The document provides an overview of the normal distribution and objectives for understanding its characteristics. It describes how the normal distribution illustrates random variables obtained through measurement. The normal distribution is symmetrical and bell-shaped, with the highest point occurring at the mean. Approximately 68%, 95%, and 99.7% of the data falls within 1, 2, and 3 standard deviations of the mean, respectively. The document also includes a multiple choice assessment to test understanding of normal distribution properties and concepts.

2. QUARTER 3 M3
THE
NORMAL
DISTRIBUTION

3. OBJECTIVES:
•illustrate a normal random variable and its
characteristics
•identify regions under the normal curve that
correspond to different standard normal values
•convert a normal random variable to a standard
normal variable and vice versa
•compute probabilities and percentiles using the
standard normal distribution

4. Directions: Choose the best answer.
Write the chosen letter on a separate
sheet of paper.
1. Which of the following
illustrations represents normal
distribution?

5. 2. What is another name for normal
distribution?
A. Gaussian distribution C. Bernoulli’s distribution
B. Poisson distribution D. Probability distribution
3. What is the total area in the distribution
under the normal curve?
A. 0 C. 2
B. 1 D. 3

6. 4.Which of the following is a parameter
of normal distribution?
A. mean C. mean and deviation
B. standard deviation D. none of the above
5. The graph of a normal distribution is
symmetrical about the ________.
A. mean C. mean and deviation
B. standard deviation D. none of the above

7. 6. What percent of the area under a normal curve
is within 2 standard deviations?
A.68.3% C. 99.7%
B. 95.4% D. 100%
7. How many standard deviations are there in each
inflection point?
A. 0 C. 2
B. 1 D. 3

8. 8.Which of the following denotes the
standard normal distribution?
A. A C. Y
B. X D. Z
9. A random variable X has a mean of 12 and a
standard deviation of 3.2. What is the
corresponding z-score for x = 8?
A. -1.25 C. -1.75
B.-1.50 D. -2.25
10.What is the area under the normal curve if
the z-score given is 2.14?
A. 0.9830 C. 0.9838
B. 0.9834 D. 0.9842

9. 11.What is the z-score if the area under
the normal curve is 0.0475?
A. -1.66 C. 1.66
B. -1.67 D. 1.67
12. What is the area under the normal curve
between the z-scores -1.99 and 1.56?
A. 0.9173 C. 0.9369
B. 0.9317 D. 0.9639

10. 13.What is the area of the P(Z > 2.58)
using the standard normal curve?
A. 0.0049 C. 0.9949
B. 0.0051 D. 0.9951
14. In an achievement test, the mean score of
normally distributed values is 70
and the standard deviation is 12. What is the
percentage of students who got
a score of 85 and above?
A. 0.1056 C. 0.8925
B. 0.1075 D. 0.8944

11. 15.What is the value of 70th percentile in
a standard normal distribution?
A. 0.51 C. 0.61
B. 0.52 D. 0.62

12. LESSON :The Normal
Distribution and Its Properties

13. After going through this lesson, you are
expected to:
•Illustrate the normal distribution
•Identify the characteristics of the normal curve
•And, find percentage of the normal curve

15. Normal Probability Distribution
is the probability distribution of continuous random
variables. It shows graphical representations of random
variables obtained through measurement such as the height
and weight of the students.
It is sometimes called “bell curve” .
It is used to describe the characteristics of population and
help us visualize the interference we make about the
population.
 It is also used to determine the probabilities and percentile
of the continuous random variables in the distribution.

18. Gaussian Distribution
• The graph is a continuous curve and has a domain
-∞ < X < ∞.
• The graph is asymptotic to the x-axis. The value of the
variable gets closer and
closer but will never be equal to 0.
*As the x gets larger and larger in the positive direction, the tail of the
curve approaches but will never touch the horizontal axis. The same
thing when the x gets larger and larger in the negative direction.
PROPERTIES FROMTHE GRAPH OFTHE NORMAL
DISTRIBUTION
also known as

19. • The highest point on the curve occurs at x = μ
(mean).
*The mean (μ) indicates the highest peak of the curve and is
found at the center.
*Take note that the mean is denoted by this symbol μ and
the standard
deviation is denoted by this symbol .
*The median and mode of the distribution are also found
at the center of the graph. This indicates that in a
normal distribution, the mean, median
and mode are equal.

20. The curve is symmetrical about the mean.
*This means that the curve will have
balanced proportions when cut in
halves and the area under the curve to
the right of mean (50%) is equal to the
area under the curve to the left of the mean (50%).

21. The total area in the normal distribution
under the curve is equal to 1.
*Since the mean divides the curve into halves,
50% of the area is to the
right and 50% to its left having a total of 100%
or 1.

22. In general, the graph of a normal distribution
is a bell-shaped curve with two
inflection points, one on the left and another
on the right. Inflection points are
the points that mark the change in the curve’s
concavity.
Inflection point is the point at which a change
in the direction of curve at mean minus
standard deviation and mean plus standard
deviation.
Note that each inflection point of
the normal curve is one standard deviation away from the mean.

23. Every normal curve corresponds to the “empirical
rule” (also called the 68 -95 - 99.7% rule):

26. In this example, the curve represents the human height measurements.
short average tall
Or
anywhere in
between

27. The y-axis represents the relative probability of observing someone
who is really short, or really tall, or who has an average height.
More likely
Less Likely

30. Please remember once more!
PROPERTIES/CHARACTERISTICS OF A
NORMAL CURVE
•1. The distribution curve is bell- shaped.
•2. The curve is symmetrical about its
center. This means that if we draw a
segment from the peak of the curve down
to the horizontal axis, the segment divides
the normal curve into two equal parts or
areas.

31. •3. The mean, median, and mode coincide
at the center. This also means that in a
normal distribution, or a distribution
described by a normal curve, the mean,
the median, and mode are equal.
•4. The width of the curve is determined by
the standard deviation of the distribution.

32. •5. The tails of the curve are plotted in both
directions and flatten out indefinitely along
the horizontal axis. The tails are asymptotic
to a reference axis or another graph is
called an asymptote, always approaching
another but never touching.
•6. The total area under a normal curve is 1.
This means that the normal curve
represents the probability, or the
proportion, or the percentage associated
with specific sets of measurement values.

34. Example no. 1- Drawing a Normal Distribution
The height of SHS students is normally
distributed with a mean of μ= 150 cm and a
standard deviation of σ=10 cm. Sketch a
normal curve that describe this distribution.
SOLUTION:
Step 1: Sketch a normal curve
Step 2: The mean of 150 cm goes in the middle.
Step 3: Each standard deviation is a distance of 10 cm.

36. Sketch a normal curve for each distribution. Label the
x-axis at one, two and three standard deviations from
the mean.
1. mean= 95 ; standard deviation=12
2. mean= 100 ; standard deviation =15
3. mean = 60 ; standard deviation = 6

37. +1 standard deviation from the mean
+3 standard deviations from the mean
-1 standard deviation from the mean
-2 standard deviations from the mean
ANSWER THE FOLLOWING QUESTIONS:
A set of data with a mean of 45 and a standard
deviation of 8.3 is normally distributed. Find each
value, given its distance from the mean.

38. FINDING PERCENTAGES IN THE NORMAL IN
A NORMAL DISTRIBUTION
The height of SHS students is normally
distributed with a mean of μ = 150cm and a
standard deviation of σ = 10cm. Approximately
what percent of these have a height greater
than 170 cm?
STEP 1:
Sketch a normal distribution with
a mean of 150 cm and a standard
deviation of σ =10cm.

39. Step 2:
The height of 170 cm is two
standard deviations above the mean.
Shade above that point.
Step 3: Add the percentages
in the shaded area.

40. FINDING THE WHOLE COUNT
The height of SHS students is normally
distributed with a mean of μ =15o cm and
a standard deviation of σ = 10cm. A certain
school has 500 students. Approximately
how many students have a height of lower
than 140 cm?
Solution:
Step 1 : Sketch a normal distribution with a mean of 150 cm
and a standard deviation of = 10 cm.

41. Step 2: The height of 140 cm is one standard
deviation below the mean. Shade below that point.
Step 3: Add the percentages
in the shaded areas.

42. Solution:
Step 4: How many students have a height
than 140 cm?

43. Standard deviations

45. a. P (39 < X < 51) = 68.3%

46. b. P (33 < X < 63) = 97.55%

47. c. P (X > 45) = 50%

48. P (X < 39) = 15.85%

52. ND
ND
NND
ND
NND
NND
ND
ND
NND
ND

54. ASSESSMENT:
Directions: Read the instructions given and write your
answers on your answer sheets.
A. Multiple Choice. Choose the letter of the best answer.
1.What is another name for normal distribution?
A.Gaussian distribution
B. Poisson distribution
C. Bernoulli’s distribution
D. Probability distribution

55. 2. What is the total area in the distribution under
the curve?
A. 0 B. 1 C. 2 D. 3
3. What marks the change in the curve’s concavity?
A. curve
B. inflection points
C. mean
D. standard deviation

56. 4. Which value is found at the center of the normal
curve?
A. mean C. mode
B. median D. all of the above
5. Which of the following is a parameter of
normal distribution?
A.Mean
B. standard deviation
C. mean and standard deviation
D. none of the above

57. 6. Which of the following symbols is used to
denote the mean?

58. 7. Which of the following does not describe a
normal curve?
A. asymptotic
B. bell-shaped
C. Discrete
D. symmetrical about the mean
8. What percent of the area under a normal
curve is within 2 standard deviations?
A. 68.3% C. 99.7%
B. 95.4% D. 100%

59. 9. What percent of the area under a normal
curve is within 1 standard deviation?
A. 68.3% C. 99.7%
B. 95.4% D. 100%
10. What percent of the area under a normal
curve is within 3 standard deviations?
A. 68.3% C. 99.7%
B. 95.4% D. 100%

60. Lesson:
The Standard
Normal
Distribution

61. The standard normal distribution, which is denoted
by Z, is also a normal distribution having a mean of
0 and a standard deviation of 1. Since the normal
distribution can have different values for its mean
and standard deviation, it can be standardized by
setting the µ = 0 and the σ = 1.

63. Directions: Observe the illustrations then answer
the questions that follow. Write your answers on
a separate sheet of paper.

65. Manually compute the approximate area.
 FORMULA:

67. Z- SCORE
Describes the position of a
raw score in terms of its
distance from the mean when
measured in standard
deviation units. The Z-score
is positive if the values lies
above the mean and negative
if it lies below the mean.

68. Z= (X- μ )
σ
How to get the z-score?
Where x is the raw score, μ
is the population mean, and
σ is the population standard
deviation.

69. HOW TO GET THE AREA WITH THE Z-
SCORE?

76.  https://www.youtube.com/watch?v=itwNvyYmPf8

81. NOW PERFORM THIS!

86. THE PERCENTILE
A percentile is a measure used in statistics
indicating the value below which a given
percentage of observations in a group of
observations fall.

87. Imagine you took a standardized test
and you scored 91 at the 89th
percentile. This means that 89% of the
examiners scored lower than 91 and 11%
scored higher than 91. This explains that
89th percentile is located where 89% of
the total population lies below and 11%
lies above that point.