NORMAL DISTRIBUTION FACTS

Many variables are nearly normal, but none are exactly normal. Thus the normal distribution, while not perfect for any single problem, is very useful for a variety of problems. We will use it in data exploration and to solve important problems in statistics.

Normal distribution model

The normal distribution always describes a symmetric, unimodal, bell-shaped curve. How- ever, these curves can look different depending on the details of the model. Specifically, the normal distribution model can be adjusted using two parameters: mean and standard deviation. As you can probably guess, changing the mean shifts the bell curve to the left or right, while changing the standard deviation stretches or constricts the curve.

−3 −2 −1 0 1 2 3

Y

7 11 15 19 23 27 31

Both curves represent the normal distribution. However, they differ in their center and spread.

If a normal distribution has mean µ and standard deviation σ, we may write the distribution as N(µ, σ). The two distributions in Figure 4.3 may be written as

N(µ = 0, σ = 1) and N(µ = 19, σ = 4)

Because the mean and standard deviation describe a normal distribution exactly, they are called the distribution’s parameters. The normal distribution with mean µ = 0 and standard deviation σ = 1 is called the standard normal distribution.

1It is also introduced as the Gaussian distribution after Frederic Gauss, the first person to formalize its mathe- matical expression.