Chapter 3

Modeling Distributions of Data

Section 3.2Normal Distributions

Normal Curve

The Normal curve has a distinctive, symmetric, single-peaked bell shape.

Normal curves have special properties

Normal curves describe Normal distributions

Normal distributions.... play an important role and are rather special, but not at all "normal", "average" or "natural"!They are important becauseThey are good descriptions for distributions of real dataThey are good approximations to results of chance outcomesMany statistical inference procedures are based on them

Figure 3.12 Two Normal curves. The standard deviation fixes the spread of a Normal curve.

We can locate the standard deviation of the distribution by eye on the curve!

Knowing the mean and standard deviation completely specifies the curve.

The mean fixes the centerThe standard deviation determines its shape

Normal curves...

Changing the mean does not change its shape, only its location on the x-axis

Changing the standard deviation does change the shape, a smaller deviation is less spread out and more sharply peaked

Are symmetric, bell-shaped curves that have these properties

completely described by giving its mean and its standard deviationthe mean determines the center of the distribution, it is located at the center of symmetry of the curvethe standard deviation determines the shape of the curve, it is the distance from the mean to the change-of-curvature points on either side

Normal curves...

The Empirical Rule (aka 68-95-99.7 Rule)

In any Normal distribution, approximately

68% of the observations fall within one standard deviation of the mean95% of the observations fall within two standard deviations of the mean99.7% of the observations fall within three standard deviations of the mean

Figure 3.14 The 68–95–99.7 rule shows that 84% of any Normal distribution lies to the left of the point one standard deviation above the mean. Here, this fact is applied to SAT scores.

Before we stated that

z = __x - mean __ standard deviation

Nowμ (mu) = mean of a density curve

σ (small sigma) = standard deviation of a density curve

We use these to represent a population distribution

So... z =

x-μσ

The standard Normal distribution

the Normal distribution with mean 0 and standard deviation 1

z has the standard Normal distribution...

The Empirical Rule states that 68% of observations fall between z = -1 and z = 1.

How can we find the percentage of observations between z = -1.25 and z = 1.25?

https://www.geogebratube.org/student/m32297

The standard Normal table (the z-score table) gives us areas under the Normal curve.

Find the proportions of observations from the standard Normal distribution that are (a) less than -1.25 and (b) greater than 0.81.

z-scores.xps

(a)

z-scores.xps

(b)

Now find the area between -1.25 and 0.81

The area under the standard Normal curve between z = -1.25 and z = 0.81 is 0.6854.

1 - (0.1056 + 0.2090) = 1 - 0.3146 = 0.6854

Finding z-scores from percentiles

z-scores.xps

To do problems with a Normal distribution instead of the Standard Normal distribution, first standardize values by finding z-scores.

Practice 3.2.xps

Attachments

z-scores.xps

Practice 3.2.xps