ACADEMIA DE SAN FRANCISCO JAVIER, BATANGAS, INC.
Unified Schools of the Archdiocese of Lipa (USAL)
Ruiz Martinez Street, Poblacion 6, Nasugbu, Batangas
0955.058.2700 /0930.632.7693
acadnasugbu@yahoo.com.ph/asfj2020@gmail.com
Learning Module:
Statistics &
Probability
Quarter 4
Prepared by: Ms. Aliyah Monique V. Benitez
Subject Teacher
Name of the Student
Statistics & Probability
SHS Learning Module
ACADEMIA DE SAN FRANCISCO JAVIER, BATANGAS, INC.
Unified Schools of the Archdiocese of Lipa (USAL)
Ruiz Martinez Street, Poblacion 6, Nasugbu, Batangas
0955.058.2700 /0930.632.7693
acadnasugbu@yahoo.com.ph/asfj2020@gmail.com
Name of the Student:
Date:
Semester 2
Module 4: Standard Normal Probability Table
Time Frame: 10 days
I. Objectives:
 Define normal curve.
 Identify regions under the normal curve.
 Converting a normal random variable to a standard normal variable and vice-versa.
II. DISCUSSION OF CONTEXT LESSON
Identifying Regions under Normal Curve Corresponds to Different Standard Normal Values
In the previous module, we discussed the properties of the normal probability distribution, and one of the things
that you learned is that the total area under the curve is equal to one. Since the standard normal distribution is defined as
normally distributed random variable having a mean of zero (μ= 0 ) and standard deviation of one (ơ = 1), the fifty (50)
percent of the region under the curve is below the mean and the other 50% is above the mean. Figure 1, as shown below
illustrate the standard normal curve.
You can easily identify the area of the regions under the normal curve by using the Table of Areas under the Normal
Curve which is also known as z-Table (see attachment at the last page of this module). This table gives the area of any value
of z from -3.99 to 3.99. The value from this table will describe the area of the specific region of the curve to the left of the
given z-value.
How to use the Table of Areas under the Normal Curve?
Example: Find the area to the left of -1.69
Solution: First, split the given z-value into hundredths. We can find the whole number and the tenths digit (-1.6) at the left
side of the table while the hundredths (0.09) located at the uppermost of the table. Then the intersection of these numbers
will be the area of the normal curve to the left of the z-value.
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Answer: The area to the left of z= -1.69 is 0.0455
STEPS ON HOW TO FIND THE AREA THAT CORRESPONDS TO Z –VALUE
1. Draw/sketch a normal curve and locate the given z-value on the normal curve
2. Shade the region of the curve according to the condition of z-value whether it is below, above, or between.
3. Use the table of the area under the normal curve to find the corresponding area.
4. Choose the appropriate operation based on step 2 and 3
4.1. When the z-value is to the left or any related terms (e.g. below, less than) just write the value we obtained in
step 3
4.2. When the z-value is to the right or any related terms (e.g. above, greater than), subtract 1 by the obtained
value in step 3
4.3. When the shaded region is in between of the two z-value, subtract the biggest by the smallest value obtained
in step 3
5. Label the shaded region and draw a conclusion
Example 1: Find the area that corresponds below z = -1.35
Solution:
Step 1. Draw/sketch a normal curve and locate the given z-value.
Step 2: Shade the region of the curve according to the condition of z-value whether it is below, above, or between.
Step 3. Use the table of the area under the normal curve to find the corresponding area.
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The intersection between -1.3 and 0.05 is 0.0885
Step 4. Choose the appropriate operation based on step 2 and 3
4.1 When the z-value is to the left or any related terms (e.g. below, less than) just write the value we obtained in
step 3. Since the shaded region of the curve is to the left of the z= -1.35 and the intersection between -1.3 and 0.05
is 0.0885. Therefore, the area of the shaded region is 0.0885
Step 5. Label the shaded region and draw a conclusion.
0.0885
Example 2: Find the area to the right of z = -1.35
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Converting a Normal Random Variable to a Standard Normal Variable and Vice-versa
In solving real-life problems involving the normal curve, it is particularly important to convert a random variable
x to a standard normal variable or z-score. This procedure is known as “standardizing” or “standardization” of a random
variable, where a standardized value is called a z-score. A z-score is a measure of the number of standard deviations (σ) a
particular data value is away from the mean (μ).
Suppose your score on a test in Probability and Statistics is 39 and the scores are normally distributed
with a mean of 33 and standard deviation of 3, then your score is exactly 2 standard deviations above the
mean. If you scored 30, then it is exactly 1 standard deviation below the mean. All values that are above the
mean have positive z-scores and all values that are below the mean have negative z-scores. If you
obtained a z-score of -3, this means that your score is 3 standard deviations below the mean, that is, 33 –
3(3) = 33 – 9 = 24.
Given any value x from a normal distribution with mean μ and standard deviation σ, to convert x to a z-score
(standard normal score), you need to;
(a) Subtract the mean μ from x.
(b) Divide this quantity, � – �, by the standard deviation σ.
The formula used in converting a random variable x to a standard normal variable z is:
Example 1.
Suppose IQ scores are normally distributed with a mean of 100 and standard deviation of 10. If your IQ is 85, what is your
z-score? (Round off your answer to the nearest hundredths)
Solution:
Example 2.
On a nationwide placement test that is normally distributed, the mean was 125 and standard deviation was 15. If you
scored 149, what was your z-score? (Round off your answer to the nearest hundredths)
Solution:
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III. EVALUATION
Assessment
A. Directions: Match the estimated probabilities with their corresponding areas under the normal curve.
B. Directions: Find and match the area that corresponds to each of the following z values. Use the attached Standard
Normal Cumulative Probability Table.
1. to the right of z = 1.28
2. below z= -0.20
3. above z= 1.57
4. at least z =-1.27
5. at most z = 1.39
a. 0.0823
b. 0.1020
c. 0.0582
d. 0.1003
e. 0.4207
C. Directions: Supply the needed information by solving the missing value. Use the formula.
1. x = 25, μ= 40, σ = 6, z = ?
2. x = 39, μ= 30, σ = 3, z = ?
3. x = 125, μ= 105, σ = 8, z = ?
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4. Each month, a Filipino household generates an average (mean) of 28 pounds of newspaper for garbage or recycling.
Assume the standard deviation is 2 pounds. Determine the z-score of a household that generates 22 pounds of newspaper.
IV. REFERENCES
Melosantos, Antonio, et. al (2016). Math Connections in the Digital Age: Statistics and Probability
Normal probability distribution. Normal Probability Distribution - an overview | ScienceDirect
Topics. (n.d.). Retrieved February 3, 2022, from
https://www.sciencedirect.com/topics/mathematics/normal-probability-distribution
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Negative scores in the z-table correspond to the values which are less
than the mean.
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Positive scores in the Z-table correspond to the values which are
greater than the mean.
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