Statistics and probability

1. Lesson 3.2
Characteristics of a Normal
Random Variable

2. Learning
Competencies
At the end of the lesson, the learners should be able
to illustrate a normal random variable and its
characteristics (M11/12SP-IIIc-1).

3. Learning
Objective
At the end of this lesson, you should be able to
● Accurately determine the characteristics of a
normal random variable.
● Correctly identify if a given score is an outlier.
● Correctly solve real-life problems involving
characteristics of a normal random variable.

4. a. A normal distribution is symmetric about its mean.
In a normally distributed set of data, the mean will be placed
at the center if the data are arranged in ascending order.
Characteristics of a Normally Distributed Set of
Data

5. a. A normal distribution is symmetric about its mean.
Moreover, approximately 50% of the data is less than the
mean and approximately 50% of the data is greater than the
mean.
Characteristics of a Normally Distributed Set of
Data

6. b. The mean, median, and mode of a normal distribution
are all equal.
Since the mean, median, and mode are all equal, the median
and the mode are also located at the middle of the
distribution.
Characteristics of a Normally Distributed Set of
Data

7. c. A normal distribution is thicker at the center and less
thick at the tails.
”Thicker” at the center means that there are more scores
located near the center of the distribution and there are fewer
scores found near both ends of the distribution.
Characteristics of a Normally Distributed Set of
Data

8. d. Approximately 68.26% of the area of a normal
distribution is within one standard deviation of the
mean.
Note that the area of a normal distribution refers to the
graphical representation of the percentage, proportion, or
probability of a normal distribution.
Characteristics of a Normally Distributed Set of
Data

9. d. Approximately 68.26% of the area of a normal
distribution is within one standard deviation of the
mean.
Characteristics of a Normally Distributed Set of
Data

10. d. Approximately 68.26% of the area of a normal
distribution is within one standard deviation of the
mean.
In a given set of data, approximately 68.26% of the scores are
located between the scores one standard deviation above and
one standard deviation below the mean.
Characteristics of a Normally Distributed Set of
Data

11. d. Approximately 68.26% of the area of a normal
distribution is within one standard deviation of the
mean.
This means that in a normally distributed set of data, a score
of that data has a 68.26% chance of falling within one
standard deviation from the mean.
Characteristics of a Normally Distributed Set of
Data

12. d. Approximately 68.26% of the area of a normal
distribution is within one standard deviation of the
mean.
We may write this as a probability notation.
𝑃 𝜇 − 𝜎 < 𝑋 < 𝜇 + 𝜎 = 0.6826
Characteristics of a Normally Distributed Set of
Data

13. e. Approximately 95.44% of the area of a normal
distribution is within two standard deviations of the
mean.
Characteristics of a Normally Distributed Set of
Data

14. e. Approximately 95.44% of the area of a normal
distribution is within two standard deviations of the
mean.
In a given set of data, approximately 95.44% of the scores are
located between the scores two standard deviations above
and two standard deviations below the mean.
Characteristics of a Normally Distributed Set of
Data

15. e. Approximately 95.44% of the area of a normal
distribution is within two standard deviations of the
mean.
This means that in a normally distributed set of data, a score
of that data has a 95.44% chance of falling within two standard
deviations from the mean.
Characteristics of a Normally Distributed Set of
Data

16. e. Approximately 95.44% of the area of a normal
distribution is within two standard deviations of the
mean.
We may write this as a probability notation.
𝑃 𝜇 − 2𝜎 < 𝑋 < 𝜇 + 2𝜎 = 0.9544
Characteristics of a Normally Distributed Set of
Data

17. f. Approximately 99.74% of the area of a normal
distribution is within three standard deviations of the
mean.
Characteristics of a Normally Distributed Set of
Data

18. f. Approximately 99.74% of the area of a normal
distribution is within three standard deviations of the
mean.
In a given set of data, approximately 99.74% of the scores are
located between the scores three standard deviations above
and three standard deviations below the mean.
Characteristics of a Normally Distributed Set of
Data

19. f. Approximately 99.74% of the area of a normal
distribution is within three standard deviations of the
mean.
This means that in a normally distributed set of data, a score
of that data has a 99.74% chance of falling within three
standard deviations from the mean.
Characteristics of a Normally Distributed Set of
Data

20. f. Approximately 99.74% of the area of a normal
distribution is within three standard deviations of the
mean.
We may write this as a probability notation.
𝑃 𝜇 − 3𝜎 < 𝑋 < 𝜇 + 3𝜎 = 0.9974
Characteristics of a Normally Distributed Set of
Data

21. This rule states that 68.26% of the scores fall within one
standard deviation away from the mean, 95.44% of the scores
fall within two standard deviations away from the mean, and
99.74% of the scores fall within three standard deviations
away from the mean.
Empirical Rule

22. Scores that are more than two standard deviations away from
the mean.
Example:
A normally distributed set of data has a mean of 𝜇 = 5 and a
standard deviation of 𝜎 = 2. The score 𝑥 = 10 is an outlier
because it lies beyond two standard deviations above the
mean.
Outliers

23. Scores that are more than three standard deviations away
from the mean.
Example:
A normally distributed set of data has a mean of 𝜇 = 15 and a
standard deviation of 𝜎 = 3. The score 𝑥 = 5 is an extreme
outlier because it lies beyond three standard deviations below
the mean.
Extreme Outliers

24. Example 1: In a normally distributed set of data containing
5 624 scores, how many scores are expected to fall within
one standard deviation away from the mean?

25. Solution:
According to the empirical rule, 68.26% of the scores would
fall within one standard deviation away from the mean.
To get the number of scores that are within that range, we
get 68.26% of 5 624.
Example 1: In a normally distributed set of data containing
5 624 scores, how many scores are expected to fall within
one standard deviation away from the mean?

26. Solution:
5 624 ∙ 0.6826 = 3 838.9424
≈ 3839
Therefore, around 3 839 of the scores are expected to fall
within one standard deviation away from the mean.
Example 1: In a normally distributed set of data containing
5 624 scores, how many scores are expected to fall within
one standard deviation away from the mean?

27. Example 2: A particular normally distributed set of data has
a mean of 𝜇 = 53.28 and a standard deviation of 𝜎 = 8.41. Is
the score 𝑋 = 34 considered an outlier?

28. Solution:
1. Make a plan in solving the problem.
A score is considered an outlier if it is more than two
standard deviations away from the mean.
Example 2: A particular normally distributed set of data has
a mean of 𝜇 = 53.28 and a standard deviation of 𝜎 = 8.41. Is
the score 𝑋 = 34 considered an outlier?

29. Solution:
Thus, we should first solve for the scores two standard
deviations away from the mean and determine whether the
given score lies beyond those points.
Example 2: A particular normally distributed set of data has
a mean of 𝜇 = 53.28 and a standard deviation of 𝜎 = 8.41. Is
the score 𝑋 = 34 considered an outlier?

30. Solution:
2. Solve for the scores two standard deviations away from
the mean.
Example 2: A particular normally distributed set of data has
a mean of 𝜇 = 53.28 and a standard deviation of 𝜎 = 8.41. Is
the score 𝑋 = 34 considered an outlier?

31. Solution:
First, solve for the score two standard deviations above the
mean.
𝜇 + 2𝜎 = 53.28 + 2 8.41
𝜇 + 2𝜎 = 53.28 + 16.82
𝜇 + 2𝜎 = 70.1
Example 2: A particular normally distributed set of data has
a mean of 𝜇 = 53.28 and a standard deviation of 𝜎 = 8.41. Is
the score 𝑋 = 34 considered an outlier?

32. Solution:
Next, solve for the score two standard deviations below the
mean.
𝜇 − 2𝜎 = 53.28 − 2 8.41
𝜇 − 2𝜎 = 53.28 − 16.82
𝜇 − 2𝜎 = 36.46
Example 2: A particular normally distributed set of data has
a mean of 𝜇 = 53.28 and a standard deviation of 𝜎 = 8.41. Is
the score 𝑋 = 34 considered an outlier?

33. Solution:
3. Compare the scores using a number line.
The mean, the scores two standard deviations away from
the mean, and the score 𝑋 = 34 are presented in the
number line on the next slide.
Example 2: A particular normally distributed set of data has
a mean of 𝜇 = 53.28 and a standard deviation of 𝜎 = 8.41. Is
the score 𝑋 = 34 considered an outlier?

34. Solution:
Example 2: A particular normally distributed set of data has
a mean of 𝜇 = 53.28 and a standard deviation of 𝜎 = 8.41. Is
the score 𝑋 = 34 considered an outlier?

35. Solution:
The score 𝑋 = 34 lies lower that the score two standard
deviations below the mean.
Therefore, the score 𝑋 = 34 is an outlier.
Example 2: A particular normally distributed set of data has
a mean of 𝜇 = 53.28 and a standard deviation of 𝜎 = 8.41. Is
the score 𝑋 = 34 considered an outlier?

36. Individual Practice:
1. In a normally distributed set of data containing 12 658
scores, how many scores are expected to be more than
two standard deviations away from the mean?
2. In a normally distributed set of data with 𝜇 = 23.6 and
𝜎 = 4.4. Is the score 𝑋 = 33.1 considered an outlier?

37. Group Practice: To be done in two to five groups
The average height of Grade 3 students in a certain campus
is 120.2 cm with a standard deviation of 5.7 cm. What is the
probability that a randomly chosen Grade 3 student has a
height between 114.5 cm and 125.9 cm?

38. ● These are the characteristics of a normally distributed
set of data:
o A normal distribution is symmetric about its mean.
o The mean, median, and mode of a normal distribution
are all equal.
o A normal distribution is thicker at the center and less
thick at the tails.

39. ● These are the characteristics of a normally distributed
set of data:
o Approximately 68.26% of the area of a normal
distribution is within one standard deviation of the
mean.
o Approximately 95.44% of the area of a normal
distribution is within two standard deviations of the
mean.

40. ● These are the characteristics of a normally distributed
set of data:
o Approximately 99.74% of the area of a normal
distribution is within three standard deviations of the
mean.

41. ● The empirical rule states that 68.26% of the scores fall
within one standard deviation away from the mean,
95.44% of the scores fall within two standard deviations
away from the mean, and 99.74% of the scores fall within
three standard deviations away from the mean.
● Outliers are scores that are more than two standard
deviations away from the mean.

42. ● Extreme outliers are scores that are more than three
standard deviations away from the mean.