Distribution........................pptx

Learning Competency:
The learner illustrates a random
variable and its characteristics.

Normal Distribution
It is the cornerstone of the
application of statistical inference in
analysis of data because the distributions
of several important sample statistics tend
towards a normal distribution as the
sample size increases.

Normal Random Variable
The normal random variable is a
continuous random variable that follows
the normal distribution with mean and
standard deviation .

Normal Equation
It is the probability density function
of the normal distribution. The value of the
random variable Y is:
where x is the normal random variable.

Normal Curve
The graph of a normal distribution is bell-shaped.
It depends on two factors: the mean and
standard deviation.
The mean determines the center of the graph
and the standard deviation determines its height
and width.
Normal distributions with higher standard
deviation create curves with smaller height and
bigger width.

Normal Distribution
It is a representative of continuous
data (such as measurements) that have
some variation but no bias above or below
the mean.

Characteristics of the Normal Probability Distribution

The curve has a single peak.

It is bell-shaped or shape like a Mexican sombrero.

The mean (average) lies at the center of the
distribution. The distribution is symmetrical about the
mean.

The two tails extend indefinitely in both directions
coming closer and closer to the horizontal axis but
never quite touching it.

It includes 100% of the data, so the area under the
curve is 1.

Empirical Rule
It is a rule that use standard
deviation so that we can measure with
great precision percentage of items that
fall within specific ranges under a normal
distribution.

Empirical Rule
50% of all data points are above the mean and
50% are below
68.27% of all data points are within 1 standard
deviation of the mean
95.45% of all data points are within 2 standard
deviations of the mean
Approximately 99.73% of all data points are
within 3 standard deviations of the mean

AREAS UNDER A
NORMAL CURVE
AND THE Z SCORE

Learning Competencies:
At the end of the lesson, learners are
expected to:
1. identify regions under the normal curve
2. convert a normal random variable to a
standard normal variable and vice versa
3. find the area of z under the normal
curve

Normal Distribution
It is the most common distribution that
applies to many real life data.
Examples:
1. heights of people
2. weights of people
3. exam scores
4. blood pressure

Standard Score or Z-Score
•A z-score also called a standard score
gives an idea of how far from the mean a
data point is.
•It’s measure of how many standard
deviations below or above the population
mean a raw score is.

Standard Score or Z-Score
•It allows us to calculate the probability of
a score occurring within our normal
distribution.
•It enables us to compare two scores that
are from different normal distributions.

Converting to a Standard Normal Variable Z and Vice
Versa
To convert a value of a normal random variable x to its
standard normal variable Z (or Z score value), we use the
formula:
where: Z = the number of standard deviations from x to the
mean of the distribution
x = the value of the random variable
= standard deviation of the distribution
= the mean of the distribution of the random variable

Converting to a Standard Normal Variable Z and Vice
Versa
To convert a standard normal variable Z (or Z score)
into its normal random variable x value, we use the
formula:
where: Z = the number of standard deviations from x to the
mean of the distribution
x = the value of the random variable
= standard deviation of the distribution
= the mean of the distribution of the random variable

Example 1
Suppose you have a set of test scores
that are normally distributed with mean
equal to 80 and standard deviation equal
to 5. If you got 75, what is your z-score?

Example 2
Math exam scores are normally
distributed with a mean of 80 and a
standard deviation of 5. If you got a z-score
of -2, what is your exam score?

Example 3
You would like to compare your
performance during the summative tests
in HOPE 2 and StatProb subjects. You are
enrolled in a class with 44 students. With
the given table below, where did you
perform better?
Subjects X
HOPE 2 38 37.5 1.5
StatProb 36 32.5 6

ACTIVITY #11
Solve the unknown.

Finding the Area Under a Normal Curve Given a Z Value
A specific proportion of the area of the region under the curve
can be calculated manually using the formula
where: Y = represents the height of the curve at a particular value
of
X = represents any score in the distribution
= represents the standard deviation of the population
= represents the population mean
= 3.1416
= 2.7183

Steps in Finding the Area (or Area Percentage) Under a
Normal Curve Given a Z Value
1. Express the given z-value into a three-digit number.
2. Using the z-Table, find the first 2 digits on the first
column.
3. Find the third digit on the first row on the right.
4. Read the area for probability at the intersection of
the row (first 2 digit number) and column (third digit
number). The value observed at the intersection
indicates the area of the given z-value.

Example 1
Find the area that corresponds to a z-
score of 1.5. (Look for 1.5 on the first
column and 0 on the first row)
A = 0.93319

Example 2
score of 1.09. (Look for 1 on the first
column and 0.09 on the first row)
A = 0.86214

Example 3
score of 0.27. (Look for 0.2 on the first
column and 0.07 on the first row)
A = 0.60642

Example 4
Find the area of the region
corresponding to the z-score = -1. (Look for
-1 on the first column and 0 on the first row)
A = 0.15866

Example 5
Find the area of the region corresponding
to the z-score = -2.36. (Look for -2.3 on the
first column and 0.06 on the first row)
A = 0.00914

ACTIVITY #12
Find the area under the normal curve given
the z-value.
1. 26. -2.24
2. -3 7. 0.4
3. 1.2 8. -0.5
4. -2.1 9. 0.67
5. 1.52 10. -0.48

SHADED REGION
UNDER THE
NORMAL CURVE

Learning Competencies:
At the end of the lesson, learners are expected
to:
1. find the shaded region under the normal
curve given specific conditions using the
standard normal table
2. define percentile
3. compute probabilities and percentiles using
the standard normal table

Standard Normal Distribution
It is a normal distribution with a mean
of zero and standard deviation of one and
has a total area under its normal curve of
one.

Probability Notations
P(z<a) P(z>a) P(a<z<b)
The probability
that the value
is…
“less than z”
“to the left of z”
“below z”
“lower than z”
“under z”
The probability
that the value
is…
“greater than z”
“to the right of
z”
“above z”
“more than z”
The probability
that the z value
is between a
and b.

Steps in Finding the Areas Under the Normal Curve
1.Compute for the z-score for
observation x.
2.Draw the Normal Distribution Curve
and shade the area.
3.Find the corresponding area of the z-
score.

Example 1
Vehicles’ speed at McArthur Hi-way
have a normal distribution with a mean
of 65 mph and a standard deviation of 5
mph. What is the probability that a
randomly selected car is going 73 mph
or less?

Example 2
The pulse rates of adult females have a
normal curve distribution with a mean of
75 beats per minute (bpm) and a standard
deviation of 8 bpm. Find the probability
that a randomly selected female has a
pulse rate greater than 85 bpm.

Example 3
The mean for IQ scores is 100 and
the standard deviation is 15. What
proportion of IQ scores falls between
100 and 130?

Example 4
Find the area of the region between
the and under the standard normal
curve.

Example 5
Compute for the probability that the
Z may take the value/s.
a. Greater than 2
b.Less than 2
c. Between 1 and 2

ACTIVITY #13
Find the area under the normal curve in each of the
following cases.
1. Less than z = 2.35 6. Greater than z = 0.89
2. Less than z = -1.31 7. Between z = 0 and z = 1.63
3. Less than z = 0.35 8. Between z = 0 and z = -1.78
4. Greater than z = 1.85 9. Between z = 1.56 and 2.51
5. Greater than z = -0.95 10. Between z = -2.76 and z =
-1.25

PERCENTILES
UNDER THE
NORMAL CURVE

Percentile
It is a measure used in statistics
indicating the value below which a given
percentage of observation in a group of
observation falls.
It is a measure of relative standing as it
measures the relationship of a
measurement of the rest of the data.

Steps in Finding the Percentiles Under the Normal
Curve
1.Draw the Normal Distribution Curve and shade the area.
2.Express percentile as probability.
3.Refer to the table of areas under the normal curve. Locate its
the corresponding area.
4.Locate the nearest value.
5.Find the x-score that lies in the percentile, using the formula .
6.Locate the z-value and draw a line through under the curve.
7.Describe the shaded region.

Example 1
Find the 95th
percentile of the normal
curve.

Example 2
What is the 33rd
percentile for the
intelligence Quotient scores when the
mean and standard deviation

Example 3
Consider the normal distribution of IQs
with a mean of 100 and a standard
deviation of 25. What percentage of IQs are:
a. greater than 95?
b.less than 120?
c. between 90 and 110?

Example 4
In an English test, the mean is 60 and the
standard deviation is 6. Assuming the scores
are normally distributed, what percent of the
score is:
a. greater than 65?
b. less than 70?
c. between 50 and 65?

Distribution........................pptx

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