OVERVIEW: Normal distributions are quite common inreal life settings. Any set of normally-distributed observations canbe examined efficiently by converting the data to standardizedobservations know as z-scores. A z-transformation changes a normalrandom variable with mean m and standarddeviation s into a standard normal randomvariable with mean 0 and standard deviation 1.

Any set of numbers has a mean and standard deviation. The set W ={5,10,20,65,80} has m = 36 and

The **standardized value** (sometimes called a **z-score**)of an observation, x, is

The following table displays the standardized values forobservations from the set W.

| z-score |

| -1.0140 |

| -0.8507 |

| -0.5235 |

| 0.9489 |

| 1.4397 |

If a variable x has a normal distribution with mean **standard normal table** that appears in statisticstextbooks. (And, which is provided on the Advanced PlacementStatistics Examination.) It is important to be able to use thistable. It's also significant to note that many useful computationscan be done on the TI-83. Here are a few examples...

Suppose a set of observations is approximately normalwith mean = 50 and standard deviation = 4.

We know that approximately 68% of the observationswill be within one standard deviation of the mean. Note that

normalcdf(46,54,50,4) = 0.6826894809normalcdf(-1,1,0,1) = 0.6826894809If we want to know what percent of the scores are above 56, we cannote that z

_{56}= (56-50)/4 = 1.5.

normalcdf(56,1E99,50,4) = 0.0668072287normalcdf(1.5,1E99,0,1) = 0.0668072287

**Important to note**: You canstandardize any set of numerical observations and obtain z-scores.The z-scores simply reflect how many standard deviations anobservation is from the mean. The z-scores will form a normaldistribution only if the original data set is normal. You cancorrectly use the normal distribution table to interpret z-scoresonly if the original data set is normal. __Standardizing scores doesnot magically convert non-normal data into normal data__.

If you have a data set in a TI-83 list, you can use the calculatorto construct a **normal probability plot**. This is one of theoptions when you do a __stat__ __plot__ on the calculator.

If you are given a numerical data set, always (I repeat,

Using the TI-83, this can be done very easily with a histogram or aboxplot.

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