Chapter 2: Normal Distribution

1. Prepared by: Roqui M. Gonzaga, Lpt
Chapter 2
11th
GRADE
Normal
Distribution

2. Lesson Objectives:
At the end of the lesson, you will be able to;
 correctly find the area of a region under the normal curve; and

3. MEMO

4. NORMAL DISTRIBUTION
a continuous probability distribution where most of the scores tend to be closer
to the mean. Normal distribution is also known as Bell-shaped distribution.
18th Century
A French Mathematician Abraham De Moivre was the first to developed the
mathematical equation for the normal curve in the normal distribution,
especially in the use of concepts and application.
19th Century
A German Mathematiician Johann Karl Friedrich Gaus was also developed the
concept s of normal curve from the study of errors on repeated measurements

5. NORMAL DISTRIBUTION
known as Gaussian Distribution in honor of the Genrman
Mathematician.
Example:
1. The height of people in a population follows a normal distribution.

6. Characteristics of a Normal
Random Variable

7. Characteristics of a Normal Random Variable
1. The curve of a normal distribution is Bell-Shaped and
has a single pick at the center.
2. Normal distributions are symmetric around their mean.
3. The mean, median and mode of a normal distribution
are equal.
4. The area under the normal curve is equal to 1-100 %.
5. It is asymptotic to the X-axis.
-The tail ends of the curve extend indefinitely in both
directions, until the curve gets closer and closer to X-axis
but never met.

8. Empirical Rule of the
Normal Distribution

9. Empirical Rules
Empirical Rule states that all values of a normal random variables lie in the interval or
between the mean which is the ± standard deviations.
The area of the region between one standard deviation away from the mean is 0.6826,
two standard deviations away from the mean is 0.9544, and three standard deviations
away from the mean is 0.9974.
1. Area of the Region between One Standard Deviation away from the Mean

10. Empirical Rules
Empirical Rule states that all values of a normal random variables lie in the interval or
between the mean which is the ± standard deviations.
The area of the region between one standard deviation away from the mean is 0.6826,
two standard deviations away from the mean is 0.9544, and three standard deviations
away from the mean is 0.9974.
2. Area of the Region between Two Standard Deviations away from the Mean

11. Empirical Rules
Empirical Rule states that all values of a normal random variables lie in the interval or
between the mean which is the ± standard deviations.
The area of the region between one standard deviation away from the mean is 0.6826,
two standard deviations away from the mean is 0.9544, and three standard deviations
away from the mean is 0.9974.
3. Area of the Region between Three Standard Deviations away from the Mean

12. Normal Distribution depends upon its Mean and Standard Deviation.
𝒛 =
𝒙 − 𝝁
𝝈
Z= Z-Values on the Z-table which is also known as table of areas under normal
curve
𝝁 =Mean
𝝈 = Standard Deviation
𝒙 = variable or the raw score

13. 1. 𝑥 = 40, 𝜎 = 8 𝑎𝑛𝑑 𝜇 = 52

14. 2. Given the mean 𝜇 = 50 and the standard deviation, 𝜎 = 4 of the
population of the reading scores. Find the z-value that corresponds
to a score x=58.

15. 3. Score in Mathematics:
Locate the z-value that corresponds to a mathematics score of 39
given that 𝜇 = 45, 𝜎 = 6.

16. 4. The test scores for a civil services exam are normally distributed
with a mean of 152 and a standard deviation of 7. Find the standard
z-score for a person with score of;
a. 161 b. 148 c. 152

17. 5. Find the test score for a person given the 𝜇 = 152, 𝜎 = 7 with
standard score of;
b. 𝑧 = −1.75
a. 𝑧 = 2.33 c. 𝑧 = 0

18. Thank You and
God bless!!!