103 Normal Random Variables (2 of 6)
Learning Objectives
- Use a normal probability distribution to estimate probabilities and identify unusual events.
Example
Beyond One Standard Deviation from the Mean
Earlier we stated that for all normal curves, the area within 1 standard deviation of the mean will equal 0.68. From this fact, we can see that the area outside of this region equals 1 − 0.68 = 0.32. And since normal curves are symmetric, this outside area of 0.32 is evenly divided between the two outer tails. So the area of each tail = 0.16.
[latex]\mathrm{area\; of\; each\; tail}=\frac{1}{2}(1-\mathrm{central\; area})=\frac{1}{2}(1-.68)=\frac{1}{2}(.32)=.16[/latex]
The outer tail areas allow us to answer related probability questions:
- Question: What is the probability that a normal random variable is more than 1 standard deviation from its mean?
- Answer: 0.32
- Question: What is the probability that a normal random variable is more than 1 standard deviation larger than its mean?
- Answer: 0.16
Before leaving this example, we highlight one more geometric fact about normal curves. Look at the arrows pointing at the normal curve in the following figure.
At these points, the curve changes the direction of its bend and goes from bending upward to bending downward, or vice versa. A point like this on a curve is called an inflection point. Every normal curve has inflection points at exactly 1 standard deviation on each side of the mean.
With the following simulation, you can look at a variety of normal curves. Use the slider to change the standard deviation. As you change the standard deviation, you will of course get different normal curves. Observe that the two properties we discussed in the examples remain true for any standard deviation you select:
- The probability that a value is within 1 standard deviation of the mean is 68%.
- The x-values of the inflection points correspond to 1 standard deviation above and below the mean.
Click here to open this simulation in its own window.
Learn By Doing
Now we extend this idea to look at the probability of a value falling within 2 standard deviations of the mean or 3 standard deviations of the mean.
If X is a normal random variable with mean [latex]\mathrm{μ}[/latex] and standard deviation [latex]\mathrm{σ}[/latex], then
- The probability that X is within 1 standard deviation of the mean equals approximately 0.68.
- The probability that X is within 2 standard deviations of the mean equals approximately 0.95.
- The probability that X is within 3 standard deviations of the mean equals approximately 0.997.
To summarize using probability notation:
[latex]\begin{array}{l}\mathrm{1.}P(\mu-\sigma These three facts together are called the empirical rule for normal curves. Let’s take a moment to look a bit deeper at what the empirical rule tells us.Comment
Learn By Doing