The Normal Distribution and Areas Under The Normal Curve

The Normal Distribution and
Areas Under the Normal Curve

Introduction
Random variables can be either discrete or
continuous.
- A discrete variable cannot assume all values
between any two given values of the variables.
- A continuous variable can assume all values
between any two given values of the variables.
Examples of continuous variables are the heights of
adult men, body temperature of rats, and
cholesterol levels of adults.
Introduction
Distributions that are bell-shaped are called
approximately normally distributed variables.
Sample size increased and class Normal distribution for the

width decreased population
Introduction
The distribution is also known as the bell curve
or the Gaussian distribution, named for the
German mathematician Carl Friedrich Gauss
(1777-1855), who derived its equation.
Introduction
• When the data values are evenly distributed
about the mean, the distribution is said to be
symmetrical.
Mean
Median
Mode
Introduction
• When the majority of the data values fall to the left or right of
the mean, the distribution is said to be skewed.
• When the majority of the data values fall to the right of the
mean, the distribution is said to be negatively skewed. The
mean is to the left of the median, and the mean and the
median are to the left of the mode.
Mean Mode
Median
Introduction
• When the majority of the data values fall to the left
of the mean, the distribution is said to be positively
skewed. The mean falls to the right of the median
and both the mean and the median fall to the right
of the mode.
Mode Mean
Median
Properties of Normal Distribution
Note:
𝜇 is the mean
𝜎 is the standard
deviation
• The figure above shows two normal distributions

with the same mean values but different standard
deviations. The larger the standard deviation, the
more dispersed, or spread out, the distribution, is.

with the same standard deviation but with
different means. These curves have the same
shapes but are located at different positions on
the x-axis.

with different means and different standard
deviations.
• The normal distribution is a continuous,
symmetric, bell-shaped distribution of a
variable.
Summary of the Properties of the
Theoretical Normal Distribution
• The normal distribution curve is bell-shaped.
• The mean, median, and mode are equal and
located at the center of the distribution.
• The normal distribution curve is unimodal
(i.e., it has only one mode).
• The curve is symmetric about the mean, which
is equivalent to saying that its shape is the
same on both sides of a vertical line passing
through the center.
• The curve is continuous – i.e., there are no
gaps or holes. For each value of X, there is a
corresponding value of Y.
• The curve never touches the x axis.
Theoretically, no matter how far in either
direction the curve extends, it never meets
the x axis – but it gets increasingly closer.
• The total area under the normal distribution
curve is equal to 1.00, or 100%.
Finding Areas Under the Normal
Distribution Curve
Situation 1:
0 +z -z 0
Between 0 and any z value:
Look up the z value in the table to get the area.
Examples:
1. Find the area under the normal distribution curve between z = 0 and z =
2.34
2. Find the area between z = 0 and z= 1.5
3. Find the area between z = 0 and z = -1.75
Answer (1)
Draw the normal curve.
2.34
Answer (1)
Locate the value of 2.34 in
the z-table.
para macomplete yung 2.34, hanapin

ang intersection ng 2.3 at ng 0.04.
Therefore, the area is 0.4904.

Answer (2)
1.5
Answer (2)
Locate the value of 1.5 in
the z-table.
para macomplete yung 1.5, hanapin


Answer (3)
-1.75
Answer (3)
Locate the value of -1.75 in
the z-table.
Since the normal curve is symmetric,

kahit negative yung z, same lang siya
ng value sa positive. So instead of
locating for the value of -1.75, we will
locate for the value of 1.75.
Para macomplete yung 1.75, hanapin


Distribution Curve
Situation 2: In any tail:
a. Look up the z value to get the area.
b. Subtract the area from 0.5000.
Example: 0 +z -z 0
4. Find the area to the right of z = 1.11

5. Find the area to the left of z = -1.93
Answer (4)
Draw the normal curve
1.11
Answer (4)
Locate the value of 1.11 in the
z-table.
Para macomplete yung 1.11, hanapin ang
intersection ng 1.1 at ng 0.01.
If you can notice, sa graph, hindi nakadikit

yung shade sa zero. If that’s the case, para
makuha ang area:
Area = 0.5 – value sa table

(bakit 0.5? Going back sa properties ng
normal curve, ang area ng buong
distribution ay 1. Since nahahati sa
dalawang side ang normal curve, left side is
0.5, right side is also 0.5. A = 0.5 – 0.3665
A = 0.1335
Answer (5)
-1.93
Answer (5)
Locate the value of -1.93 in the z-
table.
Since the normal curve is symmetric, kahit negative
yung z, same lang siya ng value sa positive. So
instead of locating for the value of -1.93, we will
locate for the value of 1.93 instead.
If you can notice, sa graph, hindi nakadikit yung

shade sa zero. If that’s the case, para makuha ang
area:
Area = 0.5 – value sa table
(bakit 0.5? Going back sa properties ng normal curve,

ang area ng buong distribution ay 1. Since nahahati
sa dalawangf side ang normal curve, left side is 0.5,
right side is also 0.5.
Para macomplete yung 1.93, hanapin ang

A = 0.5 – 0.4732
intersection ng 1.9 at ng 0.03. A = 0.0268
Distribution Curve
Situation 3: Between two values on the same side of the mean:
a. Look up both z values to get the areas.
b. Subtract the smaller area from the larger area.
0 𝑧1 𝑧2 −𝑧1−𝑧2 0
Examples:
6. Find the area between z = 2.00 and z = 2.47
7. Find the area between z = -2.48 and z = -0.83
Answer (6)
2 2.47
Answer (6)
Kapag ang itsura ng shade ay nasa
isang side lang (either left or right) at
hindi nakadikit sa zero, hanapin lang
sa table ang area ng dalawang
values tapos pagsubtractin.
Find the values of 2.00 and 2.47 in the table.
2.00 is 0.4772, while 2.47 is 0.4932.
Subtract the smaller area from the larger area.
A = 0.4932 – 0.4772
A = 0.016
Answer (7)
-2.48 -0.83
Answer (7)
Kapag ang itsura ng shade ay nasa
isang side lang (either left or right) at
hindi nakadikit sa zero, hanapin lang
sa table ang area ng dalawang
values tapos pagsubtractin.
Find the values of 2.48 and 0.83 in the table.
2.48 is 0.4934, while 2.47 is 0.2969.
Subtract the smaller area from the larger area.
A = 0.4934 – 0.2969
A = 0.1965
Distribution Curve
Situation 4: Between two z values on opposite sides of the mean:
a. Look up both z values to get the areas.
b. Add the areas.
Example: -z 0 +z
8. Find the area between z = 1.68 and z = -1.37
Answer (8)
-1.37 1.68
Answer (8)
Kapag ang itsura ng shade ay
nagmeet sa gitna, mag-aadd ng
0.5.
Find the values of 1.37 and 1.68 in the
table.
1.37 is 0.4147, while 1.68 is 0.4535.
Add the two areas.
A = 0.4147 + 0.4535
A = 0.8682
Distribution Curve
Situation 5: Less than any z value to the right of the mean.
a. Look up the z value to get the area.
b. Add 0.5000 to the area.
0 +z
Example:
9. Find the area to the left of z = 1.99
Answer (9)
1.99
Answer (9)
nashade yung kalahati ng
distribution, automatic meron na
agad 0.5.
Find the value of 1.99.
The value of 1.99 is 0.4767.
Add 0.4767 to 0.5
A = 0.5 + 0.4767
A = 0.9767
Distribution Curve
Situation 6: Greater than any z value to the left of the mean.
a. Look up the z value in the table to get the area.
b. Add 0.5000 to the area.
Example: -z 0
10. Find the area to the right of z = -1.16

Answer (10)
-1.16
Answer (10)
nashade yung kalahati ng
distribution, automatic meron na
agad 0.5.
Find the value of 1.16.
Add 0.3770 to 0.5
A = 0.5 + 0.3770
A = 0.8770
Distribution Curve
Situation 7: Find the total area under the curve in any two tails.
a. Look up the z values in the table to get the areas.
b. Subtract both areas from 0.5000
c. Add the answers.
-z 0 +z
Example:
11. Find the area to the right of z = 2.43 and to the left of z = -3.01
Answer (11)
-3.01 2.43
Answer (11)
dalawang nasa magkabilang
dulo, kunin yung area sa table
nung dalawang values tapos
isubtract pareho sa 0.5.

1st area = 0.5 – 0.4987 = 0.0013

2nd area = 0.5 – 0.4925 = 0.0075
Add the two areas.
A = 0.0013 +0.0075 = 0.0088

Summary (Determining the Area)
Situation 1: Nakadikit yung shade sa 0, kunin
lang yung value sa table.
Situation 2: Hindi nakadikit sa 0, isubtract sa 0.5

yung value sa table.
Situation 3: Nasa same side, hindi nakadikit sa 0,

pagsubtractin yung values nila sa table.
Situation 4: May negative at positive na z,
nagmeet sa 0 yung shade, pag-addin yung
values nila sa table.
Situation 5: Nashade-an yung kalahati ng

distribution, automatic may 0.5 na. Hanapin ang
area ng z sa table tapos iadd sa 0.5.
Situation 6: Katulad ng situation 5.
Situation 7: Kapag parehong side ng distribution

ay may shade tapos hindi nakadikit sa 0, kunin
pareho ang value sa table tapos isubtract sa 0.5
pareho. Lastly, pag addin yung dalawang sagot.
Note:
Kapag ang z ay negative, ititreat mo sya as
positive when looking for its area in the table.

The Normal Distribution and Areas Under The Normal Curve

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The Normal Distribution and

Areas Under the Normal Curve

Sample size increased and class Normal distribution for the

• The figure above shows two normal distributions

• The figure above shows two normal distributions

• The figure above shows two normal distributions

para macomplete yung 2.34, hanapin

Therefore, the area is 0.4904.

para macomplete yung 1.5, hanapin

Therefore, the area is 0.4332.

Since the normal curve is symmetric,

Para macomplete yung 1.75, hanapin

Therefore, the area is 0.4599.

4. Find the area to the right of z = 1.11

If you can notice, sa graph, hindi nakadikit

Area = 0.5 – value sa table

If you can notice, sa graph, hindi nakadikit yung

Area = 0.5 – value sa table

(bakit 0.5? Going back sa properties ng normal curve,

Para macomplete yung 1.93, hanapin ang

2.00 is 0.4772, while 2.47 is 0.4932.

Subtract the smaller area from the larger area.

2.48 is 0.4934, while 2.47 is 0.2969.

Subtract the smaller area from the larger area.

1.37 is 0.4147, while 1.68 is 0.4535.

Add the two areas.

The value of 1.99 is 0.4767.

Add 0.4767 to 0.5

10. Find the area to the right of z = -1.16

The value of 1.16 is 0.3770.

Add 0.3770 to 0.5

The value of 3.01 is 0.4987.

1st area = 0.5 – 0.4987 = 0.0013

Add the two areas.

A = 0.0013 +0.0075 = 0.0088

Situation 2: Hindi nakadikit sa 0, isubtract sa 0.5

Situation 3: Nasa same side, hindi nakadikit sa 0,

Situation 5: Nashade-an yung kalahati ng

Situation 7: Kapag parehong side ng distribution

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