Are you Normal?
• According to Bill Bostock of
Bussiness Insider.com(2019)
the average Filipino is
156.41cm (5
feet 1.57 inches) tall.
• The average Filipino man is
163.22cm (5
feet 4.25 inches) tall.
• The average Filipino woman
is 149.60cm (4 feet
10.89 inches) tall.
Lesson 1
Basic Concept of a Normal Distribution
2
Basic Concept of a Normal Distribution
if a researcher selects a random sample of 100 adult women,
measures their heights, and constructs a histogram, the researcher
gets a graph similar to the one shown in Figure 1.1
3
Basic Concept of a Normal Distribution
if the researcher increases the sample size and decreases the width
of the classes, the histograms will look like the ones shown in Figure
1.2 and Figure 1.3
4
Basic Concept of a Normal Distribution
if it were possible to measure exactly the heights of all adult females in
the United States and plot them, the histogram would approach what is
called a normal distribution curve, as shown in Figure 1.4
5
Basic Concept of a Normal Distribution
This distribution is also known as a bell curve or a Gaussian distribution
curve, named for the German mathematician Carl Friedrich Gauss
(1777–1855), who derived its equation.
6
Basic Concept of a Normal Distribution
✗ Carl Friedrich Gauss
named this bell-shaped,
probability density
function, with a peak in
the mean, as the
Gaussian function,
Gaussian distribution or
bell curve. Gauss used
this normal curve to
analyze astronomical data
in 1809.
7
Basic Concept of a Normal Distribution
✗ The normal Curve was
developed
mathematically in 1733
by Abraham de Moivre
(1667-1754) as an
approximation to the
binomial distribution.
8
Lesson 2
Properties of a Normal Distribution
9
Properties of a Normal Distribution
✗ In mathematics, curves can be represented
by equations. For example, the equation of
the circle shown in Figure 2.1 is
𝑥 2 + 𝑦 2 = 𝑟 2 , where 𝑟 is the radius.
✗ the equation and the properties of a circle
can be used to study many aspects of the
wheel, such as area, velocity, and
acceleration.
10
Properties of a Normal Distribution
✗ In a similar manner, the theoretical curve,
called a normal distribution curve, can be
used to study many variables that are not
perfectly normally distributed but are
nevertheless approximately normal.
11
Properties of a Normal Distribution
The mathematical equation for a normal distribution is:
Where
𝑒 ≈ 2.718 … (the exact value is identified by a scientific calculator)
𝜋 ≈ 3.14 … (the exact value is identified by a scientific calculator)
𝜇 = 𝑝𝑜𝑝𝑢𝑙𝑎𝑡𝑖𝑜𝑛 𝑚𝑒𝑎𝑛
𝜎 = 𝑠𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝑑𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛
12
Properties of a Normal Distribution
The shape and position of a normal distribution curve depend on two
parameters, the mean and the standard deviation. Each normally
distributed variable has its own normal distribution curve, which
depends on the values of the variable’s mean and standard deviation.
13
Properties of a Normal Distribution
✗ Suppose one normally distributed variable has 𝜇 = 0 and 𝜎 = 1, and
another normally distributed variable has 𝜇 = 0 and 𝜎 = 2. As shown
in Figure 2.2 the value of the standard deviation increases, the shape
of the curve spreads out
14
Properties of a Normal Distribution
If one normally distributed variable has 𝜇 = 0 and 𝜎 = 2 and another
normally distributed variable has 𝜇 = 2 and 𝜎 = 2, then the shapes of
the curve are the same, but the curve with 𝜇 = 2 moves 2 units to the
right just as shown in Figure 2.3.
15
Properties of a Normal Distribution
Summary of Properties
1. The distribution is bell shaped
2. The mean, median, and mode are equal and are located at the center
of the distribution.
3. The normal distribution curve is unimodal
4. 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.
16
Properties of a Normal Distribution
Summary of Properties
5. The Normal Distribution is continuous (no gaps or holes).
6. The Normal Curve is Asymptotic (curve never touches the x axis).
7. The total area under a normal distribution curve is equal to 1.00, or
100%.
17
Properties of a Normal Distribution
Summary of Properties
8. The area under the part
of a normal curve that
lies within 1 standard
deviation of the mean is
approximately 0.68, or
68%; within 2 standard
deviations, about 0.95,
or 95%; and within 3
standard deviations,
about 0.997, or 99.7%.
18
Properties of a Normal Distribution
Looking at Figure
2.4, it shows the
area in each region.
These values show
the empirical rule
for a Symmetric
distribution (normal
distribution is
symmetric).
19
Properties of a Normal Distribution
Comparing Normal Distributions to Skewed Distributions as shown
in Figure 2.5.
20
Properties of a Normal Distribution
The “tail” of the curve indicates the direction of skewness (right is positive, left is
negative).
21
Properties of a Normal Distribution
When the majority of the data values fall to the right of the mean, the distribution is
said to be a negatively or left-skewed distribution. The mean is to the left of the
median, and the mean and the median are to the left of the mode as shown on
Figure 2.5(b).
22
Properties of a Normal Distribution
When the majority of the data values fall to the left of the mean, a distribution is
said to be a positively or right-skewed distribution. The mean falls to the right of
the median, and both the mean and the median fall to the right of the mode. See
Figure 2.5(c).
23
Lesson 3
Standard Normal Distribution
24
Standard Normal Distribution
A normal distribution can be converted into a standard normal distribution
by obtaining the “z value.” A z value is the signed distance between a
selected value, designated 𝑥 and the mean 𝜇 divided by the standard
deviation.
25
Standard Normal Distribution
A normal distribution can be converted into a standard normal distribution
by obtaining the “z value.” A z value is the signed distance between a
selected value, designated 𝑥 and the mean 𝜇 divided by the standard
deviation.
26
27
28
Standard Normal Distribution
The normal distribution property allows to compute a probability problem
concerning 𝑥 into one concerning 𝑧.
Example:
solve practical application problems, such as finding the percentage of
adult women whose height is between 5 feet 4 inches and 5 feet 7 inches,
finding the probability that a new battery will last longer than 4 years.
29
Finding Areas under the Standard Normal Distribution Curve
For the solution of problems using the standard normal distribution, a twostep process is recommended.
Step 1 Draw the normal distribution curve and shade the area.
Step 2 Find the appropriate figure in the Procedure Table and follow the
directions
30
Finding Areas under the Standard Normal Distribution Curve
Example 3.1
Determine the area under the standard normal distribution curve between
𝑧 = 0 and 𝑧 = 1.85
Solution:
To identify the area we make use of the standard Normal Distribution Table
from 0 − 𝑧
31
32
Finding Areas under the Standard Normal Distribution Curve
Example 3.1
Determine the area under the standard normal distribution curve between 𝑧 = 0 and 𝑧 = 1.85
Solution:
To identify the area we make use of the standard Normal Distribution Table from 0 − 𝑧
33
Finding Areas under the Standard Normal Distribution Curve
The area of 𝑧 = 1.85 can be identified using the table buy looking up 1.85.
First we trace from z looking down until you reach 1.8
Since we are looking for 1.85 with the second decimal place 0.05, we trace from z going to the
right of the table until we reach 0.05.
Identifying their intersection will give you 0.4678
34
Finding Areas under the Standard Normal Distribution Curve
Draw the figure and represent the area.
P 0 < 𝑧 < 1.85 = 0.4678
Hence, the area is 0.4678 or 46.78%
35
Finding Areas under the Standard Normal Distribution Curve
Example 3.2
Determine the area under the standard normal distribution curve between 𝑧 = 0 and 𝑧 = −1.15.
Solution:
Since the normal curve is a symmetric curve finding the area on the positive side or the right side
is the same as finding the area on the left side or negative side.
36
Finding Areas under the Standard Normal Distribution Curve
𝑃 −1.15 < 𝑧 < 0 = 0.3749
Hence, the area is 0.3749 or
37.49%
37
38
Finding Areas under the Standard Normal Distribution Curve
Example 3.3
Find the area under the standard normal distribution curve to the right of 𝑧 = 1.15
Solution:
Draw the figure and represent the area as shown in the figure.
39
Finding Areas under the Standard Normal Distribution Curve
Solution:
The required area is at the right tail
of the normal curve. Since the
Standard Normal Distribution
Table gives the area between
𝑧 = 0 and 𝑧 = 1.15, first find the
area.
1.15
40
Finding Areas under the Standard Normal Distribution Curve
Solution:
𝑃 0 < 𝑧 < 1.15 = 0.3749
1.15
Then subtract 𝑃 0 < 𝑧 < 1.15 =
0.3749 from 0.5000, since half of
the are under the curve is to the
right of 𝑧 = 0.
41
Finding Areas under the Standard Normal Distribution Curve
Solution:
1.15
42
43
44
45
46
47
Learning Task
48
Learning Task
Find the area of the following Normal Distributions. Show
the bell curve representations of your answers.
1. Determine the area under the standard normal
distribution curve between 𝒛 = −𝟎. 𝟑𝟓 to 𝒛 = 𝟎. 𝟒𝟓 and
to the right of 2.57
2. Determine the area under the standard normal
distribution curve between 𝒛 = 𝟏. 𝟔𝟓 to 𝒛 = 𝟐. 𝟗𝟐 and
𝒛 = −𝟏. 𝟑𝟑 𝒕𝒐 𝒛 = −𝟏. 𝟖𝟖
49