2 ND S E M E S T E R - 3RD QUART ER
STATISTICS &
PROBABILITY
LESSON 5 – THE NORMAL DISTRIBUTION AND
ITS PROPERTIES
Learning Competency with code
The learner illustrates a normal
random
variable
and
its
characteristics (M11/12SP-IIIc-1)
NORMAL DISTRIBUTION
A normal distribution is a continuous,
symmetric, bell-shaped distribution of
a variable.
PROPERTIES OF THE NORMAL DISTRIBUTION
1. A normal distribution curve is bellshaped.
2. The mean, median, and mode are
equal and are located at the center of
the distribution.
PROPERTIES OF THE NORMAL DISTRIBUTION
3. A normal distribution curve is unimodal
(i.e., it has only one mode).
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.
PROPERTIES OF THE NORMAL DISTRIBUTION
5. The curve is continuous ; that is, there are no
gaps or holes. For each value of X, there is a
corresponding value of Y.
6.
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.
PROPERTIES OF THE NORMAL DISTRIBUTION
7. The total area under a normal distribution
curve is equal to 1.00, or 100%.
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%.
IMPORTANT NOTES ABOUT THE NORMAL DISTRIBUTION
• Normal distribution depends on two factors :
the mean
μ and
the standard deviation
σ.
• The mean determines the location of the center
of the bell-shaped curve. Thus, a change in the
value
of
the
mean
shifts
the
graph
normal curve to the right or to the left.
of
the
NORMAL AND SKEWED DISTRIBUTIONS
When the data values are evenly distributed
about the mean, a distribution is said to be a
symmetric distribution .
NORMAL AND SKEWED DISTRIBUTIONS
When the majority o f the data values fall to
the right of the mean, the distribution is said
to be a negatively or left -skewed distribution .
NORMAL AND SKEWED DISTRIBUTIONS
When the majority o f the data values fall to
the left of the mean, a distribution is said to
be a positively or right -skewed distribution .
NORMAL AND SKEWED DISTRIBUTIONS
The “tail” of the curve
indicates the direction of
skewness (right is positive,
left is negative).
IMPORTANT NOTES ABOUT THE NORMAL DISTRIBUTION
(a) The mean represents the balancing point of the
graph of the distribution.
(b) The mode represents the “high point” of the
probability density function.
(c) The median represents the point where 50% of
the area under the distribution is to the left and
50% of the area under the distribution is to the
right.
IMPORTANT NOTES ABOUT THE NORMAL DISTRIBUTION
▪ For symmetric distributions with a single peak,
such as the normal curve, Mean = Median = Mode.
▪ The standard deviation determines the shape of
the graphs (particularly, the height and width of
the curve). When the standard deviation is large,
the normal curve is short and wide, while a small
value for the standard deviation yields a skinnier
and taller graph.
The curve above on the left is shorter and wider
than the curve on the right, because the curve on
the left has a bigger standard deviation
In figure 2, the two normal curves are identical in
form but are centered at different positions along
the horizontal axis.
In figure 3, the two normal curves centered at
exactly the same position on the horizontal axis,
but the curve with larger standard deviation is
lower and spreads out farther.
In
figure
4,
two
curves
centered
at
different
positions on the horizontal axis and their shapes
reflect
the
deviation.
two
different
values
of
standard
HISTORICAL NOTES ON THE NORMAL CURVE
▪ The French-English mathematician Abraham de Moivre first
described the use of the normal distribution in 1733 when he
was developing the mathematics of chance, particularly for
approximating the binomial distribution. Marquis de Laplace
used the normal distribution as a model of measuring errors.
Adolphe Quetelet and Carl Friedrich Gauss popularized its use.
Quetelet used the normal curve to discuss “the average man”
with the idea of using the curve as some sort of an ideal
histogram
while
Gauss
used
astronomical data in 1809.
the
normal
curve
to
analyze
HISTORICAL NOTES ON THE NORMAL CURVE
▪ In some disciplines, such as engineering, the normal
distribution is also called the Gaussian distribution (in
honor of Gauss who did not first propose it!). The first
unambiguous use of the term “normal” distribution is
attributed to Sir Francis Galton in 1889 although Karl
Pearson's consistent and exclusive use of this term in
his
prolific
writings
led
to
its
eventual
throughout the statistical community.
adoption
WHY IS THE NORMAL DISTRIBUTION CONSIDERED
THE MOST IMPORTANT CURVE IN STATISTICS?
(a) Many random variables are either normally
distributed
normally
or,
at
least,
distributed .
approximately
Heights,
weights,
examination scores, the log of the length of
life
of
random
some
equipment
variables
that
normally distributed .
are
among
a
few
are
approximately
WHY IS THE NORMAL DISTRIBUTION CONSIDERED
THE MOST IMPORTANT CURVE IN STATISTICS?
(b) It is easy for mathematical statisticians to
work with the normal curve . A number of
hypothesis tests and the regression model are
based on the assumption that the underlying
data have normal distributions .
EXAMPLE
Assume that 68.3 % of grade 11 students
have score between 34 and 48 out of 50-
point
quiz
and
the
data
are
normally
distributed.
a. Find the mean and standard deviation
b. Construct
the
normal
normal distribution
curve
of
the
ANSWER
EXAMPLE
ANSWER
TRY THIS!
Construct a normal curve .
TRY THIS!
a. Find the mean and standard deviation.
b. Construct
the normal curve of the
normal distribution.