This document discusses converting between normal random variables and z-scores (standard normal variables). It provides the formula for calculating z-scores and examples of finding z-scores from raw scores and raw scores from z-scores. It also gives an example word problem involving comparing two students' scores on different exams by converting them to z-scores. The document aims to teach how to convert between normal random variables and z-scores.

1. Converting a Normal Random Variable to a
Standard Normal Variable & Vice-Versa
QUARTER 3 – MELC 12

2. • The z-score is stated to be a measure of relative
standing. It describes a value's relationship to the
mean of a group of values.
• A Z-score can reveal if a value is typical for a
specified data set or if it is atypical. Z-scores also
make it possible for analysts to adapt scores from
various data sets to make scores that can be
compared to one another more accurately.
• Basically, a z-score gives you an idea how far a
data is from the mean.

3. • z-scores are a way to compare results to a “normal”
population. A lot of tests and surveys reveal thousands
of possible results and units; those results can often
seem meaningless.
• For example, knowing that someone’s height is 5 ft
might be good information, but if you want to compare it
to the “average” person’s height, looking at a vast table
of data can be overwhelming.
• A z-score can tell you where that person’s weight is
compared to the average population’s mean height.

4. •Most raw scores comprise large values, so
for convenience, these raw scores are
converted to a more manageable scores
known as z-scores. In this lesson, you will
learn to convert a normal random variable to
a standard normal random variable (z-
values) and vice versa.

5. The formula for calculating z is.

6. Example 1. Finding the z-value

7. Given the mean, 𝜇 = 30 and the standard deviation, 𝜎 = 4
of a population of scores in Statistics and Probability. Find
the z-value that corresponds to a score 𝑋 = 38.

8. Example 2. Finding the raw
score

9. Steps Solution
1, Modify the formula to solve for X. X = X + zs
2. Substitute in the formula. X = 80 + (0.52)(15)
3. Compute the raw scores. X = 80 + 7.8
X = 87.8
Thus the raw score is 87.8 when z =
0.52

10. Example 3. Problem Solving

11. entrance exam in different universities. Ana’s score is
140 (𝜇 = 125, 𝜎 = 15) while John’s score is 90 (𝜇 = 80, 𝜎
= 5). Who among them scored better than their peers?
Solution: To solve this problem, we need to
standardize the two scores. Thus, we need to convert
these to z-scores.

12. Activity Proper

13. Exercies 1. Complete the table by providing the
correct answers. Show your process on your paper.

14. 1. Zeleo and Zoey are close buddies who took
the national achievement test. The mean score
and standard deviation are 120 and 10,
respectively. If Zeleo obtained a z-score of 1.5
and Zoey got -0.95 z-score. What were their
scores in the examination? Who scored
higher?
2. If the scores in a Statistics and Probability
test are normally distributed with a mean of 35
and standard deviation of 4.5. What is the z-
score for a score of 30?

15. The End