Improvements in Graphics from 1967 - present at Hope College,

Prepared by

Elliot A. Tanis

August, 2009

IBM 1130 Computer Purchased in 1967• A web site: http://ibm1130.org/• This computer was used for student

projects is statistics. Simulation presented a problem because we could generate only 8,192 “random numbers” before it repeated. And successive pairs of random numbers fell on at most 256 parallel lines.

Examples of pairs of “random numbers” falling

on parallel lines.

Pairs of “random numbers” generated by our improved version of a pseudo random number generator.

Graphics were a challenge with the line printer. But Deanna was able to

illustrate the CLT empiricallyIn her independent project report on May 25, 1968, she included an illustration of the Central Limit Theorem using a U-shaped distribution with p.d.f. f(x) = (3/2)x^2, -1 < x < 1. Not only did she do the simulation, but she also wrote some graphics programs to see what was happening. Some of her output from the line printer is included on the next slide.

This example is used to illustrate improvements in graphics.

Here is a graph of the p.d.f. and the FORTRAN program that will do the simulation.

This graph shows output for samples of size 1 to confirm that we are indeed sampling from this U-shaped distribution.

This graph shows the distribution of x-bar when the sample size is 3.

This parent distribution has been used in many presentations throughout my tenure at Hope College, culminating is the use of MAPLE and finding the theoretical p.d.f.

Deanna Gross illustrated the CLT empirically using a U-shaped

distribution, March, 1969 := f

3 x2

2For -1 < x < 1,

A plotter is used for the histogram and the N(0,1) p.d.f. is superimposed.

Microcomputer developments• Spring, 1980 – Developed and offered the

course “Statistics for Scientists” under the CAUSE grant.

• February 18, 1980: The 10 TRS-80 Model I computers were sold and replaced with 6 TRS-80 Model III computers using a grant from NSF.

TRS-80 graphics: (1984)

This figure shows the distribution of the

sample mean when sampling from a

U-shaped distribution for n = 2, 5, and 8.

Moving from TRS-80 to IBM PCs

• January, 1985 – “A Computer Based Laboratory for Introductory Statistics” was presented at the MAA Annual Meeting. This was TRS-80 based using BASIC.

• Summer, 1985: David Kraay worked on the “Translation and Adaptation of Statistics Programs to the IBM-PC.”

• January, 1986 – We have laboratory materials for Math 212 using IBM PCs.

IBM PC graphics: Comparison of t and z confidence intervals for the mean. How many rolls of a 20-sided die are needed to observe at least one of its faces twice? Graph compares theoretical and empirical histograms.

“Computer Simulations to Motivate and/or Confirm Theoretical Concepts” (1987)

MAPLE

The laboratory for mathematical statistics and

probability incorporated MAPLE and

MINITAB. This manual was

published in 1995 by Prentice Hall.

Maple and Minitab continued to influence the mathematical statistics lab

• This second edition was published in 1998.

Here is output from a talk given at the

Joint Statistics Meetings in Dallas, 1998.

This was prepared using

MAPLE V, comparing

simulated data and the

theoretical p.d.f.

Use MAPLE to find p.d.f.s of sums of independent random variables. This example uses the U-shaped

distribution introduced by Deanna Gross in 1968, finding p.d.f.s of the sums of random samples of sizes 2, 3, 4.

:= f3 x

2

2

SumX1X2 , ,::expr algebraic a bproc( ) :=

piecewise [ ]y 1 2 a 0 [ ]y 1 a b ( )int ,( )subs ,x [ ]y 1 [ ]y 2 expr ( )subs ,x [ ]y 2 expr [ ]y 2 .. a [ ]y 1 a [ ]y 1 2 b, , , , ,(

( )int ,( )subs ,x [ ]y 1 [ ]y 2 expr ( )subs ,x [ ]y 2 expr [ ]y 2 .. [ ]y 1 b b 0, )

end proc

For -1 < x < 1,

The p.d.f. of the sum of a random sample of size 3 from this U-shaped distribution.

Sum of a sample of size 4.

:= g4

0 x -4

2367

350x

1926

385

1

492800x

11 197

40x

2 99

112x

4 423

160x

3 3

20x

5 9

1120x

7 3

2240x

8 9

400x

6 x -2

3

492800x

11 153

385

77

40x

2 99

112x

4 81

32x

3 27

1120x

7 3

2240x

8 9

400x

6 x 0

3

492800x

11 153

385

77

40x

2 99

112x

4 81

32x

3 27

1120x

7 3

2240x

8 9

400x

6 x 2

2367

350x

1926

385

1

492800x

11 197

40x

2 99

112x

4 423

160x

3 3

20x

5 9

1120x

7 3

2240x

8 9

400x

6 x 4

0 4 x

Additional Material

• To see more of this material and some animated output, go to

http://www.math.hope.edu/tanis/