## 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

2nd VARS DISTR 1

Do this while in the Y= menu.

Y1=normalpdf(X,0,1)

WINDOW:

Xmin=-3

Xmax=3

Xscl=1

Ymin=-.01

Ymax=.4

Yscl=.1

Then GRAPH.

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.