Normal Distribution

Statistics & Probability Quarter 3 Module 3:

JAY R R. BALDELOVAR

Teacher II

OBJECTIVES

• Describe a normal random variable and its properties

• Draw a normal curve; and

• State the empirical rule

What I Need To Know

What I Know

Directions: Choose the best answer

Multiple Choice

1. Which of the following illustrations represents normal distribution?

Multiple Choice

2. What is another name for normal distribution?

• A. Gaussian distribution

• B. Poisson distribution

• C. Bernoulli’s distribution

• D. Probability distribution

Multiple Choice

3. What is the total area in the distribution under the normal curve?

• A. 0

• B. 1

• C. 2

• D. 3

Multiple Choice

4. 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

Multiple Choice

5. The graph of a normal distribution is symmetrical about the ________.

• A. mean

• B. standard deviation

• C. mean and standard deviation

• D. none of the above

Multiple Choice

6. 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%

Multiple Choice

7. How many standard deviations are there in each inflection point?

• A. 0

• B. 1

• C. 2

• D.3

Multiple Choice

8. Which of the following denotes the standard normal distribution?

• A. A

• B. X

• C. Y

• D.Z

What's In

Directions: Anticipation-Reaction Guide

Multiple Choice

1. The normal curve of the distribution is bell-shaped.

Multiple Choice

2. In a normal distribution, the mean, median and mode are of equal values

Multiple Choice

3. The normal curve gradually gets closer and closer to 0 on one side

Multiple Choice

4. The normal curve is symmetrical about the mean.

Multiple Choice

5. The distance between the two inflection points of the normal curve is equal to the value of the mean.

What's New

Consider the random event of tossing four coins once, then follow these steps:

•  1. List all the possible outcomes using the tree diagram.

• 2. Determine the sample space.

• 3. Determine the possible values of the random variables.

• 4. Assign probability values P(X) to each of the random variable.

• 5. Construct a probability histogram to describe the P(X). 

Tree Diagram

*Note: Each coin has Head (H) and Tail (T)

• 1. List all the possible outcomes using the tree diagram.

• 2. Determine the sample space.

Sample space

*Note: Count all the possible outcomes to determine the sample space

• 1. List all the possible outcomes using the tree diagram.

• 2. Determine the sample space.

• 3. Determine the possible values of the random variables.

• 4. Assign probability values P(X) to each of the random variable.

Let T be the random variable representing the number of tails

Let T be the random variable representing the number of tails

The values of the random variable T are 0, 1, 2, 3 and 4

• 1. List all the possible outcomes using the tree diagram.

• 2. Determine the sample space.

• 3. Determine the possible values of the random variables.

• 4. Assign probability values P(X) to each of the random variable.

• 5. Construct a probability histogram to describe the P(X). 

Multiple Choice

2. What composes the sample space?

• A. 0,1,

• B. 0, 1, 2,

• C. 0, 1, 2, 3,

• D. 0, 1, 2, 3, 4

3. How will you describe the histogram?

• The histogram looks like a curve.

NORMAL DISTRIBUTION

• a function that represents the distribution of many random variables as a symmetrical bell-shaped graph.

• it was first discovered by Carl Friedrich Gauss.

• it also called Gaussian distribution

• The normal distribution is a probability distribution.

• The normal distribution is a continuous probability distribution that is very important in many fields of science.

PROPERTIES OF A NORMAL DISTRIBUTION

1. The graph is a continuous curve and has a domain -∞ < X < ∞.

• This means that X may increase or decrease without bound.

2. 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.

3. 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  $\sigma$ .

• 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.

 4. 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%). 

5. 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.

 6. 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. 

7. Every normal curve corresponds to the “empirical rule”

(also called the 68% - 95% - 99.7% rule)

 • about 68.3% of the area under the curve falls within 1 standard deviation of the mean

 • about 95.4% of the area under the curve falls within 2 standard deviations of the mean

 • about 99.7% of the area under the curve falls within 3 standard deviations of the mean.

Example 1. Suppose the mean is 60 and the standard deviation is 5, sketch a normal curve for the distribution.

• This is how it would look like. 

Example 2. A continuous random variable X is normally distributed with a mean of 45 and standard deviation of 6. 

Illustrate a normal curve and find the probability of the following;

• a. P (39 < X < 51) = 68.3%

• b. P (33 < X < 63) = 97.55%

• c. P (X > 45) = 50%

• d. P (X < 39) = 15.85%

Example 2a. Illustrate a normal curve and find the probability of P (39 < X < 51) = 68.3%

 *Since the area covered is 1 standard of the deviation to the left and to the right.

Example 2b. Illustrate a normal curve and find the probability of P (33 < X < 63) = 97.55%

  *Since the area covered is 2 standard of the deviation to the left and to the right.

Example 2c. Illustrate a normal curve and find the probability of P (X > 45) = 50% 

  * Since the area covered is half of the curve

Example 2d. Illustrate a normal curve and find the probability of P (X < 39) = 15.85%

  

What's More

Directions: Read the following statements carefully.

Choose ND if the statement describes a characteristic of a normal distribution, and NND if it does not describe a characteristic of a normal distribution.

Multiple Choice

1. The curve of the distribution is bell-shaped.

Multiple Choice

2. In a normal distribution, the mean, median and mode are of equal values.

Multiple Choice

3. The normal curve gradually gets closer and closer to 0 on one side.

Multiple Choice

4. The curve is symmetrical about the mean.

Multiple Choice

5. The distance between the two inflection points of the normal curve is equal to the value of the mean.

Multiple Choice

6. A normal distribution has a mean that is also equal to the standard deviation.

Multiple Choice

7. The two parameters of the normal distribution are the mean and the standard deviation.

Multiple Choice

8. The normal curve can be described as asymptotic.

Multiple Choice

9. Two standard deviations away from the left and right of the mean is equal to 68.3%.

Multiple Choice

10. The area under the curve bounded by the x-axis is equal to 1.

What I Have Learned

Direction: Complete the given diagram below by filling up the necessary details about normal distribution. 

What I Can Do

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. 

Assessment

Multiple Choice. Choose the letter of the best answer.

Multiple Choice

1. What is another name for normal distribution?

• A. Gaussian distribution

• B. Poisson distribution

• C. Bernoulli’s distribution

• D. Probability distribution

Multiple Choice

2. What is the total area in the distribution under the curve?

• A.0

• B. 1

• C. 2

• D. 3

Multiple Choice

3. What marks the change in the curve’s concavity?

• A. curve

• B. inflection points

• C. mean

• D. standard deviation

Multiple Choice

4. Which value is found at the center of the normal curve?

• A. mean

• B. median

• C. mode

• D. all of the above

Multiple Choice

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

Multiple Choice

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

• A.  $\sigma$  

• B. µ

• C. α

• D. ∞

Multiple Choice

7. Which of the following does not describe a normal curve? 

• A. asymptotic   

• B. bell-shaped

• C. discrete

• D. symmetrical about the mean
  

Multiple Choice

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%  
  

Multiple Choice

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%  
  

Multiple Choice

10. What percent of the area under a normal curve is within 3 standard deviations?

• A. 68.3%  

• B. 95.4%

• C. 99.7% 

• D. 100%  
  

Thank You!