Chapter 6

The Normal Distribution and Other Continuous Distributions

6.1: Continuous Probability Distributions

• Continuous Random Variables– If X is a continuous RV, then P(X=a) = 0,

where “a” is any individual unique value– Because X has individual unique values– P(a X b) = “something nonzero” where

“a” to “b” represents an interval

• Normal is most important continuous probability distribution.

6.2: Normal Distribution• Also known as “Gaussian Distribution”

• Works close enough for a lot of continuous RVs.

• Works close enough for a few discrete RVs.

• Necessary for our inferential statistics.

• Bell-shaped and symmetric.

• All measures of central tendency are equal.

• In theory, X is continuous and unbounded.

Normal RV

• Probabilities for discrete RV were given by a probability distribution function.

• Probabilities for continuous RV are given by a probability DENSITY function (pdf).

• Normal pdf requires you to know two parameters to find probabilities: and .

Finding Normal Probabilities

• Equation 6.1: fun but not useful.

• Like to have a table for each combination of and .– Can’t.

• Generate 1 table that can be used by everyone.

– Get everyone to convert or transform data so that it works with that one table!

– Transform X into Z

6.3: Evaluating Normality

• The assumption of Normality is made all the time: sometimes correctly so, and sometimes incorrectly so.

• Said another way: not all continuous random variables are normally distributed.

Checking Normality

• Text discusses two ways in this section (other ways discussed in Stat 2!)

1 Compare what you know about the data to what you know about the normal distribution.

2 Construct a normal probability plot.

Comparing actual data to theory

• Central tendency: actual data mean, median, and mode should be similar.

• Variability:– Is the interquartile range about equal

to 1.33*the standard deviation?

– Is the range about equal to 6 times the standard deviation?

Comparing actual data to theory

• Shape:– plot the data and check for symmetry.

– check to determine if the Empirical Rule applies.

• Sometimes samples are small--is the data non-normal or do you have a non-representative sample?

Normal Probability Plot

• Best left to software.

• The straighter the line, the better the sample approximates a normal distribution.

• Systematic deviation from a straight line indicates non-normality.

Plot Construction

• Order the data• Use inverse normal scores

transformation to find the standardized normal quantile for each data point.° P(Z < Oi) = i/(n+1)° i.e. solve for Oi for the 1st data point and

the second data point, etc.

Plot Construction (cont.)

• Plot the data points:– actual values on the Y axis

– Standardized Normal Quantiles on the X axis

• A straight line demonstrates normality.

• A non-straight line demonstrates non-normality.