16 Mathematics of Normal Distributions 16.1 Approximately Normal Distributions of Data 16.2 Normal Curves and Normal Distributions 16.3 Standardizing Normal Data 16.4 The 68-95-99.7 Rule 16.5 Normal Curves as Models of Real-Life Data Sets 16.6 Distribution of Random Events 16.7 Statistical Inference

The 68-95-99.7 Rule When we look at a typical bell-shaped distribution, we can see that most of the data are concentrated near the center. As we move away from the center the heights of the columns drop rather fast, and if we move far enough away from the center, there are essentially no data to be found.

The 68-95-99.7 Rule These are all rather informal observations, but there is a more formal way to phrase this, called the 68-95-99.7 rule. This useful rule is obtained by using one, two, and three standard deviations above and below the mean as special landmarks. In effect, the 68-95-99.7 rule is three separate rules in one.

THE 68-95-99.7 RULE 1. In every normal distribution, about 68% of all the data values fall within one standard deviation above and below the mean. In other words, 68% of all the data have standardized values between z = –1 and z = 1. The remaining 32% of the data are divided equally between data with standardized values z ≤ –1 and data with standardized values z ≥ 1.

THE 68-95-99.7 RULE

THE 68-95-99.7 RULE 2. In every normal distribution, about 95% of all the data values fall within two standard deviations above and below the mean. In other words, 95% of all the data have standardized values between z = –2 and z = 2. The remaining 5% of the data are divided equally between data with standardized values z ≤ –2 and data with standardized values z ≥ 2.

THE 68-95-99.7 RULE

THE 68-95-99.7 RULE 3. In every normal distribution, about 99.7% (i.e., practically 100%) of all the data values fall within three standard deviations above and below the mean. In other words, 99.7% of all the data have standardized values between z = –3 and z = 3. There is a minuscule amount of data with standardized values outside this range.

THE 68-95-99.7 RULE

Practical Implications For approximately normal distributions, it is often convenient to round the 99.7% to 100% and work under the assumption that essentially all of the data fall within three standard deviations above and below the mean. This means that if there are no outliers in the data, we can figure that there are approximately six standard deviations separating the smallest (Min) and the largest (Max) values of the data.

Practical Implications Earlier in the text, we defined the range R of a data set (R = Max – Min) and, in the case of an approximately normal distribution, we can conclude that the range is about six standard deviations. Remember that this is true as long as we can assume that there are no outliers.