normal distribution

1. STATISTICS AND
PROBABILITY

2. GAMETIME!
▫ Using the app or the printed z
table,locate the probabilities
of each area.
▫ For android user ,always
round off your answer to
4 decimal places.
▫ First to give the correct
answer will have an
additional bonus point.

3. How to use ?
▫ Using the Z distribution Table
Locate Z=0.15
First
column
First
row

4. 5
FINDING THE AREA
OF STANDARD SCORES

5. 66
USING THE Z DISTRIBUTION
TABLE, FIND THE AREA OF
THE FF:
z= 1.63

6. 7
7
1. FIND THE AREA TO THE LEFT OF z = 1.63

7. 88
USING THE Z DISTRIBUTION
TABLE, FIND THE AREA OF
THE FF:
z= 0.05

8. 9
9
2. FIND THE AREA TO THE LEFT OF z = 0.05
9

9. 10
10
USING THE Z DISTRIBUTION
TABLE, FIND THE AREA OF
THE FF:
z= 0.38

10. 11
3. FIND THE AREA TO THE LEFT OF z = 0.38

11. 12
12
USING THE Z DISTRIBUTION
TABLE, FIND THE AREA OF
THE FF:
z= 1.08

12. 13
4. FIND THE AREA TO THE LEFT OF z = 1.08
13

13. 14
14
USING THE Z DISTRIBUTION
TABLE, FIND THE AREA OF
THE FF:
z= 0.94

14. 5. FIND THE AREA TO THE LEFT OF z = 0.94
15

15. NORMAL
DISTRIBUTION

16. LEARNING OBJECTIVES:
1
Understand the
concept of the
normal curve
distribution.
17
2
Identifies regions
under the normal
curve corresponding
to different standard
normal values.
3
computes
probabilities and
percentiles using the
standard normal
table.

17. 18

18. NORMAL
DISTRIBUTION

19. Data can be distributed “spread out” in different ways.
Spread-out to the left No definite pattern
Spread-out to the right

20. But there are many cases where the data tends to be around a
central value with no bias left or right, and it gets close to a
"Normal Distribution" like this:
The Bell Curve is a Normal Distribution.
The yellow histogram shows data that follows it closely, but not
perfectly.

21. The graph of a normal distribution depends on two factors
1. Mean - determines the location
of the center of the graph.
2. Standard Deviation –
determines the height and width
of the graph.
The red curve has larger value of standard deviation,
therefore more spread-out.

22. PROPERTIES OF THE NORMAL CURVE
1. It is bell shaped
2. The mean, median and mode are
equal. They are located at the
center of the distribution.
3. It is symmetric about the mean.
4. The total area under the curve is
1.
5. The curve is asymptotic to the x-
axis. This means the curve gets
nearer the x-axis but never
touches it. It extends indefinitely
to the left and to the right.
Above the mean
Below the mean

23. UNDERSTANDING THE
STANDARD NORMAL CURVE

24. 25
The Standard Normal Curve
Is a normal probability distribution that has a mean 𝜇 = 0 and a
standard deviation 𝜎 = 1.

25. Standard Normal Distribution
Why convert Normal Distribution into Standard Normal Distribution?
1. To standardized the distribution so that only one table of areas can be used for
all normal distribution.
2. To compare value from one normal distribution to another.

26. Standard Score (Z- score)
The standard score is the distance of the score from the mean (𝒙) in terms of the
standard deviation (s).
To change an observed values (x)
into standard score, use the
equation:
𝑧 =
𝑋 − 𝜇
𝜎
𝑧 =
𝑋 − 𝑋
𝑠
(z-score for population data) (z-score for sample data)
X = raw score or given
measurement
𝜇= population mean
x̄ = sample mean
𝜎= population 𝑠𝑡𝑎𝑛𝑑𝑎𝑟𝑑
𝑑𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛
S= sample standard
deviation

27. Problem 1:
EXAMPLE 1 :
In the final examination in
Mathematics, the mean
grade 82 and the standard
deviation was 8. In English,
the mean grade was 86 and
the standard deviation was
10. Joseph scored 88 in
Mathematics and 92 in
English. In which subject
was his standing higher?
Given:
𝜇(math) = 82
𝜎 = 8
x = 88
𝜇(eng) = 86
𝜎 = 10
x = 92
Formula:
Solution:
Zmath = 88 − 82
8
𝑧 =
𝑋 − 𝜇
𝜎
Zmath = 0.75
Zenglish = 92 − 86
10
Zenglish = 0.60

28. IDENTIYING REGIONS OF AREAS
UNDER THE NORMAL CURVE

29. Finding Probabilities of Normally
Distributed Random Variables
𝑃 𝑧 < 𝑎 denotes the probability that the z score is
less than a.
𝑃(𝑧 > a) denotes the probability that the z score is
greater than a.
𝑃 𝑎 < 𝑧 < 𝑏 denotes the probability that the z score is
between a and b

30. Finding Probabilities of Normally
Distributed Random Variables
𝑃(𝑧 > a)
Greater than z
At least z
More than z
To the right of z
Above z
𝑃 𝑧 < 𝑎
Less than z
At most z
No more than z
Not greater than z
To the left of z
𝑃 𝑎 < 𝑧 < 𝑏
Between z1 and z2
Case 1 Case 2 Case 3

31. Normal curves may have different heights and widths, but
in all cases the following characteristics may apply.
EMPIRICAL RULE
For a normal distribution:
1. 68% of data falls within the first
standard deviation from the mean.
2. 95% fall within two standard
deviations.
3. 99.7% fall within three standard
deviations.

32. 33
Example:
Find the area from 𝒛 = 𝟎 and 𝒛 = 𝟏. 𝟐𝟓
The area between 𝑧 = 0 and
𝑧 = 1.25 is the area between
the mean ( center of the
curve ) and 1.25. Locate the
area for 𝑧 = 1.25 from the
table.

33. 34
Example:
Find the area from 𝒛 = 𝟏. 𝟐𝟓 and 𝒛 = 𝟐. 𝟎
To find the area, subtract the area of 1.25 from the area of 2.0.

34. 35
Determine each of the following areas of specific regions under the normal curve
using probability notation.then show each of these regions graphically.
Between z =0.76 and z = 2.88 Probability notation : P( 0.76<z<2.88) )
0.76 2.88

35. 36
Determine each of the following areas of specific regions under the normal curve
using probability notation.then show each of these regions graphically.
Between z =0.76 and z = 2.88 Probability notation : P( 0.76<z<2.88)
Solution:
0.76 = 0.2764
0.4980 - 0.2764
0.2216
2.88 = 0.4980

36. 37
Example:
Find the area from and to the right of 𝒛 = 𝟐. 𝟎

37. 38
Determine each of the following areas of specific regions under the normal curve
using probability notation.then show each of these regions graphically.
Above z = 1.46 Probability notation : P( z > 1.46 )
1.46

38. 39
Determine each of the following areas of specific regions under the normal curve
using probability notation.then show each of these regions graphically.
Above z = 1.46 Probability notation : P( z > 1.46 )
1.46
Solution:
1.46 = 0.4279
0.50 - 0.4279 =
0.0721

39. 40
Example:
Find the area from and to the left of 𝒛 = 𝟐. 𝟎

40. 41
Determine each of the following areas of specific regions under the normal curve
using probability notation.then show each of these regions graphically.
below z = -0.58 Probability notation : P( z < -0.58
)
-0.58

41. 42
Determine each of the following areas of specific regions under the normal curve
using probability notation.then show each of these regions graphically.
Above z = -0.58 Probability notation : P( z < -0.58 )
Solution:
-0.58 = 0.2190
0.50 - 0.2190 =
0.2810

42. 43
Example:
Find the area to the left of 𝒛 = −𝟏. 𝟓𝟎

43. DETERMINING PROBABILITIES

44. Any
question?

45. Closing Prayer
Glory be to the Father,
and to the Son,
and to the Holy Spirit,
as it was in the beginning,
is now, and ever shall be,
world without end.
Amen.
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