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1. E
Abstract Reasoning Test
x = 65 µ = 73 σ = 8.6
𝑧 =
𝑥−𝜇
𝜎
=
65−73
8.6
=
−8
8.6
z = −0.93 (rounded off to two decimal places)
Aptitude Test
x = 56.5 µ = 54.5 σ = 5.1
𝑧 =
𝑥−𝜇
𝜎
=
56.5−54.5
5.1
=
2
5.1
z = 0.39 (rounded off to two decimal places)
ANSWER. The z-score indicates the deviation of the score from the mean in each
distribution.
Highest: Aptitude Test (it has the highest z-score)
Lowest: Abstract Reasoning (it has the lowest z-score)
Computing Probabilities and Percentile Using Standard Normal Table
Standard Normal Table (Z-score Table)
Directions: Draw a normal curve. Shade the area that is being asked in each item under
the standard normal curve.
1. To the left of z=0.5 2. To the left of z=1.43
7
4. The two tails extend indefinitely in both directions coming closer and closer to the
horizontal axis but never quite touching it.
5. The curve is symmetrical about the vertical line x = µ.
6. The mean, median, and mode coincide at the exact center of the distribution.
7. The total area under the curve is 1.00, and since the curve is symmetrical about x = µ, it
follows that the area on either side of the vertical line x = µ is 0.5. This area represents the
total number of cases (N).
The Normal Curve
Reading Activity. Facts about Normal Distribution/Curve
In the above normal curve point P is the maximum where the mean, median and
mode are located. The left side is symmetrical to the right side about the mean (x = µ).
Points A and B are inflection points, the point where the curve changes directions from
downward to upward. The horizontal Distance from A (or B) to the line x = µ is one unit
of standard deviation σ.
Normal curves can describe a large number of groups of data differentiated only
by their mean and their tendency to spread.
Any normal distribution is defined by two measures: the mean which locates the
center and the standard deviation which measures the spread around the center.
The standard deviation permits us to determine quite accurately where the
values in a distribution are located relative to their mean. With the use of the standard
deviation, we can measure with great precision percentage of items that fall within
specific ranges under a normal distribution.
2
Activity 4. Lhenylene’s final examination results in the three subjects are as
follows:
Subject Grade Mean S.D.
Business Math 86 81 5.75
College Algebra 76 73 6
Business Statistics 91 93 6.5
On which test did Lhenylene perform well? worst?

2. Week # 3 Inclusive Dates: ____________ Quarter: Quarter 3 Score: _______
Competencies:
1. Illustrate a normal random variable and its characteristics (M11/12SP-IIIc-1);
2. Identify regions under the normal curve that correspond to different standard
normal values (M11/12SP-IIIc-3);
3. Convert a normal random variable to a standard normal variable and vice versa
(M11/12SP-IIIc-4); and
4. Compute probabilities and percentiles using standard normal distribution
(M11/12SP-IIIc-d-1);
NORMAL DISTRIBUTION CONCEPTS
A normal distribution is an arrangement of a data set in which most values cluster in the
middle of the range and the rest taper off symmetrically toward either extreme.
Characteristics of Normal Probability Distribution:
1.The curve has a single peak.
2.It is bell-shaped or shape like Mexican hat.
3.The mean lies at the center of the distribution. The distribution is symmetrical about the
mean.
3. To the right of z= - 2.54 4. In between z=0.9 and z= 2.0
 To calculate the area under the normal curve, we use the standard normal or z-score
table.
 In a z-score table, the left most column tells you how many standard deviations above
the mean to 1 decimal place, the top row gives the second decimal place, and the
intersection of a row and column gives the probability.
 For the case of negative z-score, the same manner as with the computation of
probabilities of positive z-score since the curve is symmetrical along the mean.
Activity 5
A. Directions: Find the area under the standard normal curve which lies to the following
conditions. Refer to the z-table and to your concept notes for you to be guided.{Hint: To
the right means P(x>a), to the left means P(x<a), and between means P(a<x<b)}.
1. to the right of z=0.66
2. to the left of z=-1.53
3. to the right of z=-2.34
4. to the right of z=1.30
5. between z=-0.78 and z=0.56
B. Directions: Solve the following problem that involves area under normal curve.
You are a farmer about to harvest the crop. To describe the uncertainty in the size of
the harvest, you feel that it may be described as normal distribution with a mean value of
80,000 bushels with a standard deviation of 2,500 bushels. Find the probability that your
harvest will exceed 84,400 bushels.
1
8
Division of Nueva Ecija
NEHS – Senior High School
Quezon District, Cabanatuan City
STATISTICS AND PROBABILITY
Creative & Critical Thinking Activities
Name: ____________________________________________
Grade & Section: ___________________________________
Parent / Guardian Name & Signature: __________________
Teacher: __________________________________________

3. 1. z =− 1.5
Solution:
𝑥 = 𝑧𝑠 + 𝑥 = −1.5 8.5 + 75 = −12.75 + 75
𝒙 = −𝟔𝟐. 𝟐𝟓
2. z = 2.30
Solution:
𝑥 = 𝑧𝑠 + 𝑥 = 2.30 8.5 + 75 = 19.55 + 75
𝒙 = 𝟗𝟒. 𝟓𝟓
Application
Apply the concept of converting normal random variable to standard normal variable to
solve verbal problems.
1. Shown below are Mr. Rivera’s scores, the mean, and the standard deviation of each
three tests given to 500 applicants.
Test Mr. Rivera’s Score Mean S.D.
Personality 72.2 68.2 10.5
Abstract Reasoning 65 73 8.6
Aptitude 56.5 54.5 5.1
On which test did Mr. Rivera stand highest? lowest?
Solution:
To determine the answer, convert the scores of each test in standard score.
Personality Test
x = 72.2 µ = 68.2 σ = 10.5
𝑧 =
𝑥−𝜇
𝜎
=
72.2−68.2
10.5
=
4
10.5
z = 0.38 (rounded off to two decimal places)
Three Mathematical Facts about Normal Distribution:
1. About 68% of the values in the population fall within ±1 std. deviation from the mean
(actual value 68.27%)
2. About 95% of the values in the population within ±2 std. deviation from the mean
(actual value 95.45%)
3. About 99% of the values are in an interval ranging from 3 std. deviation below the
mean to 3 std. deviation above the mean (actual value 99.74%).
Activity 1. Directions: True or False. Write T if the statement is correct and F if
otherwise. Write you answer on the space provided.
______1. Heights and weights of people, blood pressure, test score and error in
measurements are examples of things that follow normal distribution.
______2. The curve in normal distribution is divided into two by the line x = µ.
______3. The horizontal distance of both inflection points to the line x = µ is 2 units of
standard deviation.
______4. The three important measures in normal distributions are mean, median and
mode.
______5. If you have 500 observations and it represents a normal distribution, there are
340 values in the population fall within ±2 std. deviation.
Activity 2. Directions: Choose the correct word inside the parenthesis to make the
statement correct. Underline the correct answer.
1. The normal distribution is centred at its (mean, median, standard deviation).
2. The total area under the normal curve is (1, 2, 3).
3. For a normal random variable, the probability of observing a value less than or equal
to its mean is (0, 0.5, 1).
4. The mean, median and mode in normal distribution are (equal, unequal).
5. The one that measures the location in the normal curve is (mean, variance, standard
deviation).
6 3
Activity 2. Using the derived formula, convert the following standard score (z-score) to real
score:
Given: x = 52 s = 7
1. z = 1.5 x = ____________
2. z = −2 x = ___________
3. z = −0.25 x = ___________

4. 2. x = 50
Solution:
𝑧 =
𝑥−𝜇
𝜎
=
50−60
4
=
−10
4
𝒛 = −𝟐. 𝟓
3. x = 60
Solution:
𝑧 =
𝑥−𝜇
𝜎
=
60−60
4
=
0
4
𝒛 = 𝟎
Note that in converting normal random variable x, you have to substitute the
values given and the random variable x in the formulas: 𝒛 =
𝒙−𝝁
𝝈
or 𝒛 =
𝒙−𝒙
̅
𝒔
Converting standard normal random variable to normal random variable
Note: For convenient way of converting standard normal random variable to normal
random variable, derive the formula to solve for normal random variable x.
𝑧 =
𝑥−𝜇
𝜎
𝑧 =
𝑥−𝑥
𝑠
𝑧𝜎 = 𝑥 − 𝜇 𝑧𝑠 = 𝑥 − 𝑥
𝑧𝜎 + 𝜇 = 𝑥 𝑧𝑠 + 𝑥 = 𝑥
𝒙 = 𝒛𝝈 + 𝝁 𝒙 = 𝒛𝒔 + 𝒙
̅
Determine the real score of the following z-score.
Given: x = 75 s = 8.5
3. z = 0.35
Solution:
𝑥 = 𝑧𝑠 + 𝑥 = 0.35 8.5 + 75 = 2.975 + 75
𝒙 = 𝟕𝟕. 𝟗𝟖 (rounded off to two decimal places)
The Standard Normal Distribution
Standard normal distribution is a special case of the normal distribution. It is the
distribution that occurs when a normal random variable has a mean of zero and a standard
deviation of one.
The normal random variable of a standard normal distribution is called standard score
or z-score. Every normal random variable X can be transformed into a z-score via the following
equation:
Population: 𝒛 =
𝒙−𝝁
𝝈
Sample: 𝒛 =
𝒙−𝒙
̅
𝒔
Z-score of a normal random variable (raw score x) measures the number of standard
deviations between a raw score x and the mean of the distribution where the raw score x
came from.
Normal Curve Standardized Normal Curve
Converting normal random variable to standard normal variable
Study the following examples. Pay careful attention to the results and how the normal
random variable was converted to standard normal variable.
Covert the following real score into standard score.
Given: µ = 60 σ = 4
1. x = 68
Solution:
𝑧 =
𝑥−𝜇
𝜎
=
68−60
4
=
8
4
𝒛 = 𝟐
4 5
Activity 3. Compute for the corresponding z-score for each of the normal
random variable. (Round off your answer into two decimal places)
Given: µ = 75 σ = 5.1
1. x = 80 z = _______________
2. x = 63 z = _______________
x = 82 z = _______________