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 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.

 Normal DistributionView animation

Normal Distribution

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.

These statements are illustrated in the normal distribution shown 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?

3. Consider the following three distributions.

Which of the three distributions would you consider the most normally distributed?

4. Consider the following three distributions.

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)?