Standard Scores ZScores and the Normal Distribution

1
Standard Scores (Z-Scores) and the Normal
Distribution
2
We have previously discussed the concepts of
variance and standard deviation. This lessons
continues to expand the uses of these measures.
In review, we calculated standard deviation using
the formula S
An important property of the standard deviation
is that in a set of data most of the data are
located within 2 standard deviations of the mean
of the data.
3
Whenever you attempt to compare results from 2 or
more tests that may have different means and
standard deviations, you cant just directly
compare these scores. For example a student
scoring 70 on exam 1 and 54 on exam 2, doesnt
tell you how well she did relative to her
classmates. The 70 might rank her in the lower
third of the class on exam 1, while the 54 might
rank her in the upper third of the class on exam
2. To combat this problem we can convert both
measures to standard scores (z-scores) using the
mean and standard deviation for the two exams.
Then, a comparison begins to have more meaning.
4
Z-Score formula A score X from a distribution
with mean , and standard deviation S has a
z-score given by
5
Teresa scores a 78 on the entrance exam at school
A and an 82 at school B. At which school did she
have the better score?
To answer this you need to know that the mean for
school A was 70 with S 12, and for school B the
mean was 76 with S 16.
Then, ZA (78-70)/12 8/12 0.75
Z (x-mean)/S and, ZB (82-76)/16 6/16
0.375 So Teresa actually scored better at school
A.
6
The concept of z-scores is closely related to the
so-called NORMAL DISTIBUTION (also known as the
Bell-curve or Gaussian distribution).
7
(No Transcript)
8
(No Transcript)
9
(No Transcript)
10
(No Transcript)
11
(No Transcript)
12
(No Transcript)
13
(No Transcript)
14
(No Transcript)
15
(actually13.5 )
16
(No Transcript)
17
(No Transcript)
18
The total area under the curve is equal to 1 (or
100). The picture below represents the area
between the mean and 1 standard deviation. The
value 1 on the numberline corresponds to a
z-score1. So this represents the area between 0
and 1, or 0ltZlt1.
19
To find the value .3413, you look up a Z-value of
1 in a normal curve table such as Table 10-22.
The decimal value associate with Z1.0 is .3413 .
This decimal represents the fraction of the
normal curve (or the , or the probability later
on) that is between the mean and the Z-value you
look up in the table.
0
1
P(0 lt Z lt 1) .3413
20
The picture below represents the probability that
corresponds to a z-score between 0 and 1.79,
0ltZlt1.79.
21
There are also probabilities associated with the
left half of the NC. The picture below shows a
probability that a score will be between Z0 and
Z -1.28, written -1.28 lt Z lt 0. You get this
by looking up positive 1.28 in the table.
22
Looking up a Z-value in the table tells you the
area or probability between the mean (0) and the
Z-value. Suppose you want to know the
probability that a value is greater than Z 1.5.
Then look up Z 1.5 and you will find .4332 in
the table. From the mean (0) to the right is
.5000 or 50. To find the area to the right of
Z1.5, subtract .5000 - .4332 (the .4332
represents the area from 0 to 1.5). This gives
P(Z gt 1.5) .0668
23
The picture below represents the probability Z is
between -1.27 and 2.15. To find this you look up
-1.27 in the table and find .3980. This
represents the area from 0 to -1.27. Then look
up 2.15 and find .4822. This represents the area
from 0 to 2.15. You now combine (add) these two
regions to get the final probability, .8822 .
24
The picture below represents the probability a
Z-value is greater than -1.38, P(Zgt -1.38). To
find this look up 1.38 in the table to get .4162.
This represents the amount from -1.38 to 0.
Then to the right of 0 there is an additional
50, .5000. Together these make up the combined
probability of .9162.