6.1: Properties of the Normal Distribution

Observations of Normal Distributions

There are many normal distributions. Even though all of them have the bell-shape, they vary in their center and spread.

More specifically, the center of the distribution is determined by its mean ($\mu$) and the spread is determined by its standard deviation ( $\sigma$).

Some observations we can make as we look at this graph are:

• The black and the red normal curves have means or centers at $\mu$ = 10. However, the red curve is more spread out and thus has a larger standard deviation.

  As you look at these two normal curves, notice that as the red graph is squished down, the spread gets larger, thus allowing the area under the curve to remain the same.

• The black and the green normal curves have the same standard deviation or spread (the range of the black curve is 6.5-13.5, and the green curve’s range is 10.5-17.5).

The Standard Deviation Rule for Normal Random Variables

We began to get a feel for normal distributions in the Exploratory Data Analysis (EDA) section, when we introduced the Standard Deviation Rule (or the 68-95-99.7 rule) for how values in a normally-shaped sample data set behave relative to their mean ($\overline{x}$) and standard deviation (s). This is the same rule that dictates how the distribution of a normal random variable behaves relative to its mean $\mu$ and standard deviation $\sigma$. Now we use probability language and notation to describe the random variable’s behavior. For example, in the EDA section, we would have said “68% of pregnancies in our data set fall within 1 standard deviation (s) of their mean ($\overline{x}$).” The analogous statement now would be “If X, the length of a randomly chosen pregnancy, is normal with mean ($\mu$) and standard deviation ($\sigma$), then $0.68=P(\mu -\sigma <X<\mu +\sigma )$.”

In general, if X is a normal random variable, then the probability is

68% that X falls within 1 $\sigma$ of $\mu$ , that is, in the interval $\mu \pm \sigma$

95% that X falls within 2 $\sigma$ of $\mu$ , that is, in the interval $\mu \pm 2\sigma$

99.7% that X falls within 3 $\sigma$ of $\mu$ , that is, in the interval$\mu \pm 3\sigma$

Using probability notation, we may write

$0.68=P(\mu -\sigma <X<\mu +\sigma )$

$0.95=P(\mu -2\sigma <X<\mu +2\sigma )$

$0.997=P(\mu -3\sigma <X<\mu +3\sigma )$

Comment

Notice that the information from the rule can be interpreted from the perspective of the tails of the normal curve: since .68 is the probability of being within 1 standard deviation of the mean, (1 – .68) / 2 = .16 is the probability of being further than 1 standard deviation below the mean (or further than 1 standard deviation above the mean). Likewise, (1 – .95) / 2 = .025 is the probability of being more than 2 standard deviations below (or above) the mean; (1 – .997) / 2 = .0015 is the probability of being more than 3 standard deviations below (or above) the mean. The three figures below illustrate this.