Z 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09
0.0 0.0000 0.0040 0.0080 0.0120 0.0160 0.0199 0.0239 0.0279 0.0319 0.0359
0.1 0.0398 0.0438 0.0478 0.0517 0.0557 0.0596 0.0636 0.0675 0.0714 0.0753
0.2 0.0793 0.0832 0.0871 0.0910 0.0948 0.0987 0.1026 0.1064 0.1103 0.1141
：
1.8 0.4641 0.4649 0.4656 0.4664 0.4671 0.4678 0.4686 0.4693 0.4699 0.4706
1.9 0.4713 0.4719 0.4726 0.4732 0.4738 0.4744 0.4750 0.4756 0.4761 0.4767
... ... ... ... ... ... ... ... ... ... ...
... ... ... ... ... ... ... ... ... ...
Take
Z = 1.96,
And think
about this.
OK.
As
z = 1.9 + 0.06
Consider z = 1.96
Then go back to
the table.
The line and row for 1.9
And 0.06, Respectively,
cro each other
at...0.4750!
0.4750!
AREA = 1
Yes. That is the area when
z = 1.96.
Oh, and I forgot to mention—The
area betwn the probability density
function graph and the horizontal
axis is 1, regardle of whether it
is a standard normal distribution or
something else.
You separate the
number betwn
the first decimal
and the second
decimal?
Aha!
Let's Obtain the Probability 93
Now, pay aention, because
what I am going to explain
next is today's main dish.
I can't wait
to have it.
The area bounded by the standard
normal distribution and the horizontal
axis is the same as the probability!
I am now going show you two examples. Try
to foow along.
Wha...?
94 Chapter 5
Example I
1. In a normal distribution with an average of 45 and a standard deviation of 10, the shaded area in the
chart below is 0.5.
2. The ratio of students who scored 45 points or more is 0.5 (50% of all students tested).
3. When one student is randomly chosen from all students tested, the probability that the student’s score
is 45 or more is 0.5 (50%).
4. In a normal distribution of standardized “math test results,” the ratio of students with a standard score of
0 or more is 0.5 (50% of all students tested).
5. When one student’s results are randomly chosen from all of those tested in a normal distribution of
standardized “math test results,” the probability that the selected student’s standard score is 0 or more is
0.5 (50%).
All high school freshmen in prefecture B took a math test. After the tests were
marked, the test results turned out to follow a normal distribution with a mean
of 45 and a standard deviation of 10. Now, think carefully. The five sentences
below all have the same meaning.
0.5
0.4
0.3
0.2
0.1
0
0−1−2−3−4 1 2 3 4
0.05
0.04
0.03
0.02
0.01
0
454035302520151050 50 55 60 65 70 75 80 85 90 95 100
Let's Obtain the Probability 95
The mean score is
45, so we can draw a
syetric graph whose
top score is 45.
45 Or more points is
exactly equal to the
right half of this graph...
That’s 50%!
I’m smart
enough to
understand that!
Exactly!
Coect.
I am glad you are!
Then I’ give you
another example, a
trickier version of
our first one.
96 Chapter 5
Example II
1. In a normal distribution with a mean of 45 and a standard deviation of 10, the shaded area in the chart
below is 0.5 - 0.4641 = 0.0359.
2. The ratio of students who scored 63 points or more is 0.5 − 0.4641 = 0.0359 (3.59% of all students
tested).
3. When one student is randomly chosen from all those tested, the probability that the student’s score is 63
or more is 0.5 − 0.4641 = 0.0359 (3.59%).
4. In a normal distribution of standardized test results, the ratio of students with standard scores
(or z-scores) of 1.8 or more [(each value − average) ÷ standard deviation = (63 − 45) ÷ 10 = 18 ÷ 10 =
1.8] is 3.59% (0.5 − 0.4641 = 0.0359). You can also obtain this value from a table of standard normal
distribution.
5. When one student is randomly chosen from all those tested in a normal distribution of standardized
“math test results,” the probability that the student’s standard score is 1.8 or more is 0.5 − 0.4641 =
0.0359 (3.59%).
All high school freshmen in prefecture B took a math test. Now, think carefully.
The five sentences below all have the same meaning.
0.5
0.4
0.3
0.2
0.1
0
0−1−2−3−4 1 2 3 4
1.8
0.05
0.04
0.03
0.02
0.01
0
454035302520151050 50 55 60 65
63
70 75 80 85 90 95 100
Let's Obtain the Probability 97
Get The Manga Guide to Statistics now with the O’Reilly learning platform.
O’Reilly members experience books, live events, courses curated by job role, and more from O’Reilly and nearly 200 top publishers.
