Convert a normal random variable to a standard normal variable and vice versa. Compute probabilities and percentiles using the standard normal table. Visit the website for more services: https://cristinamontenegro92.wixsite.com/onevs

1. Standard Scores

2. Learning Competencies
The learner will be able to:
1. Convert a normal random variable to a
standard normal variable and vice versa; and
2. Compute probabilities and percentiles using
the standard normal table.

3. The standard score or z-score measure how
many standard deviation a given value (x) is
above or below the mean. The z-scores are
useful in comparing observed values.
A positive z-score indicates that the score or
observed value is above the mean.
Whereas a negative z-score indicates that the
score or an observed value is below the mean.

4. The z-score is found by using the
following equations.
A. For Sample
where z=standard score, x=raw score or
observed value, =sample mean and
s=sample standard deviation.
B. For population
where z=standard score, x= raw score or
observed value, =population mean and
=population standard

5. Example 1. On a final examination on Biology,
the mean was 75 and the standard deviation
was 12. Determine the standard score of a
student who received a score of 60 assuming
that the scores are normally distributed and
sketch the graph.

6. Solution:
This indicates that 60 is 1.25 standard deviation
below the mean

7. Example 2. On the first periodic exam in
Statistics, the population mean was 70 and
the population standard deviation was 9.
Determine the standard score of a student
who got a score of 88 assuming that the
scores are normally distributed.

8. Solution:
This indicates that 88 is 2 standard deviations
above the mean.

9. Example 3. Luz scored 90 in an English test and
70 in a Physics test. Scores in the English test
have a mean of 80 and a standard deviation of
10. Scores in the Physics test have a mean of
60 and standard deviation of 8. In which
subject was her standing better assuming that
the scores in her English and Physics class are
normally distributes?

10. Solution:
For English
For Physics
Her standing in physics was better than her
standing in English. Her score in English was
one standard deviation above the mean of the
scores in English whereas in Physics, her score
was 1.25 standard deviation above the mean.

11. Example 4. In a Science test, the mean score is
42 and the standard deviation is 5. Assuming
the scores are normally distributed, what
percent of the score is a.) greater than 48? b.)
less than 50? c.) between 30 and 48?

12. Solution:
From the table,
A1=0.3849
A=A2-A1
=0.5-0.3849=0.1151
Hence, 11.51% of the scores is greater
than 48.

13. From the table,
A2=0.4452
A=A1+A2
=0.5-0.4452=0.9452 or 94.52%
Hence, 94.52% is less than 50

14. From the table, A1=0.4918 and
A2=0.3849
A=A1+A2=0.4918+0.3849=0.8767 or
87.67%
Hence, 87.67% of the scores are
between 30 and 48.

15. Example 5. The mean height of grade 9 students
at a certain high school is 164 centimeters and
the standard deviation is 10 centimeters.
Assuming the heights are normally are
normally distributed, what percent of the
heights is greater than 168 centimeters?
Solution

16. From the table, A1=0.1554
A=A2-A1=0.5-0.1554=0.3446
Hence, 34.46% of the heights are
greater than 168 centimetres.

17. Quiz: In a math test, the mean score is 45 and
the standard deviation is 4. Assuming
normality, what is the probability that a score
picked random will lie
a. Above score 50?
b. Below score 38?