Learning Goal 5:
The Standard Normal Model
• Once we have standardized, we
need only one model:
– The N(0,1) model is called the
Standard Normal model (or the
Standard Normal distribution).
• Be careful—don’t use a Normal
model for just any data set, since
standardizing does not change the
shape of the distribution.
Learning Goal 5:
Properties of the Standard Normal Dist.
•
•
•
•
•
•
Shape – normal curve
Mean (μ) = 0
Standard Deviation (σ) = 1
Horizontal axis scale – Z score
No vertical axis
Notation: N(0, 1)
Learning Goal 5:
Standard Normal Dist. Problem
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.
Learning Goal 5:
Standard Normal Dist. Problem
If you knew that the  = 0 and  = 3, which
normal curve would match the data?
a) Dataset 1
b) Dataset 2
Learning Goal 5:
Standard Normal Dist. Problem
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.
Learning Goal 5:
Standard Normal Dist. Problem
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