3. Normal Probability Distribution is a
probability distribution of continuous random
variables. It shows graphical representations of
random variables obtained through measurement
like the height and weight of the school children,
the percentile ranks of the students in their
National Career Assessment Examination
(NCAE) result, or any data with infinite values.
4. PROPERTIES OF NORMAL CURVE
1. The normal curve is bell-shaped.
2. The curve is symmetrical about its center. This means
that, the segment at the center divides the curve into two
equal parts or areas.
3. The mean, median, and mode coincide at the center.
This also means that in a normal distribution, the mean,
median, and mode are equal.
5. PROPERTIES OF NORMAL CURVE
4. The width of the curve is determined by the standard
deviation, of the distribution.
𝝈
5. The tails of the curve flatten out indefinitely along the
horizontal axis, always approaching the axis but never
touching it. That is, the curve is asymptotic to the baseline.
6. The total area under a normal curve is 1. Thus, it
represents the probability or proportion, or the percentage
associated with specific sets of measurement values.
7. The normal distribution, also known as the Gaussian Distribution,
has the following formula,
where:
f(x)= height of the curve particular values of X
𝜇 = mean of the population
𝜎 = standard deviation of the population
𝜋 = 3.14159…
𝑒 = 2.71828…
X= any score in the distribution
8. There are many normal
distributions. A normal
distribution is determined by two
parameters: mean and the
𝜇
standard deviation . If the
𝜎
mean 𝜇 is 0 and the standard
deviation 𝜎 is 1, then the normal
distribution is a standard
normal distribution.
9. However, the mean is not
𝜇
always equal to 0 and the
standard deviation is not
𝜎
always equal to 1. In the
normal curve as shown in
Figure 3, 𝜇 = 40 and 𝜎 = 12
13. 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 (1), two (2), and three (3)
of the standard deviations from the mean are approximately 68%, 95%, and
99.70%, respectively. 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. For instance, the distribution
of the grades of the Senior High School students in Statistics and Probability
for the Third Quarter is shown in the next figure.
15. Example 1. What is the frequency and relative frequency of babies weights that are within:
4.94 4.69 5.16 7.29 7.19 9.47 6.61 5.84 6.83
3.45 2.93 6.38 4.38 6.76 9.01 8.47 6.8 6.4
8.6 3.99 7.68 2.24 5.32 6.24 6.19 6.63 5.37
5.26 7.35 6.11 7.34 5.87 6.56 6.18 7.35 4.21
16. Example 1. What is the frequency and relative frequency of babies weights that are within:
a) One standard deviation from the mean
b) Two standard deviation from the mean
4.94 4.69 5.16 7.29 7.19 9.47 6.61 5.84 6.83
3.45 2.93 6.38 4.38 6.76 9.01 8.47 6.8 6.4
8.6 3.99 7.68 2.24 5.32 6.24 6.19 6.63 5.37
5.26 7.35 6.11 7.34 5.87 6.56 6.18 7.35 4.21
17. Example 2. 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.
Answer the following questions:
a) What percent of the scores are between 30 to 40?
b) What scores fall within 95% of the distribution?
18. Example 2. Use the Empirical Rule to complete the following the
table. Write on the respective column the range or interval of the
scores based on the given parameters.
MEAN STANDARD
DEVIATION
68% 95% 99.7%
20 2
35 9
12 5.5
6.5 10
55 3.5
