standard and nonstandard normal curve

1. The Most Important Probability Distribution in Statistics

2. The Normal Curve
• In the figure below, we can visualize these different properties of a
normal distribution
• The mean, median, and mode are equal.
• It is symmetrical about the center
• The total area under the normal curve is 100% or 1.0.
• The tails of curve are asymptotical relative to the horizontal axis. In
other words, the curve never touches the horizontal axis.
• The spread of the data from the middle of the curve is represented by
the standard deviation(σ) and is measured along the baseline.

5.  Suppose X~N(
 Form a new random variable by subtracting
the mean  from X and dividing by the
standard deviation :
(X
 This process is called standardizing the
random variable X.

6.  (X is also a normal random variable;
we will denote it by Z:
Z = (X
 has mean 0 and standard deviation 1:
E(Z) =  = 0; SD(Z) =  =1.
1
 The probability distribution of Z is called
the standard normal distribution.

7.  If X has mean  and stand. dev. , standardizing a
particular value of x tells how many standard
deviations x is above or below the mean .
 Exam 1: =80, =10; exam 1 score: 92
Exam 2: =80, =8; exam 2 score: 90
Which score is better?
1
exam
on
92
than
better
is
2
exam
on
90
1.25
8
10
8
80
90
z
1.2
10
12
10
80
92
z
2
1
=
=

=
=
=

=

8. X
8
3 6 9 15
0
µ = 9 and  = 3
Z
0 1 2 3
-1
-2
-3
.5
.5
µ = 0 and  = 1
(X-9)/3
Nonstandard Normal Curve
Standardized Normal
Curve
18

11. Segmentation under a Standard Normal Distribution

12. For the data above on IQ
• Assume that:
• Mean= 150 and sd=40
• So, z(110)= (110-150)/40=-1 ;
z(190)=(190-150)/40= +1

13. Discussion:
Based on the results,
68% or the greater
majority of students
have an IQ level of at
least 110 but no
more than 190. It
should be noted that
the top 16% have an
IQ level of a least
190, while the
bottom 16% do have
IQ level less than 110.
Further, around 95%
of them have IQ level
of at least 70 but no
more than 230.
Mean=150
Sd=40

14. Areas Under the Normal Curve
 Around 68 percent of the area under the normal curve is
within one standard deviation of the mean.
μ ± σ
• Around 95 percent is within two standard
deviations of the mean.
μ± 2σ
• Practically all is within three standard deviations of
the mean.
μ ± 3σ

15. EXAMPLE
 Suppose that Adversity Quotient(AQ) of Senior High
School students in Zamboanga City follows a normal
distribution with a mean of 150 and a standard
deviation of 5.
Without further calculation we can say that:
 Around 68% of the population of senior high
school students have no less than 145 AQ level but not
more than 155. Also, the top 16% have at least 155
AQ level, while the bottom 16% have less than 145 AQ
level.
 And around 95% of them have at least 140 AQ level
but no more than 160.

16. Example
 The distribution with respect to the level of
awareness of adult Filipinos on the issue of
autism is approximately normal with mean 2.91
and a standard deviation of 1.01. Interpret the
and discuss the results.
 Note: 1- Not aware, 2-Slightly aware, 3- Moderately aware, 4-Much aware, 5-Very Much Aware
 Description:
 About 68% or a greater majority of adult Filipinos are at
least slightly aware to no more than much aware level on
the issue of autism.
 While, around 16% of adult Filipinos are somewhat
slightly or not aware at all on the issue of autism and
around the same percentage(16%) are no less than
much aware.
That is, from 1.9 to 3.92