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Final Topics for Exam 1
(Week 4)
Z-Score, Population Parameters
Z-Score
Consider the following 5 test scores
(out of 25 total points): 17, 18, 20, 22, 23
What are the units for this data?
“points”
What are the units for mean?
“points”
17
18
19
20
21
22
23
24
Z-Score
Consider the following 5 test scores
(out of 25 total points): 17, 18, 20, 22, 23
What are the units for variance?
y
y y
17
-3
9
18
-2
4
20
0
0
22
2
4
23
3
9
17
( y  y )2
18
s 
2
2
(
y

y
)

n 1
94049

 8.7
4
NOT “points”!!
19
20
21
22
23
24
Z-Score
Consider the following 5 test scores
(out of 25 total points): 17, 18, 20, 22, 23
What are the units for standard deviation?
y
y y
17
-3
9
18
-2
4
20
0
0
22
2
4
23
3
9
17
( y  y )2
18
s 
2
2
(
y

y
)

n 1
s  s2  3
19
20
21
22
94049

 8.7
4
“points”
23
24
Z-Score
Consider the following 5 test scores
(out of 25 total points): 17, 18, 20, 22, 23
How do we represent “s” graphically?
Think of “s” as a distance or a length
“These two values are
3 points away from
each other”
17
18
19
20
21
22
23
24
s  s2  3
Z-Score
Consider the following 5 test scores
(out of 25 total points): 17, 18, 20, 22, 23
How do we represent “s” graphically?
Think of “s” as a distance or a length
“These two values are
1 standard deviation
away from each other”
17
18
19
20
21
22
23
24
s  s2  3
Z-Score
Consider the following 5 test scores
(out of 25 total points): 17, 18, 20, 22, 23
How do we represent “s” graphically?
Think of “s” as a distance or a length
“These two values are
2 standard deviation
away from each other”
17
18
19
20
21
22
23
24
s  s2  3
Z-Score
Consider the following 5 test scores
(out of 25 total points): 17, 18, 20, 22, 23
Statisticians often want to answer the
question: “How many standard deviations
away from the mean is a certain value”
y y
z
s
17
18
19
The Z-Score calculates this
20
21
22
23
24
s  s2  3
Z-Score
Consider the following 5 test scores
(out of 25 total points): 17, 18, 20, 22, 23
Statisticians often want to answer the
question: “How many standard deviations
away from the mean is a certain value”
Unlike “r”, “z” can take on any values and is NOT restricted between
-1 and 1 (this example has smaller z values than most data sets)
-1
-.67
17
18
0
19
20
21
.67
1
22
23
24
y y
z
s
s  s2  3
Population Parameters
Consider the same 5 test scores
(out of 25 total points): 17, 18, 20, 22, 23
This is actually a sample of five observations
from a class with 50 students.
SAMPLE
POPULATION
Mean:
y
Mean:
μ
Variance:
s2
Variance:
σ2
Standard Dev:
s
Standard Dev:
σ
Sample Size:
n
Population Size: N
“mu”
“sigma”
We sample because we usually don’t know the population information (μ, σ2, and σ)
We call μ, σ2, and σ “parameters” – there are more and are usually “greek” letters
Other Topics to Know
Another name for Q1 is “25th percentile”,
because 25% of the values in the sample
fall below this line.
Similarly, Q2 is called the “50th percentile”
and Q3 is called the “75th percentile”
Other Topics to Know
For symmetric data, the mean is the best
measure of center.
For skewed data, the mean is affected by
outliers, so it is NOT the best measure of
center. Instead, use the median to
measure center.
Other Topics to Know
Sampling Frame: A list of elements from
which a sample is drawn.