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Advanced Algebra II
Normal Distribution
In probability theory, the normal (or Gaussian)
distribution is a continuous probability distribution
that has a bell-shaped probability density function,
known as the Gaussian function or informally the bell
curve:[nb 1]
Note that a normally-distributed variable has a
symmetric distribution about its mean.
Dark blue is less than one standard deviation from
the mean. For the normal distribution, this accounts
for about 68% of the set, while two standard
deviations from the mean (medium and dark blue)
account for about 95%, and three standard deviations
(light, medium, and dark blue) account for about
99.7%.
Figure 1. A simple bimodal distribution, in this case a
mixture of two normal distributions with the same
variance but different means. The figure shows the
probability density function (p.d.f.), which is an
average of the bell-shaped p.d.f.s of the two normal
distributions
The bimodal distribution of sizes of weaver ant
workers shown in Figure 2 arises due to
existence of two distinct classes of workers,
namely major workers and minor workers.
Bimodal distributions are a commonly
used example of how summary statistics
such as the mean, median, and standard
deviation can be deceptive when used on
an arbitrary distribution. For example, in
the distribution in Figure 1, the mean and
median would be about zero, even though
zero is not a typical value. The standard
deviation is also larger than deviation of
each normal distribution.
μ is the mean of the population;
σ is the standard deviation of the population.
In statistics, a standard score indicates by how
many standard deviations an observation or datum
is above or below the mean. It is a dimensionless
quantity derived by subtracting the population
mean from an individual raw score and then
dividing the difference by the population standard
deviation.
Standard scores are also called z-values, z-scores,
normal scores, and standardized variables; the use
of "Z" is because the normal distribution is also
known as the "Z distribution".
Standard Normal Distributions and Z Scores
A normal distribution that is standardized (so that it has
a mean of 0 and a SD of 1) is called the standard normal
distribution, or the normal distribution of z-scores. If
we know the mean m ("mu"), and standard deviation s
("sigma") of a set of scores which are normally
distributed, we can standardize each "raw" score, x, by
converting it into a z score by using the following
formula on each individual score:
If a Z-Score….
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Has a value of 0, it is equal to the group mean.
Is positive, it is above the group mean.
Is negative, it is below the group mean.
Is equal to +1, it is 1 Standard Deviation above the mean.
Is equal to +2, it is 2 Standard Deviations above the mean.
Is equal to -1, it is 1 Standard Deviation below the mean.
Is equal to -2, it is 2 Standard Deviations below the mean.
Q1: Suppose that SAT scores among U.S.
college students are normally distributed with
a mean of 500 and a standard deviation of
100. What is the probability that a randomly
selected individual from this population has
an SAT score at or below 600?
Using the formula to calculate the z value, we find z = (x - m)/s =
(600 - 500)/100 = +1.00. Recall that the z-score is the number of
standard deviations that the score of interest differs from the
mean. A score of 600 is one standard deviation above the
mean, so it has a z value of 1.00.
Using either a z table or the p-z converter, we find that the
probability that a randomly selected z score in a normal
distribution will exceed z = 1.00 is .159, the right-tailed p value,
or about 16%.
The probability that z will exceed 1.00 for a randomly selected
score is equivalent to the probability that a randomly selected
individual from this population will have an SAT score over 600,
about 16%. The probability that a randomly selected individual
from this population will have an SAT score below 600 is 100% 16% = 84%
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