Continuous Probability Distributions:Normal Distribution

Any discussion about continuous random variables leads naturally
into a discussion of a widely used probability distribution—the normal distribution.
Many variables, such as exam scores, the height and weight of a
population, and customer satisfaction surveys, have distributions that
closely resemble a normal distribution.

The following animation outlines the important features of a normal
distribution. The information that follows will describe these features
in greater detail.

If you viewed the animation, you
saw that a normal distribution takes the form of a bell-shaped curve,
which is sometimes referred to as a "bell curve." This distribution is
unimodal (having a single modal peak) and symmetrical, which means that
its mean, median, and mode all have the same value and that a line
drawn through the curve at the mean, median, and mode will divide the
curve into two symmetrical halves. The value of the mean is often
called the expected value of the distribution, and it is represented by
the Greek letter m. An example of a normal distribution is shown below.

If a random variable, x, is normally distributed, you can use
the distribution's expected value and standard deviation to predict the
probabilities associated with a range of values for x. Using expected value and standard deviation, you can predict that

The probability that the variable, x, will fall within one standard deviation (s) from the mean (m) is approximately .6826. The expression of the probability for one standard deviation appears below.

The probability that the variable, x, will fall within two
standard deviations from the mean is approximately .9544. The expression
of the probability for two standard deviations appears below.

The probability that the variable, x, will fall within three
standard deviations from the mean is approximately .9974. The expression
of the probability for three standard deviations appears below.

To better understand the importance of normal distribution, recall
the plastics manufacturer discussed in "Continuous Random Variables."
This manufacturer needs to have a sufficient amount of raw material on
hand to compensate for waste generated during the weekly production
cycle. The amount of waste the manufacturer generates is shown in the
probability distribution below.

This distribution approximates a normal distribution; therefore, you
can use its expected value and standard deviation to predict the
probabilities associated with the amount of waste generated. There is a
probability of about .68 that a week's production will result in an
amount of waste that is within one standard deviation of the mean level
of waste. (Note: This claim can be made only because the distribution
of waste produced is a very good approximation of a normal
distribution. If the distribution of waste produced were not normal,
such a claim could not be made.) The expression of the probability of
values of x falling within one standard deviation is

This expression can be stated as "the probability that a continuous random variable x will fall within the interval of the mean, plus or minus one standard deviation, is .68."

In this example,

x = pounds of waste m = 1,100 s = 300

Substitute this information into the formula, as shown below.

The interval between 800 and 1,400 pounds of waste is one standard
deviation above and below the mean. The shaded area under the curve
represents the probability density associated with a range of waste of
one standard deviation above and below the mean of 1,100 pounds.

While this information is useful for the operations executive
running this factory, he needs to know more about the full range of
this variable, since a plant shutdown due to insufficient raw materials
is prohibitively expensive. In order to be fully prepared for the
probability of a wider range of shortages, the operations executive
decides to calculate the range of values for x within two standard deviations.

To determine the values of x that fall within two standard deviations of the mean, use the following equation and solve.

There is roughly a .95 probability that a production week will
produce between 500 and 1,700 pounds of waste, which is the mean plus
or minus two standard deviations. The operations executive could use
this information to make an informed decision regarding the amount of
raw materials to have on hand for each production shift.

Normal Distributions with Different Standard Deviations

Although normally distributed values will tend to have the same
general bell-shape around the expected value, they will not all have
the same "spread," or standard deviation. The exercise below shows a
number of normal distributions that have the same expected value.
However, as you will see, these distributions do not look the same due
to their different standard deviations.

As you saw in the above exercise, if a normal distribution's
standard deviation is relatively low, the shape of the distribution
will be more peaked and more concentrated around its expected value. A
normal distribution with a large standard deviation is flatter, or less
peaked, with values that are less concentrated around the expected
value. Although it is still possible to predict that .68 of the values
will fall within one standard deviation of the expected value, the
range of those values will be broader in the distribution with a larger
standard deviation.

1. What is the approximate probability that the value of a random
variable will fall within the interval of the mean, plus or minus two
standard deviations, in a normally distributed random variable?

2. What is the approximate probability that a value of a normally
distributed random variable will fall beyond the second standard
deviation on the high side of the distribution?

If these three distributions all have the same mean, would you
expect there to be a different proportion of the distributions centered
around the expected value? Why or why not?

5. The following continuous graphic illustrates the probability
distribution associated with the session-ending price of an Internet
retailer's stock price. Consider the lightly shaded area under the
curve. What is the probability that the price of the stock will be
between 70 and 85 dollars at the end of the trading session?

6. Consider the darker shaded sections under the curve. What is the
probability that the ending stock price will be less than 25 or greater
than 85? In other words, what is P(x < 25) or P(x > 85)?