The Normal Distribution

In this lesson, we describe the normal distribution, a continuous probability distribution that is widely used in statistics to compute probabilities associated with various naturally-occurring events.

The Normal Equation

The normal distribution is defined by the following equation:

The random variable X in the normal equation is called the normal random variable. It can range in value from minus to plus infinity. The normal equation is the probability density function for the normal distribution.

The Normal Curve

When X and Y values from the normal equation are graphed on an X-Y scatter chart, all normal distributions look like a symmetric, bell-shaped curve, as shown below.

The location and shape of the curve for the normal distribution depends on two factors - the mean and the standard deviation.

• The center of the curve is located on the X-axis at the mean of the distribution.
• The standard deviation determines the shape (height and width) of the curve. When the standard deviation is small, the curve is tall and narrow; and when the standard deviation is big, the curve is short and wide (see above)

Probability and the Normal Curve

The normal distribution is a continuous probability distribution. This has several implications for probability.

• The total area under the normal curve is equal to 1. And, the probability that a normal random variable X is less than positive infinity is equal to 1; that is, P(X < ∞) = 1.
• The probability that a normal random variable X equals any particular value is 0. Here's why. The normal random variable X can take any value between minus and plus infinity - an infinite number of values. The probability of selecting any single value from an infinitely large set of values is always zero.
• Let a equal any real number. The probability that X is greater than a equals the area under the normal curve bounded by a and plus infinity (as indicated by the red shaded area in the figure below).

• The probability that X is less than a equals the area under the normal curve bounded by a and minus infinity (as indicated by the red shaded area in the figure below).

• The normal curve changes from curving up to curving down at "points of inflection" exactly one standard deviation above and below the mean (μ).

• Every normal curve (regardless of its mean or standard deviation) conforms to the empirical rule (aka, the 68-95-99.7 rule), which states:
  • About 68% of the area under the curve falls within 1 standard deviation of the mean.
  • About 95% of the area under the curve falls within 2 standard deviations of the mean.
  • About 99.7% of the area under the curve falls within 3 standard deviations of the mean.

  For an example of how to use the empirical rule, see Problem 4.

Why the Normal Curve is Useful

The values observed for some natural phenomena (height, weight, IQ, blood pressure, etc.) follow an approximate normal distribution. For those phenomena, the normal distribution provides a useful frame of reference for computing probability.

To illustrate how the normal distribution provides a useful frame of reference for probability, consider this example. Suppose we weighed all of the brown mushrooms harvested on a farm in a single planting season. We might find that the mean weight of a mushroom was 60 grams, and the standard deviation was 4 grams. We could plot mushroom weight on a histogram, like so:

Notice that this bell-shaped histogram is not too different from the bell-shaped curve of a normal distribution. If we display the histogram above a normal curve having a mean of 60 and standard deviation of 4, it is easy to see the resemblance.

Given the similarities, you might suspect that the normal distribution could be useful in predicting probabilities involving mushroom weight. And you would be right! (For an example that shows how to predict probabilities associated with mushrooom weight, see Problem 3 below.)

Bottom line: Suppose X is a random variable that is distributed roughly normally in the population. If you know the mean and standard deviation of X, you can compute a cumulative probability for X. Specifically, you can compute P(X < x) and P(X > x).

How to Find Probability

To find a cumulative probability for a normal random variable, world-class statisticians can use the normal equation described earlier (plus a little calculus). However, the rest of us use one of the following:

• A graphing calculator.
• An online probability calculator, such as Stat Trek's Normal Distribution Calculator.
• A normal distribution probability table (found in the appendix of most introductory statistics texts).

In the examples below, we use Stat Trek's Normal Distribution Calculator to calculate probability. In the next lesson, we use normal distribution probability tables.

Test Your Understanding

Problem 1

An average light bulb manufactured by the Acme Corporation lasts 300 days with a standard deviation of 50 days. Assuming that bulb life is normally distributed, what is the probability that an Acme light bulb will last at most 365 days?

Solution: Given a mean score of 300 days and a standard deviation of 50 days, we want to find the cumulative probability that bulb life is less than or equal to 365 days. Thus, we know the following:

• The value of the normal random variable is 365 days.
• The mean is equal to 300 days.
• The standard deviation is equal to 50 days.

We enter these values into the Normal Distribution Calculator and compute the cumulative probability.

The answer is: P( X < 365) = 0.90319. Hence, there is about a 90% chance that a light bulb will burn out within 365 days.

Problem 2

Suppose scores on an IQ test are normally distributed. If the test has a mean of 100 and a standard deviation of 10, what is the probability that a person who takes the test will score between 90 and 110?

Solution: Here, we want to know the probability that the test score falls between 90 and 110. The "trick" to solving this problem is to realize the following:

P( 90 < X < 110 ) = P( X < 110 ) - P( X < 90 )

We use the Normal Distribution Calculator to compute both probabilities on the right side of the above equation.

• To compute P( X < 110 ), we enter the following inputs into the calculator: The raw score value of the normal random variable is 110, the mean is 100, and the standard deviation is 10. We find that P( X < 110 ) is about 0.84.

• To compute P( X < 90 ), we enter the following inputs into the calculator: The raw score value of the normal random variable is 90, the mean is 100, and the standard deviation is 10. We find that P( X < 90 ) is about 0.16.

We use these findings to compute our final answer as follows:

P( 90 < X < 110 ) = P( X < 110 ) - P( X < 90 )
P( 90 < X < 110 ) = 0.84 - 0.16
P( 90 < X < 110 ) = 0.68

Thus, about 68% of the test scores will fall between 90 and 110 (within one standard deviation of the mean), as predicted by the 68-95-99.7 rule.

Problem 3

Suppose a farmer collects a random sample of fully-developed mushrooms. He finds that the mean weight of a mushroom in his sample is 60 grams, and the standard deviation is 4 grams. Suppose further that his buyer will only purchase mushrooms bigger than 57 grams.

What is the probability that a mushroom harvested by this farmer will be smaller than 57 grams?

Solution: Given a mean weight of 60 grams and a standard deviation of 4 grams, we want to find the cumulative probability that a mushroom will weigh less than or equal to 57 grams. Thus, we know the following:

• The raw mushroom weight of interest is 57 grams.
• The mean mushroom weight is 60 grams.
• The standard deviation is 4 grams.

We enter these values into the Normal Distribution Calculator and compute the cumulative probability.

The answer is: P( X ≤ 57) = 0.22663. Hence, there is about a 23% chance that a mushroom in this farmer's crop will weigh less than 57 grams.

Problem 4

Worldwide, the average height of an adult woman is about 63 inches with a standard deviation of 3 inches.

1. What percent of women are between 60 inches and 66 inches tall?
2. What percent of women are shorter than 60 inches tall?

Solution: The solution to this problem is easy if you know the empirical rule (aka, the 69-95-99.7 rule). The empirical rule states that about 68% of all outcomes fall within 1 standard deviation of the mean; about 95%, within 2 standard deviations; and about 99.7%, within 3 standard deviations.

• Solution to Question 1. Since 60 inches is 1 standard deviation below the mean of 63 inches and 66 inches is 1 standard deviation above the mean, we know by the empirical rule that about 68% of woman have heights between 60 inches and 66 inches.
• Solution to Question 2. According to the empirical rule 68% of all outcomes fall between 1 standard deviation of the mean. This means about 32% of outcomes fall above and below 1 standard deviation. Because the normal distribution is symmetrical, about half of the 32% fall below 1 standard deviation of the mean. Therefore, we know that about 16% of all women are shorter than 60 inches.