# Powerpoint [WEB] 3.1 - z-scores and the Normal Model

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First found May 22, 2018

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```Standardized scores and the
Normal Model
Chapter 6
standardizing with z-scores
Walrus weights
The mean weight for an adult male
walrus is 1215 kg, with a standard
deviation of 82 kg.
Chuck (pictured
here) weighs 1321
kg. Find Chuck’s
standardized score
(z-score), then
interpret this in
context.
Standardizing with z-scores
or “exp”
obs  mean
z
st dev
z
x

We call the resulting values
standardized scores, or
z-scores.
more walrus weights
The mean weight for an adult FEMALE
walrus is 812 kg, with a standard
deviation of 67 kg.
Delilah weighs 680
kg (she’s been
watching her
figure!). Find
Delilah’s
standardized score
(z-score), then
interpret this in
context.
comparing walrus weights…
Based on the z-scores that we
calculated, who’s weight is MORE
UNUSUAL for their
Delilah?
z = 1.29
gender – Chuck or
z = -1.97
In an AP Comic Design class…
• Melody scored 84 on a test where the class
mean = 80 and a standard deviation of 4.
• Josh scored 90 on another test where the
mean = 87 with a standard deviation of 3.
Who scored better relative to the
other students in their AP Comic
Design class?
the Normal Model
(back to your own notes )

There is a model that shows up over and over in
Statistics, called the
Normal model
We use the Normal model to
APPROXIMATE
actual distributions
that are unimodal
and roughly
symmetric.
(it’s not exact)
Adult FEMALE walrus weights are APPROXIMATELY
NORMALLY DISTRIBUTED, with a mean of 812
kg, and a standard deviation of 67 kg.
Draw the Normal model for these female
weights.
68% of observations fall within 1 of 
95% of observations fall within 2 of 
99.7% of observations fall within 3 of 
The 68-95-99.7 Rule
(a.k.a. “the empirical rule”)
Adult FEMALE walrus weights are approximately
normally distributed, with a mean of 812 kg,
and a standard deviation of 67 kg.
What proportion of walrus weights is…
a) between 812 and 879 kg?
Adult FEMALE walrus weights are approximately
normally distributed, with a mean of 812 kg,
and a standard deviation of 67 kg.
What proportion of walrus weights is…
b)
less than 879 kg?
Adult FEMALE walrus weights are approximately
normally distributed, with a mean of 812 kg,
and a standard deviation of 67 kg.
What proportion of walrus weights is…
c)
between 745 and 946kg?
d)
greater than 946kg?
e)
less than 611kg?
Quick facts about the Normal model
1) The total area under the standard
normal curve is 1.0 (or 100%)
2) In theory, the normal curve
extends FOREVER in both
directions (the height never reaches zero)
Once we have standardized…
(converted everything into z-scores)
The
N(0,1) model is called the
standard Normal model
when we can’t use the 68-95-99.7 rule…?
one option is to use CALCULUS…
1
P( x) 
e
 2
 ( x )
2
(2 )
Gasp!
2
…take the
integral
under the curve…
so instead, we’ll use the z-table!
walruses revisited
Adult FEMALE walrus weights are approximately
normally distributed, with a mean of 812 kg,
and a standard deviation of 67 kg.
If we select an adult female walrus at
random, what is the probability that her
weight is…
a) less than 879 kg?
You are expected to communicate your process.
(what about MORE than 879 kg?)
Your work and drawings are an important part of that
Adult FEMALE walrus weights are approximately
normally distributed, with a mean of 812 kg,
and a standard deviation of 67 kg.
If we select an adult female walrus at
random, what is the probability that her
weight is…
b)
more than 780 kg?
You are expected to communicate your process.
Your work and drawings are an important part of that
Adult FEMALE walrus weights are approximately
normally distributed, with a mean of 812 kg,
and a standard deviation of 67 kg.
If we select an adult female walrus at
random, what is the probability that her
weight is…
c)
less than 600 kg?
You are expected to communicate your process.
Your work and drawings are an important part of that
Adult FEMALE walrus weights are approximately
normally distributed, with a mean of 812 kg,
and a standard deviation of 67 kg.
If we select an adult female walrus at
random, what is the probability that her
weight is…
d)
between 700 and 800 kg?
You are expected to communicate your process.
Your work and drawings are an important part of that
Adult FEMALE walrus weights are approximately
normally distributed, with a mean of 812 kg,
and a standard deviation of 67 kg.
If we select an adult female walrus at
random, what is the probability that her
weight is…
e) between 720 kg and 785
kg?
You are expected to communicate your process.
Your work and drawings are an important part of that
using the z-table in reverse
(working backwards with the Normal model)
What z-score corresponds with the 60th
percentile?
Adult FEMALE walrus weights are approximately
normally distributed, with a mean of 812 kg,
and a standard deviation of 67 kg.
f) Approximately what
weight represents the cutoff for the TOP 5% of adult
female walrus weights?
Adult FEMALE walrus weights are approximately
normally distributed, with a mean of 812 kg,
and a standard deviation of 67 kg.
g) Approximately what
weight represents the cutoff for the BOTTOM 20% of
Adult FEMALE walrus weights are approximately
normally distributed, with a mean of 812 kg,
and a standard deviation of 67 kg.
*h) What is the IQR for
Would the Normal model be appropriate
for this distribution?
No – to use the Normal model, the
distribution should be unimodal and
approximately symmetric.
(standardizing DOES NOT change the shape of the
distribution)
Hair Lengths
Students
zscore
1
32.0
0.378557
2
2.0
-1.01631
3
1.0
-1.0628
4
11.0 -0.597848
5
1.0
-1.0628
6
2.0
-1.01631
7
4.0 -0.923317
8
6.0 -0.830326
9
4.0 -0.923317
10
35.0
0.518044
11
2.0
-1.01631
12
40.0
0.750521
13
60.0
1.68043
14
47.0
1.07599
15
31.0
0.332062
16
6.0 -0.830326
17
17.0 -0.318875
18
7.0
-0.78383
19
33.0
0.425053
20
57.0
1.54095
21
34.0
0.471548
<new>
If a distribution is NOT
approximately normal…
• We
CAN
calculate a z-score
(and it still represents a # of SD’s from the
MEAN)
• We CANNOT use the normal
model (or the z-table) to find
a probability with that zscore.
California condors have a mean wingspan of
9.1 feet, with a standard deviation of 0.63
feet.
If the distribution of these wingspans is
approximately normal, what is the probability
that a randomly selected condor has a
a) less than
8 feet
wingspan
of…
b) at least 9.9 feet
c) between 8 feet and 10 feet
d) Find the cut-off (in feet)
for the largest 25% of
wingspans.
a) 0.0404
b) 0.1021
c) 0.8830
6-month old male babies have
a mean weight of 16.5 pounds.
My little nephew weighs 20
pounds, which places him at
the 95TH PERCENTILE for babies
(meanwhile he was at the 50 PERCENTILE for height…)
his age.
TH
What is the standard
deviation of weights for male
babies at 6 months of age?
the end
for now…
(homework #14 is VERY IMPORTANT,
and due next time)
```