A QUESTION OF TERMINOLOGY

One very important idea in statistics is the idea of a distribution. For a discrete variable, a distribution is a lot like a histogram-it tells how often a given value occurs as a probability between 0 and 1. For instance, a uniform distribution says that all values are equally represented. An example of a uniform distribution would occur in a business where customers pay by credit card and the same number of customers pays with American Express, Visa, and MasterCard.

The normal distribution, which plays a very special role in statistics, is an example of a distribution for a continuous variable. The following figure shows the normal (sometimes called Gaussian or bell-shaped) distribution with a mean of 0 and a standard deviation of 1. The way to read this curve is to look at areas between two points. For a value that follows the normal distribution, the probability that the value falls between two values-for example, between 0 and 1-is the area under the curve. For the values of 0 and 1, the probability is 34.1 percent; this means that 34.1 percent of the time a variable that follows a normal distribution will take on a value within one standard deviation above the mean. Because the curve is symmetric, there is an additional 34.1% probability of being one standard deviation below the mean, and hence 68.2% probability of being within one standard deviation above the mean.

The probability density function for the normal distribution looks like the familiar bell-shaped curve.

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A QUESTION OF TERMINOLOGY (continued)

The previous paragraph showed a picture of a bell-shaped curve and called it the normal distribution. Actually, the correct terminology is density function (or probability density function). Although this terminology derives from advanced mathematical probability theory, it makes sense. The density function gives a flavor for how dense a variable is. We use a density function by measuring the area under the curve between two points, rather than by reading the individual values themselves. In the case of the normal distribution, the values are densest around the 0 and less dense as we move away.

The following figure shows the function that is properly called the normal distribution. This form, ranging from 0 to 1, is also called a cumulative distribution function. Mathematically, the distribution function for a value X is defined as the probability that the variable takes on a value less than or equal to X. Because of the less than or equal to characteristic, this function always starts near 0, climbs upward, and ends up close to 1. In general, the density function provides more visual clues to the human about what is going on with a distribution. Because density functions provide more information, they are often referred to as distributions, although that is technically incorrect.

The (cumulative) distribution function for the normal distribution has an S-shape and is antisymmetric around the Y-axis.

From Standardized Values to Probabilities

Assuming that the standardized value follows the normal distribution makes it possible to calculate the probability that the value would have occurred by chance. Actually, the approach is to calculate the probability that something further from the mean would have occurred-the p-value. The reason the exact value is not worth asking is because any given z-value has an arbitrarily

small probability. Probabilities are defined on ranges of z-values as the area under the normal curve between two points.

Calculating something further from the mean might mean either of two things:

The probability of being more than z standard deviations from the mean.

The probability of being z standard deviations greater than the mean (or alternatively z standard deviations less than the mean).

The first is called a two-tailed distribution and the second is called a one-tailed distribution. The terminology is clear in Figure 5.4, because the tails of the distributions are being measured. The two-tailed probability is always twice as large as the one-tailed probability for z-values. Hence, the two-tailed p-value is more pessimistic than the one-tailed one; that is, the two-tailed is more likely to assume that the null hypothesis is true. If the one-tailed says the probability of the null hypothesis is 10 percent, then the two-tailed says it is 20 percent. As a default, it is better to use the two-tailed probability for calculations to be on the safe side.

The two-tailed p-value can be calculated conveniently in Excel, because there is a function called NORMSDIST, which calculates the cumulative normal distribution. Using this function, the two-tailed p-value is 2 * NORMS-DIST(-ABS(z)). For a value of 2, the result is 4.6 percent. This means that there is a 4.6 percent chance of observing a value more than two standard deviations from the average-plus or minus two standard deviations from the average. Or, put another way, there is a 95.4 percent confidence that a value falling outside two standard deviations is due to something besides chance. For a precise 95 percent confidence, a bound of 1.96 can be used instead of 2. For 99 percent confidence, the limit is 2.58. The following shows the limits on the z-value for some common confidence levels:

90% confidence - z-value > 1.64

95% confidence - z-value > 1.96

99% confidence - z-value > 2.58

99.5% confidence - z-value > 2.81

99.9% confidence - z-value > 3.29

99.99% confidence - z-value > 3.89

The confidence has the property that it is close to 100 percent when the value is unlikely to be due to chance and close to 0 when it is. The signed confidence adds information about whether the value is too low or too high. When the observed value is less than the average, the signed confidence is negative.