Sep 7/8

Sep 7 —  Statistics

Normal Distributions.

Gauss. 1775—1846. Gottingen, Germany.

1801: discovery of the asteroid Ceres.

Sun has 8 planets (1801).  2-dim measurements, must find 3-dimensional equation.

the distribution associated with measurement; the effect of the error of measurement on any sampling.

The graph of the normal distribution: normal distributions are the result of chance occurrences. Measurements of a single object are usually distributed in a normal pattern. For instance, if each student independently measures the length of the table at the front of the room, the measures will follow a normal distribution.

The vast majority of measurements follow a normal distribution.

The function is normalpdf(x, [μ, σ]) where x is a dummy variable and the mean and standard deviation are 0 and 1 and can optionally be changed to whatever you need.

The formula for the function with mean 0 and std. dev. 1 is

y=\frac{{{e}^{-\frac{{{x}^{2}}}{2}}}}{\sqrt{2\pi }}

No matter what value are given to the mean and standard distribution, the total area under the curve is always 1.

On the screen: To obtain the graph of the function, go to


Do this while in the Y= menu.










For this graph, 68% is within one standard deviation, 95% within two standard deviations, and 99.7% within three standard deviations.

The areas can be computed with shadenorm(lowerbound, upperbound, [μ, σ]).

This function will shade the indicated area, and display the percentage of area shaded.

One must set the window as above: [-3,3] and [-.1,.4].

In your text you will see frequent reference to z-values.

The z-value is a normalized value, intended principally for use with tables of the standard normal.

z=\frac{x-\mu }{\sigma } reduces any x value to a z value, where the mean and standard deviation of the distribution are known.

With the TI, one can insert μ and σ directly, and not bother with computing z.

How does one find the x or z value which corresponds to a given area?

Two methods: 1) table; 2) TI-83+.

On the TI, we use invNorm(area, m, s) .

Use of this function is similar to other statistical operations.

Unfortunately this only gives us the boundary of the “left tail”, similar to the output of Normcdf.

  1. No comments yet.
  1. No trackbacks yet.

Leave a Reply

Fill in your details below or click an icon to log in: Logo

You are commenting using your account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s

%d bloggers like this: