The Empirical Rule
Imagine two movable vertical lines, one on either side of the vertical line x = μ. Suppose these vertical lines, x = a and x = b, are such that the one on the left is the same distance from x = μ as the one on the right. The proportion of data points in the part of the distribution a < x < b is defined by the proportion of the area under the curve between the two movable lines x = a and x = b. Figure 310 illustrates this situation. A wellknown theorem in statistics, called the empirical rule, states that all normal distributions have the following three characteristics:
 Approximately 68% of the data points are within the range ±σ of μ.
 Approximately 95% of the data points are within the range ±2σ of μ.
 Approximately 99.7% of the data points are within the range ±3σ of μ.
Two Common Distributions Practice Problems
Practice 1
Suppose you want it to rain so your garden will grow. It's a gloomy morning. The weather forecasters, who are a little bit weird in your town, expect a 50% chance that you'll see up to 1 centimeter (1 cm) of rain in the next 24 hours, and a 50% chance that more than 1 cm of rain will fall. They say it is impossible for more than 2 cm to fall (a dubious notion at best), and it is also impossible for less than 0 cm to fall (an absolute certainty!). Suppose the radio disc jockeys (DJs), who are even weirder than the meteorologists, announce the forecast and start talking about a distribution function called R(x) for the rain as predicted by the weather experts. One DJ says that the amount of rain represents a continuous random variable x, and the distribution function R(x) for the precipitation scenario is a normal distribution whose value tails off to 0 at precipitation levels of 0 cm and 2 cm. Draw a crude graph of what they're talking about.
Solution 1
See Fig. 311. The portion of the curve to the left of the vertical line, which represents the mean, has an area of 0.5. The mean itself is x = σ = 1 cm.
Fig. 311. Illustration for Practice 1.
Practice 2
Imagine the scenario above, as the DJs continue to expound. They start talking about the extent to which the distribution function is spread out around the mean value of 1 cm. One of them mentions that there is something called standard deviation, symbolized by a lowercase Greek letter called sigma that looks like a numeral 6 that has fallen over. Another DJ says that 68% of the total prospective future wetness of the town falls within two values of precipitation defined by sigma on either side of the mean. Draw a crude graph of what they're talking about now.
Solution 2
See Fig. 312. The shaded region represents the area under the curve between the two vertical lines represented by x = μ – σ and x = μ + σ. This is 68% of the total area under the curve, centered at the vertical line representing the mean, x = μ.
Fig. 312. Illustration for Practice 2 and 3.
Practice 3
What does the standard deviation appear to be, based on the graph of Fig. 312?
Solution 3
It looks like approximately σ = 0.45, representing the distance of either the vertical line x = μ – σ or the vertical line x = μ + σ from the mean, x = μ = 1 cm. Note that this is only the result of the crude graph we've drawn here. The DJs have not said anything about the actual value of σ. We want to stick around and find out if they'll tell us what it is, but then there is a massive lightning strike on the broadcast tower, and the station is knocked off the air.
Practice problems for these concepts can be found at:
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