11th grade

Good Morning!
(Week 3, Day 1)
 11th grade

The Normal Curve

Learning objectives:
01 I can illustrate the normal distribution;

02 I can identify characteristics of a normal

curve; and

03 I can apply the empirical rule in finding the

percentage of normal distribution.
Normal Probability Distribution
A probability distribution of continuous random
variables.
It shows graphical representations of random
variables obtained through measurement.
Properties of Normal Curve
The graphical representation of the normal
distribution is popularly known as a normal curve.
The normal curve is described clearly by the
following characteristics.
Properties of Normal Curve
1. The normal curve is bell-shaped.
Properties of Normal Curve
1. The normal curve is bell-shaped.
Properties of Normal Curve
2. The curve is symmetrical about its center.
Properties of Normal Curve
3. The mean, median and mode coincide at the center

Mean
Median
Mode
Properties of Normal Curve
4. The width of the curve is determined by the standard
deviation of the distribution.

σ
Properties of Normal Curve
5. The tails of the curve are plotted in both directions and
flatten out indefinitely along the horizontal axis. The tails
are thus asymptotic to the baseline.
Properties of Normal Curve
6. The total area under a normal curve is 1. This means
that the normal curve represents the probability.
Empirical Rule
The empirical rule is better known as 68% - 95% -
99.70% rule.
This rule states that the data in the distribution lies
within one, two, and three of the standard deviation
from the mean are approximately 68%, 95%, and
99.70%, respectively.
Empirical Rule
The empirical rule is better known as 68% - 95% -
99.70% rule.

σ
Empirical Rule
The empirical rule is better known as 68% - 95% -
99.70% rule.

2σ
Empirical Rule
The empirical rule is better known as 68% - 95% -
99.70% rule.

3σ
Empirical Rule
The empirical rule is better known as 68% - 95% -
99.70% rule.

68%
Empirical Rule
The empirical rule is better known as 68% - 95% -
99.70% rule.

95%
Empirical Rule
The empirical rule is better known as 68% - 95% -
99.70% rule.

99.70%
Empirical Rule

Since the area of a normal curve is equal to 1 or 100% as

stated on its characteristics, there are only a few data
which is 0.30% falls outside the 3-standard deviation
from the mean.
Example:
The distribution of the grades of the Senior High School students in
Statistics and Probability for the Third Quarter is shown below

68%

34% 34%

75 79 83 87 91 95 99
Example:
The distribution of the grades of the Senior High School students in
Statistics and Probability for the Third Quarter is shown below

95%

34% 34%
13.5% 13.5%
75 79 83 87 91 95 99
Example:
The distribution of the grades of the Senior High School students in
Statistics and Probability for the Third Quarter is shown below

99.70%

34% 34%
13.5% 13.5%
2.35% 2.35%
75 79 83 87 91 95 99
Example:
Using the empirical rule, you can say that the distribution of the
grades are distributed normally. The shape of the graph
is bell-shaped and
symmetric about its
99.70%
mean, which is 87.

34% 34%
13.5% 13.5%
2.35% 2.35%
75 79 83 87 91 95 99
The distribution can be summarized using the following percentages:

68% of data which lies within 1 standard deviation from the mean
have a grade of 83 to 91.

68%

34% 34%

75 79 83 87 91 95 99
The distribution can be summarized using the following percentages:

95% of data which lies within 2 standard deviation from the mean
have a grade of 79 to 95.

95%

34% 34%
13.5% 13.5%
75 79 83 87 91 95 99
The distribution can be summarized using the following percentages:

99.70% of data which lies within 3 standard deviation from the

mean have a grade of 75 to 99.

99.70%

34% 34%
13.5% 13.5%
2.35% 2.35%
75 79 83 87 91 95 99
Example 1:
The scores of the Senior High School students in their Statistics and Probability
quarterly examination are normally distributed with a mean of 35 and a
standard deviation of 5.
35 + 𝜎 = 40
35 − 𝜎 = 30
35 + 2𝜎 = 45
35 − 2𝜎 = 25 99.70%
35 + 3𝜎 = 50
35 − 3𝜎 = 20
34% 34%
13.5% 13.5%
2.35% 2.35%

20 25 30 35 40 45 50
What percent of the scores are between 30 and 40?

The scores 30 to 40 falls within the first standard deviation from

the mean.

34% 34%
13.5% 13.5%
2.35% 2.35%

20 25 30 35 40 45 50
What percent of the scores are between 30 and 40?

The scores 30 to 40 falls within the first standard deviation from

the mean.
The scores are between
30 and 40 and it is
approximately 68%.

34% 34%
13.5% 13.5%
2.35% 2.35%
20 25 30 35 40 45 50
What scores fall within 95% of the distribution?

Since 95% of the distribution lies within 2 standard deviations

from the mean,

34% 34%
13.5% 13.5%
2.35% 2.35%

20 25 30 35 40 45 50
What scores fall within 95% of the distribution?

Since 95% of the distribution lies within 2 standard deviations

from the mean, The scores
corresponding to this
area of distribution are
scores from 25 to 45.
34% 34%
13.5% 13.5%
2.35% 2.35%

20 25 30 35 40 45 50
Example 2:
The district nurse of Santa Rosa East needs to measure
the BMI (Body Mass Index) of the Alternative Learning
System students. She found out that the heights of male
students are normally distributed with a mean of 160 cm
and a standard deviation of 7 cm. Find the percentage of
male students whose height is within 153 cm to 174 cm.
Find the percentage of male students whose height is within 153
cm to 174 cm.
Heights of Male Students

34% 34%
13.5% 13.5%
2.35% 2.35%
139 146 153 160 167 174 181
Find the percentage of male students whose height is within 153
cm to 174 cm.
Heights of Male Students

34.0%
34.0%
+13.5%
81.5%
34% 34%
13.5% 13.5%
2.35% 2.35%
139 146 153 160 167 174 181
Find the percentage of male students whose height is within 153
cm to 174 cm.
Heights of Male Students
81.5% of the male
students have a height
between
153cm to 174cm.
34% 34%
13.5% 13.5%
2.35% 2.35%

139 146 153 160 167 174 181

Find the percentage of male students whose height is within 139
cm to 167 cm.
Heights of Male Students

34% 34%
13.5% 13.5%
2.35% 2.35%
139 146 153 160 167 174 181
Find the percentage of male students whose height is within 139
cm to 167 cm.
Heights of Male Students
2.35%
13.5%
34.0%
+34.0%
83.85%
34% 34%
13.5% 13.5%
2.35% 2.35%
139 146 153 160 167 174 181
Find the percentage of male students whose height is within 139
cm to 167 cm.
Heights of Male Students
83.85% of the male
students have a height
between
139cm to 167cm.
34% 34%
13.5% 13.5%
2.35% 2.35%
139 146 153 160 167 174 181