A Variety of Distributions

David C. Howell

SPSS Program to Generate Samples from Populations

This demonstration can be used in several ways. Either the instructor or the student can enter the relevant program and simply run it off as given. Students can compare the shapes of the various distributions and note the relationship between the descriptive statistics and the characteristics of the plots. Note in particular that even with large samples, several of these distributions look to the eye like the normal distribution, even though they are not normal. This is especially true when the normal curve is not superimposed.

In addition to just using the program as written, you can

• Decrease the sample size to 100 and note the bumpiness of the distributions.
• Increase the sample size to 20,000 and see how much smoother the distributions are. This gives the students some idea of the value of larger samples and the random noise that we find in small samples.
• Change the mean and variance for the normal distribution and see what happens.
• Increase the degrees of freedom for the chi-square distribution and note how it approaches normal. (See Chapter Six in Methods or Chapter nineteen in Fundamentals.)
• Change the rate of decay for the exponential distribution and see how the shape changes. Ask students what naturally occurring data might have a shape that is similar to an exponential distribution.
• More advanced students might want to look up the logistic distribution in another source and see how it differs from the normal distribution.
• Play around with the binomial distribution. This particular one models data that you would expect from a five item true/false test where the probability of being correct on any item is 0.80. What happens to the shape of the distribution when we have 10 or 50 items? What happens to the shape when the probability of being correct moves toward .50? What does this example tell us about using statistical procedures that assume some kind of normality in the dependent variable?

Copy the following program exactly as it is written. Be careful about putting in all the periods and don't have unquoted periods in comments.

* This SPSS program generates samples from a variety of distributions, calculates 



* descriptive statistics, and plots the results.



* I have superimposed a normal curve on each histogram for comparison purposes.



* Note the descriptive statistics and how they are revealed in the plots.



* You will see several of these distributions later, and can come back to this program,



* alter the parameters, and see what happens. I



* In fact, why not alter them now and see what happens?



* Last modified 3/27/96  David C Howell.







Input program.



	Loop #i = 1 to 5000.



			* Data from a normal distribution with mean = 0 and std = 1.



		Compute xnorm = rv.normal(0,1).



			* Data from a chi-square distribution with 2 degrees of freedom.



		Compute xchisq = rv.chisq(2).



			* Data from an exponential distribution with rate of decay = 2.



		Compute xexpon = rv.exp(2). 



			* Data from a logistic distribution.



		Compute xlogist = rv.logistic(0,1).



			* Data from a binomial distribution with n = 5 and p = .80.



		Compute xbinom = rv.binom(5, 0.8).



		End Case.



	 End Loop.



	End File.



End Input Program.







Examine Vars = all



	/Plot StemLeaf Boxplot  /*Boxplots hard to read because outliers are identified.



	/Statistics.







Graph



	/Histogram(normal) = xnorm.



Graph



	/Histogram(normal) = xchisq.	



Graph



	/Histogram(normal) = xexpon.



Graph



	/Histogram(normal) = xlogist.



Graph



	/Histogram(normal) = xbinom.	

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Last updated 6/30/98