The document discusses the normal distribution and its key properties. It describes how the normal distribution is a bell-shaped curve that is symmetrical around the mean. It also notes that the mean, median, and mode are equal at the center of the distribution. Additionally, it explains how the standard deviation determines the width of the curve and that the total area under the curve is 1.

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The Normal
Distribution and
Its Properties

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NORMAL DISTRIBUTION
• Is a probability distribution of continuous
random variables.
• It shows a graphical representations of obtained
through measurement such as height and weight
of the students.
• It is also known as Gaussian Distribution and
sometimes called the Bell Curve.
• It is used to describe the characteristics of a
populations and help us visualize the inferences
we make about the population.
• It is also used to determine the probabilities and
percentile of the continuous random variables in
the distribution. 3

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Properties of a Normal Distribution / Normal Curve
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1. Distribution curve is bell-shaped.
2. The curve is symmetrical about its
center. This means that, if we draw
segment from the peak of the curve
down to the horizontal axis, the
segment divides the normal into two
equal parts or areas.
3. The mean, median and mode coincide
at the center. This also means that in
the normal distribution, or distribution
described by normal curve, the mean,
median, mode are equal.

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4. The width of the curve is determined
by the standard deviation of the
distribution.
5. The tails of the curve are plotted in
both direction and flatten out
indefinitely along the horizontal axis.
The tails are thus asymptotic to the
baseline. A portion of the of the graph
that is asymptotic to the reference axis
or another graph is called an
asymptote, always approaching one
another but never touching it.
Properties of a Normal Distribution / Normal Curve
Standard deviation
Has small value
Standard deviation
Has large value

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6. The total area of a normal curve is 1.
This means that the normal curve
represents probability, or the proportion,
or the percentage associated with
specific set of measurement values.
Properties of a Normal Distribution / Normal Curve

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• are the points that mark the change in the curve’s
concavity.
• 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.
Inflection Point

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Let’s try this!
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Sketch the normal curve of each
distribution
1. Mean = 95, Standard deviation = 12
2. Mean = 60 ; Standard deviation = 6
3. Mean = 23.8; Standard deviation = 5.2

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Let’s try this!
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A continuous random variable X is
normally distributed with a mean of 45 and
standard deviation of 6. Find the
percentage of the following:
1. P( 39 < X > 51)
2. P ( X > 45)
3. P ( X < 39)

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Evaluation
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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
2. What is the total area in the distribution under the curve?
A. 0
B. 1
C. 2
D. 3

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Evaluation
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Multiple Choice. Choose the letter of the
best answer.
3. What marks the change in the curve’s concavity?
A. curve
B. inflection points
C. mean
D. standard deviation
4. Which value is found at the center of the normal curve?
A. mean
B. median
C. mode
D. all of the above

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Evaluation
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Multiple Choice. Choose the letter of the
best answer.
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
6. Which of the following symbols is used to denote the mean?
A.ó
B. μ
C. α
D. ∞

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Evaluation
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Multiple Choice. Choose the letter of the
best answer.
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%
B. 95.4%
C. 99.7%
D. 100%

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Evaluation
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Multiple Choice. Choose the letter of the
best answer.
9. What percent of the area under a normal curve is within 1
standard deviation?
A. 68.3%
B. 95.4%
C. 99.7%
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%

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Additional Activity
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Directions: Make a sketch for each of the 3 areas under
the normal curve as stated in the empirical rule. Using a
mosaic art, shade the area that corresponds to the area
under the normal curve. You may use eggshells, old
magazines, dried leaves or any materials available at
home.