NORMAL
DISTRIBUTION
Continuous Probability Distribution
Objectives
• Identify the properties of a normal distribution
• Identify distributions as symmetric or skewed
• Find the area under the standard normal distribution,
given various z values
• Find probabilities for a normally distributed variable by
transforming it into a standard normal variable
• Use the central limit theorem to solve problems involving
sample means for large samples
Normal Distributions
• If a random variable has a probability distribution whose
graph is continuous, bell-shaped, and symmetric, it is
called a normal distribution. The graph is called a normal
distribution curve.
Properties of Normal Distribution
1. A normal distribution is bell-shaped.
2. The mean, median, and mode are equal and are
located at the center of the distribution.
3. A normal distribution is unimodal.
4. The curve is symmetric about the mean, which is
equivalent to saying that its shape is the same on both
sides of a vertical line passing through the center.
5. The curve is continuous; that is, there are no gaps or
holes. For each value of X, there is a corresponding
value of Y.
Properties of Normal Distribution
6. The curve never touches the x-axis. Theoretically, no
matter how far in either direction the curve extends, it
never meets the x-axis – but it gets increasingly close.
7. The total area under a normal distribution is equal to
1.00 or 100%.
8. The area under the part of a normal curve that lies
within 1 standard deviation of the mean is
approximately 0.68 or 68%; within 2 standard
deviations, about 0.95 or 95%; and within 3 standard
deviation 0.997 or 99.7%.
Normal vs Skewed Distributions
• When the data values are evenly distributed about the
mean, a distribution is said to be symmetric.
Normal vs Skewed
• When the majority of the data fall
to the left or right of the mean, the
distribution is said to be skewed.
• Negatively or left-skewed distribution
• When majority of the data values fall to
the right of the mean
• Positively or right-skewed distribution
• When majority of the data values fall to
the left of the mean
Standard Normal Distribution
• It is a normal distribution with a mean of 0 and a standard
deviation of 1.
Standard Normal Distribution Formula
−𝑧 2 /2
𝑒
𝑦=
√2𝜋
• All normal distributed values can be transformed into the
standard normally distributed variable by using the
formula for the standard score:
Finding Areas Under the Standard Normal
Distribution Curve
1. To the left of any z value:
Look up the z value in the table and use the area
given.
Example
• Find the area under the standard normal distribution curve
to the left of z = 2.09.
SOLUTION
• Step 1: Draw the figure.
• Step 2: Look up area in the table. The area to the left of
z = 2.09 is 98.17% or 0.9817.
Finding Areas Under the Standard Normal
Distribution Curve
2. To the right of any z value:
Look up the z value and subtract the area from 1.
Example
• Find the area under the standard normal distribution curve
to the right of z = -1.14.
SOLUTION
• Step 1: Draw the figure.
• Step 2: Look up area in the table. The area to the right of
z = -1.14 is 87.29% or 0.8729.
Finding Areas Under the Standard Normal
Distribution Curve
3. Between any two z values:
Look up both z values and subtract the
corresponding areas.
Example
• Find the area under the standard normal distribution curve between z =
1.62 and z = -1.35.
SOLUTION
• Step 1: Draw the figure.
• Step 2: Look up areas corresponding to the two z-values and subtract
smaller area from the larger area. Do not subtract the z values!
Area for z = 1.62 is 0.9474
Area for z = 1.35 is 0.0885
Therefore, the area between these z values is 0.9474-0.0885 = 0.8589 or
85.89%.
Applications of the Normal Distributions
• Finding the area under any normal curve
• To solve problems by using the standard normal distribution,
transform the original variable to a standard normal distribution
variable by using the formula:
Example
• An adult has on average 5.2 liters of blood. Assume the
variable is normally distributed and has a standard deviation of
0.3. Find the percentage of people who have less than 5.4
liters of blood in their system.
SOLUTION
Step 1: Draw a normal curve and
shade the desired area.
Step 2: Find the z value corresponding to 5.4
Step 3: Find the corresponding area in the table. The area under
the standard normal curve to the left of z = 0.67 is 0.7486.
Therefore, 0.7486, or 74.86%, of adults have less than 5.4 liters
of blood in their system.
Applications of the Normal Distributions
• Finding data values given specific probabilities
Example
To qualify for a police academy, candidates must score in the top
10% on a general abilities test. Assume the test scores are
normally distributed and the test has a mean of 200 and a
standard deviation of 20. Find the lowest possible score to
qualify.
SOLUTION
Step 1: Draw a normal distribution curve and shared the desired
area that represents the probability.
Solution ni siya
Solution
Solution
solution
Work backward to solve this problem: Subtract 0.1000 from
1.0000 to get the area under normal distribution to the left of X.
STEP 2: Find the z value that corresponds to the desired area.
Hey
Hey
Hey
Hey
STEP 3: Find the X value:
Therefore, a score of 226 should be used as a cutoff.
Anybody scoring 226 or higher qualifies for the academy.
The Central Limit Theorem
• As the sample size n increases without limit, the shape of
the distribution of the sample means taken with
replacement from a population with mean µ and standard
deviation will approach a normal distribution.
Formula Clarifications
This formula should be used This formula should be used
to gain information about a to gain information about
sample mean. an individual data value
obtained from the
population.
Example
The average time spent by construction workers who work
on weekends is 7.93 hours (over 2 days). Assume the
distribution is approximately normal and has a standard
deviation of 0.8 hour.
a. Find the probability that an individual who works at that
trade works fewer than 8 hours on the weekend
b. If a sample of 40 construction workers is randomly
selected, find the probability that the mean of the
sample will be less than 8 hours.
Solution A
Step 1: Draw a normal distribution and shade the desired
area.
Solution A
Step 2: Find the z value
Step 3: Find the area to the left of z = 0.09.
It is 0.5359. Hence, the probability of selecting a
construction worker who works less than 8 hours on a
weekend is 0.5359 or 53.59%.
Solution B
Step 1: Draw a normal curve and shade the desired area.
Solution B
Step 2: Find the z value for a mean of 8 hours and a
sample size of 40.
Step 3: Find the area corresponding to z = 0.55.
The area is 0.7088. Hence, the probability of getting a
sample mean of less than 8 hours when the sample size is
40 is 0.7088, or 70.88%.
