normal distribution under the curve

1. Statistics and Probability
Quarter 3: Normal Distribution
Ma’am Neomy Angela L. Tolentino

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3. Statistics and Probability

4. Let’s have
a warm
up!

5. “
Statistics and Probability
What is a normal distribution?
A normal distribution (Gaussian distribution) is a
probability distribution that is symmetric about the
mean, showing that data near the mean are more
frequent in occurrence than data far from the mean.

6. “
Statistics and Probability
𝑓 𝑥 =
1
𝜎 2𝜋
𝑒
−
1
2
𝑥−𝜇
𝜎
2
f(x) = the height of the curve
particular values of x
X = any score in the
distribution
𝜎 = standard deviation of the
population
𝜇 = mean of the population
𝜋 = 3.1416
𝑒 = 2.7183

7. “
Statistics and Probability
In graph form, normal
distribution will appear as a bell
curve.

8. Properties of the normal distribution:
Statistics and Probability
1
The graph is a
continuous curve
and has a domain
− ∞ < 𝑋 < ∞.

9. Properties of the normal distribution:
Statistics and Probability
As the x gets larger in either positive direction, the tail of the curve approaches but will never
touch the horizontal axis. The same thing happens when x gets smaller in the negative
direction.
2
The graph is asymptotic
to the x-axis. The value
of the variable gets closer
and closer but will never
be equal to 0.

10. Properties of the normal distribution:
Statistics and Probability
▹ The mean (𝜇) indicates the highest peak of the
curve and is found at the center.
▹ Note:
3The highest point on
the curve occurs at
𝑥 = 𝜇 (mean)
𝜇 = 𝑚𝑒𝑎𝑛
𝜎 = 𝑠𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝑑𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛

11. Properties of the normal distribution:
Statistics and Probability
4
The curve is
symmetrical about
the mean.

12. Properties of the normal distribution:
Statistics and Probability
5
The total area in the
normal distribution
under the curve is
equal to 1.
100% or 1

13. Properties of the normal distribution:
Statistics and Probability
6
In general, the graph of
a normal distribution is
a bell-shaped curve
with two inflection
points, one on the left
and another on the
right. Inflection points
are the points that
mark the change in the
curve’s concavity

14. Properties of the normal distribution:
Statistics and Probability
7
Every normal
curve
corresponds
to the
“empirical
rule” (also
called the 68 –
95 – 99.7%
rule):
0.3413
or
34.13%
0.4772
or
47.72%
0.4987
or
49.87%
0.4987
or
49.87%
0.4772
or
47.72%
0.3413
or
34.13%
68.26%
95.44%
99.74%

15. Example #1:
Statistics and Probability
Suppose the
mean is 60 and
the standard
deviation is 5.
Sketch a normal
curve for the
distribution.
60
𝝁 =
5
𝝈 =
65 70
55
50
0 1 2
-1
-2
45
-3
75
3
5 5
Z scores

16. Statistics and Probability
How to
compute for
the z-score
given the raw
score and
standard
deviation?
1. Use the formula 𝑧 =
𝑥−𝜇
𝜎
2. Substitute the values of the mean, raw
score and standard deviation to the
formula.
3. Leave your final answer in decimal form.

17. Example #1:
Statistics and Probability
Q1:What is the z score if
x = 66? x = 57? x = 46?
𝑧 =
𝑥 − 𝜇
𝜎
𝜎 = 5, 𝜇 = 60
x = 66
𝑧 − 𝑧 𝑠𝑐𝑜𝑟𝑒
𝑥 – raw score/ observed
value
𝜇 − 𝑚𝑒𝑎𝑛
𝜎 − 𝑠𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝑑𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛
𝑧 =
66 − 60
5
𝑧 =
6
5
𝑧 = 1.2
x = 57
𝜎 = 5, 𝜇 = 60
𝑧 =
57 − 60
5
𝑧 =
−3
5
𝑧 = −0.6
x = 46
𝜎 = 5, 𝜇 = 60
𝑧 =
46 − 60
5
𝑧 =
−4
5
𝑧 = −0.8

18. Example #2:
Statistics and Probability
Find for the z score of the following:
𝝈 = 𝟓
𝑧 =
50 − 40
5
𝑧 =
10
5
𝑧 = 2
𝒙 𝝈 𝝁
50 5 40
40 8 52
36 6 28
60 10 74
75 15 82
x = 50 𝝁 = 𝟒𝟎
𝒛 =
𝒙 − 𝝁
𝝈
𝝈 = 𝟖
x = 40 𝝁 = 𝟓𝟐
𝑧 =
40 − 52
8
𝑧 =
−12
8
= −
3
2
𝑧 = 1.5
𝑧 = 1.33
𝑧 = −1.4
𝑧 = −0.47

19. How to sketch
a normal
distribution
and its area or
compute its
probability? Table of Areas under the Normal
Curve is also known as the z-table.

20. Four steps in Finding the Areas Under
the Normal Curve Given a z-value
Step 1: Express the given z-
value into a three-digit form
Step 2: Using the z-table, find
the first two digits on the left
column
Step 3: Match the third digit
with the appropriate column
on the right
Step 4: Read the area (or
probability) at the intersection
of the row and the column
Example: 0.78
Area: 0.2823

21. 𝑧 = 1.2
Sketch the normal distribution of each
scores and indicate its area.
Area: 0.3849
60 65 70
55
50
45 75
1.2
65
38.49%

22. 𝑧 = −0.6
Sketch the normal distribution of each
scores and indicate its area.
Area: 0.2257
60 65 70
55
50
45 75
-0.6
57
22.57%

23. 𝑧 = −0.8
Sketch the normal distribution of each
scores and indicate its area.
Area: 0.2881
60 65 70
55
50
45 75
-0.8
46
28.81%

24. Place your answer in a 1 whole sheet of paper to be submitted on the day of
retrieval
Sketch the normal distribution of each z-scores and indicate its area.
𝑧 = 2 𝑧 = 1.5 𝑧 = 1.33 𝑧 = −1.4 𝑧 = −0.47
Assignment #1

25. Statistics and Probability
Compute probability of
P(z>1.27),
P(z<1.31) ,
P(-1.4<z<2.71) &
P(1.5<z<2).

26. Statistics and Probability
Compute
probability of
P(z>1.27)
1.27
1. Get the probability of
the given z score
2. Subtract the
probability from
0.5000

27. Statistics and Probability
Compute
probability of
P(z>1.27)
1.27
1. Get the probability of
the given z score
2. Subtract the
probability from
0.5000
0.3980
0.5000 Solution:
= 0.5000 – 0.3980
= 0.102 or 10.2%

28. Statistics and Probability
Compute
probability of
P(z<1.31)
1.31
1. Get the probability of
the given z score
2. Add the probability
of the z score to
0.5000
0.4049
0.5000
Solution:
= 0.5000 + 0.4049
= 0.9049 or 90.49%

29. Statistics and Probability
Compute
probability of
P(-1.4<z<2.71)
2.71
1. Get the probability of
the given z score
2. Add the probability
of the two z scores
0.4966
0.4192
-1.4
Solution:
= 0.4192 + 0.4966 = 0.9158 or 91.58

30. Statistics and Probability
Compute
probability of
P(1.5<z<2)
2
1. Get the probability of
the given z score
2. Subtract the
probability of the two
z scores
0.4772
0.4332
1.5
Solution:
= 0.4772 - 0.4332
= 0.044 or 4.4%

31. Statistics and Probability
Summary in computing the probability given:
a. Area less than +z
ex. P(z<1) = ADD
the area of the z
score to 0.5000
c. Area greater than
+z ex. P(z>1) =
SUBTRACT the
area of the z score
from 0.5000
b. Area greater than
-z ex. P(z>-1) = ADD
the area of the z
score to 0.5000
d. Area less than -z ex.
P(z<-1) = SUBTRACT
the area of the z score
from 0.5000
e. Area between –z and
+z ex. P(-1<z<-1) = ADD
the area of the two z
scores
e. Area between –z and –z / +z
and +z
ex. P(1<z<2) / P(-2<z<-1) =
SUBTRACT the area of the
two z scores

32. Place your answer in a 1 whole sheet of paper to be submitted on the day of
retrieval.
Compute for the probability of the following and sketch the graph.
P(z>2.13)
Assignment #2
P(-2.19<z<0.16) P(-1.74<z<-0.25) P(z<0.15)

33. Let’s have
an
activity!

34. Statistics and Probability
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everyone!