CHAPTER 4
The Normal Distribution
INTRODUCTION:
This lesson will focus on the normal distribution. This is a continuous
distribution which is considered as the most important distribution because of its
application. The normal curve which represents the graph of the normal distribution
generally applies to the larger population. For this reason, normal distribution is used
in many problems in business, engineering, education, and many other fields.
OBJECTIVES:
: AtA.theUse
end of this lesson, you should be able to:
normal curve to calculate probabilities;
B. Determine areas under the normal curve;
C. Convert raw scores to standard scores; and
D. Apply normal distribution to different situation.
The Normal Distribution
A normal distribution is a continuous distribution which become the basis of
the entire theory of statistics. This is also known as Gaussian distribution which is
named after Carl Friedrich Gauss. This is a bell shaped curve is dependent on two
parameters – mean and standard deviation.
Properties of Normal Curve
1. The peak of the distribution lies at the center of the curve which represents the
mean, mode and median. This means that the mean, median and mode are equal.
2. The curve is continuous.
3. The curve is symmetric about the mean.
4. The tails of the curve are asymptotic with the horizontal axis.
5. The total area of the curve is equal to 1 or 100%.
6. Approximately 68% of the area lie between ±σ standard deviation from the
mean; approximately 95.5% of the area lie between ±2σ standard deviation from the
mean and approximately 99.7% of the area lie between ±3σ standard deviation from
the mean.
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, The Standard Normal Distribution
As discussed earlier, the normal distribution is characterized by two
parameters which are the mean and the standard deviation. This mean that every
unique pair of mean and standard deviation generates a different normal distribution.
However, it is impossible to have different normal distribution tables for all family of
normal curves. To result this issue, we can use a mechanism which we can covert
any normal distribution into a single distribution – the z distribution. Any normal
distribution can be converted into a standard normal distribution by solving for the z
value. The z value can be obtained by the following formula:
x−μ
z =
σ
where: z = z value
x = the value of any measurement
μ = the man of the distribution
σ = standard deviation of the distribution
Example 4.1 On a Midterm exam in Statistics, the mean is 55 and standard
deviation is 10. A) determine the score of a student who got a raw score of 60. B)
find the grade of a student with corresponding z score of -1.5.
Solution:
A) The z score of a student who got 55 is
x−μ
z =
σ
60 − 55 5
z = = = 0.5
10 10
B) The corresponding raw score of a student who has z score equals -1.5 is
x = σz + μ
x = 10(−1.5) + 55
x = −15 + 55
x = 40
In standard normal distribution the mean μ is always equal to 0 and standard
deviation σ is always equal to 1. As stated in the properties of the normal curve, the
total area is equal to 1 or 100%. This area also corresponds to the probability of the
distribution. Using the table of the Areas Under the Normal curve, we can compute
probability problems.
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The Normal Distribution
INTRODUCTION:
This lesson will focus on the normal distribution. This is a continuous
distribution which is considered as the most important distribution because of its
application. The normal curve which represents the graph of the normal distribution
generally applies to the larger population. For this reason, normal distribution is used
in many problems in business, engineering, education, and many other fields.
OBJECTIVES:
: AtA.theUse
end of this lesson, you should be able to:
normal curve to calculate probabilities;
B. Determine areas under the normal curve;
C. Convert raw scores to standard scores; and
D. Apply normal distribution to different situation.
The Normal Distribution
A normal distribution is a continuous distribution which become the basis of
the entire theory of statistics. This is also known as Gaussian distribution which is
named after Carl Friedrich Gauss. This is a bell shaped curve is dependent on two
parameters – mean and standard deviation.
Properties of Normal Curve
1. The peak of the distribution lies at the center of the curve which represents the
mean, mode and median. This means that the mean, median and mode are equal.
2. The curve is continuous.
3. The curve is symmetric about the mean.
4. The tails of the curve are asymptotic with the horizontal axis.
5. The total area of the curve is equal to 1 or 100%.
6. Approximately 68% of the area lie between ±σ standard deviation from the
mean; approximately 95.5% of the area lie between ±2σ standard deviation from the
mean and approximately 99.7% of the area lie between ±3σ standard deviation from
the mean.
60
, The Standard Normal Distribution
As discussed earlier, the normal distribution is characterized by two
parameters which are the mean and the standard deviation. This mean that every
unique pair of mean and standard deviation generates a different normal distribution.
However, it is impossible to have different normal distribution tables for all family of
normal curves. To result this issue, we can use a mechanism which we can covert
any normal distribution into a single distribution – the z distribution. Any normal
distribution can be converted into a standard normal distribution by solving for the z
value. The z value can be obtained by the following formula:
x−μ
z =
σ
where: z = z value
x = the value of any measurement
μ = the man of the distribution
σ = standard deviation of the distribution
Example 4.1 On a Midterm exam in Statistics, the mean is 55 and standard
deviation is 10. A) determine the score of a student who got a raw score of 60. B)
find the grade of a student with corresponding z score of -1.5.
Solution:
A) The z score of a student who got 55 is
x−μ
z =
σ
60 − 55 5
z = = = 0.5
10 10
B) The corresponding raw score of a student who has z score equals -1.5 is
x = σz + μ
x = 10(−1.5) + 55
x = −15 + 55
x = 40
In standard normal distribution the mean μ is always equal to 0 and standard
deviation σ is always equal to 1. As stated in the properties of the normal curve, the
total area is equal to 1 or 100%. This area also corresponds to the probability of the
distribution. Using the table of the Areas Under the Normal curve, we can compute
probability problems.
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