TI-30Xa Algorithms: Linear Regression

 TI-30Xa Algorithms: Linear Regression

Welcome to the March 2024 installment of TI-30Xa Algorithms.

Linear Regression with a TI-30Xa?

Today’s task is to fit bi-variate data to the line:

y = m * x + b

with the TI-30Xa. What? This calculator does not have a linear regression mode. Yes, with it is possible. We are going to use the technique shown in by the TI-36 Solar Guidebook from 1985 (see Source below). The TI-36 Solar was part of Texas Instruments’ line in the late 1980s and the early 1990s.

You can read my review on the TI-36 Solar from September 26, 2020 here:

https://edspi31415.blogspot.com/2020/09/retro-review-ti-36-solar.html

Where the TI-36 Solar only had one memory register, the TI-30Xa has three memory registers, and they are going to come in handy here.

Procedure

Caution: Be sure the calculator is on the entire time. Turning off the calculator will clear out statistics mode.

1. Enter the y data. Store the mean (y-bar) into memory register 2. Store the population deviation (σy) into memory register 3.

Keystrokes:

Mean of y-data: [ 2^nd ] [ x^2 ] {x-bar} [ STO ] [ 2 ]

Population of y-data: [ 2^nd ] [ ÷ ] { σxn } [ STO ] [ 3 ]

2. Clear the stat registers and enter the x data. As we are entering the x data, use memory register 1 to calculate Σxy.

Keystrokes:

Clear Stat Registers: [ 2^nd ] [ 7 ] {CSR}

1^st Point:

x1 [ × ] y1 [ = ] [ STO ] [ 1 ]

x1 [ Σ+ ]

Every point there after:

x_i [ × ] y_i [ = ] [ 2^nd ] [ RCL ] {SUM} [ 1 ]

x_i [ Σ+ ]

3. Calculate the slope, and replace Σxy with the slope.

m = ( y-bar * Σx – Σxy) / ( x-bar * Σx - Σx^2)

Keystrokes:

[ ( ] [ RCL ] [ 2 ] [ × ] [ 2^nd ] [ ( ] [ - ] [ RCL ] [ 1 ] [ ) ]

[ ÷ ] [ ( ] [ 2^nd ] [ x^2 ] [ × ] [ 2^nd ] [ ( ] [ - ] [ 2^nd ] [ ) ] [ = ] [ STO ] [ 1 ]

4. Calculate the y-intercept, replace y-bar with the y-intercept.

b = -m * x-bar + y-bar

Keystrokes:

[ RCL ] [ 1 ] [ +/- ] [ × ] [ 2^nd ] [ x^2 ] [ + ] [ RCL ] [ 2 ] [ = ] [ STO ] [ 2 ]

5. Calculate the correlation. If the correlation is close to -1 or +1, the linear fit will be excellent.

r = m * σx / σy

Keystrokes:

[ RCL ] [ 1 ] [ × ] [ 2^nd ] [ ÷ ] [ ÷ ] [ RCL ] [ 3 ] [ = ]

6. Use slope (m) and intercept (b) to predict x and y values:

y’ = m * x0 + b

Keystrokes: [ RCL ] [ 1 ] [ × ] x0 [ + ] [ RCL ] [ 2 ] [ = ]

x’ = (y0 – b) / m

Keystrokes: [ ( ] y0 [ - ] [ RCL ] [ 2 ] [ ) ] [ ÷ ] [ RCL ] [ 1 ] [ = ]

Key Map

[ RCL ] [ 1 ]: first Σxy, then m

[ RCL ] [ 2 ]: first y-bar, then b

[ RCL ] [ 3 ]: σy

[ 2^nd ] [ x^2 ]: x-bar

[ 2^nd ] [ ÷ ]: σxn

Example

Fit a line to the data:

X	Y
12	100
13	98.7
14	97.1
16	94.9
18	92.6
20	90

1. Enter the y data. Store the mean (y-bar) into memory register 2. Store the population deviation (σy) into memory register 3.

100 [ Σ+ ]

98.7 [ Σ+ ]

97.1 [ Σ+ ]

94.9 [ Σ+ ]

92.6 [ Σ+ ]

90 [ Σ+ ] (display n = 6)

Mean of y-data: [ 2^nd ] [ x^2 ] [ STO ] [ 2 ] (y-bar = 95.55)

Population of y-data: [ 2^nd ] [ ÷ ] [ STO ] [ 3 ] (σy = 3.465424457)

2. Clear the stat registers and enter the x data. As we are entering the x data, use memory register 1 to calculate Σxy.

Keystrokes:

Clear Stat Registers: [ 2^nd ] [ 7 ] {CSR}

12 [ × ] 100 [ = ] [ STO ] [ 1 ]

12 [ Σ+ ]

13 [ × ] 98.7 [ = ] [ 2^nd ] [ RCL ] [ 1 ] (SUM 1)

13 [ Σ+ ]

14 [ × ] 97.1 [ = ] [ 2^nd ] [ RCL ] [ 1 ] (SUM 1)

14 [ Σ+ ]

16 [ × ] 94.9 [ = ] [ 2^nd ] [ RCL ] [ 1 ] (SUM 1)

16 [ Σ+ ]

18 [ × ] 92.6 [ = ] [ 2^nd ] [ RCL ] [ 1 ] (SUM 1)

18 [ Σ+ ]

20 [ × ] 90 [ = ] [ 2^nd ] [ RCL ] [ 1 ] (SUM 1)

20 [ Σ+ ]

RCL 1: Σxy = 8827.7

[ 2^nd ] [ x^2 ]: x-bar = 15.5

3. Calculate the slope, and replace Σxy with the slope.

[ ( ] [ RCL ] [ 2 ] [ × ] [ 2^nd ] [ ( ] [ - ] [ RCL ] [ 1 ] [ ) ]

[ ÷ ] [ ( ] [ 2^nd ] [ x^2 ] [ × ] [ 2^nd ] [ ( ] [ - ] [ 2^nd ] [ ) ] [ = ] [ STO ] [ 1 ]

Slope: m = -1.230526316

4. Calculate the y-intercept, replace y-bar with the y-intercept.

[ RCL ] [ 1 ] [ +/- ] [ × ] [ 2^nd ] [ x^2 ] [ + ] [ RCL ] [ 2 ] [ = ] [ STO ] [ 2 ]

Intercept: b = 114.6231579

5. Calculate the correlation.

[ RCL ] [ 1 ] [ × ] [ 2^nd ] [ ÷ ] [ ÷ ] [ RCL ] [ 3 ] [ = ]

Correlation: r = -0.999092386

The line is:

y = -1.230526316 * x + 114.6231579

6. Predict values.

If x = 15, predict the y value (y’):

[ RCL ] [ 1 ] [ × ] x0 [ + ] [ RCL ] [ 2 ] [ = ]

y’ = 96.16526316

If y = 95, predict the x value (x’):

[ ( ] y0 [ - ] [ RCL ] [ 2 ] [ ) ] [ ÷ ] [ RCL ] [ 1 ] [ = ]

x’ = 15.94696322

Source

Alley, Chris M., Brenda M. Cornitius, et al. TI-36 Solar Guidebook Texas Instruments Incorporated. Dallas, TX. 1985, 1986, 1987. pp. 4.6 – 4.13

Eddie

All original content copyright, © 2011-2024. Edward Shore. Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited. This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author.

Retro Review: TI-36 Solar

Retro Review:  TI-36 Solar 

Just the Facts:

Model:  TI-36 Solar

Company: Texas Instruments

Type: Scientific

Display:  10 digit with 2-digit exponents

Battery:  Solar

Logic: AOS

Memory Registers: 1

Years: 1984-1990

Features:

*  Complex Number Arithmetic

*  Rounding Numbers (to fixed number setting)

*  Base Conversions

*  Normal Distribution

*  Single-Variable Statistics

Examples with Complex Numbers

MODE 5:  CPLX

There are two separate registers to hold parts of complex numbers:

[ a ]:  Real part (rectangular),  Radius (polar)

[ b ]:  Imaginary part (rectangular), Angle (polar)

Example 1:  (4 + 3i) * (11 - 2i)

4 [ a ] 3 [ b ] [ × ] 11 [ a ] 2 [ +/- ] [ b ] [ = ]

Display:  50  

Press [ b ], Display:  25

Result:  50 + 25i

Example 2:  Convert 3 + 2.4i to polar form, degrees

3 [ a ] 2.4 [ b ] [ INV ] (R>P) 

Display:  3.841874542

Press [ b ],  Display:  38.65980826

3 + 2.4i = 3.841874542 ∠ 38.65980826°

Examples with Normal Distribution

MODE 6:  STAT

There are three functions that determine the area under the normal distribution curve:

P(t) from - ∞ to t,  lower tail curve

R(t) from 0 to t

Q(t) from t to ∞, upper tail curve

Note that for any t, P(t) + Q(t) = 1.

Also note that P(t), R(t), and Q(t) will operate on the standard normal curve, where μ = 0 and σ = 1, regardless of the amount of data points entered in Statistics mode through Σ+.

For z = t =2:

2 [ INV ] ( P(t) ) returns 0.97725

2 [ INV ] ( R(t) ) returns 0.47725

2 [ INV ] ( Q(t) ) returns 0.02275

Comparison:  TI-36 Solar vs. TI-35 Plus

The TI-36 Solar and TI-35 Plus have the same set of functions and features.  You can see my retro review from 2017 of the TI-35 Plus here:  http://edspi31415.blogspot.com/2017/09/retro-review-texas-instruments-ti-35.html

There are several keyboard differences:

Shift key is marked [ INV ] (with 2nd above it) for the TI-36 Solar, and the shift key is marked [ 2nd ] for the TI-35 Plus.

The equals key is twice in vertical height, covering the spans of the bottom two rows on the TI-35 Plus.

The top three rows* (all of the shift markings remain intact):

TI-36 Solar:

[CE/C] [ 1/x ] [ a ] [ b ] [ AC ]

[ hyp ] [ sin ] [ cos ] [ tan ] [ DRG ]

[ INV ] [ x^2 ] [ log ] [ ln x ] [ y^x ]

TI-35 Plus:

[ 2nd ] [ x^2 ] [ log ] [ ln x ] [ OFF ]

[ hyp ] [ sin ] [ cos ] [ tan ] [ DRG ]

[ y^x ] [ 1/x ] [ a ] [ b ] [ ÷ ]

The [ hyp ] key has the hyp^-1 label over it on the TI-35 Plus.  

*not counting the [ON/C] key above all the rows on the TI-35 Plus

Comparison:  TI-36 Solar vs. the original TI-36X Solar 

Here is my review from 2018 for the TI-36X Solar:  http://edspi31415.blogspot.com/2018/09/retro-review-texas-instruments-ti-36x.html

The TI-36X Solar:  

*  does not have the complex arithmetic mode of the TI-36 Solar

*  adds the Boolean functions AND, OR, XOR, XNOR, and NOT to the BIN, OCT, and HEX mode

*  adds linear regression

*  adds a second shift key, [ 3rd ]

*  there is no [ MODE ] key, every mode is selected through [ 3rd ] (key) combos

*  adds a fraction/decimal conversions

*  adds 8 scientific constants and 10 metric/US conversions

Verdict

Like the TI-35 Plus, the TI-36 Solar is a step up from the TI-30 series (1980s versions of TI-30).   Again, the TI-34 (1987) has the Boolean functions and fraction functions that the TI-36 Solar doesn't.  You don't have to worry about batteries at all since the TI-36 Solar runs entirely on solar/light power.

Eddie

All original content copyright, © 2011-2020.  Edward Shore.   Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited.  This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author. 

Retro Review: Texas Instruments TI-36X Solar

Retro Review:  Texas Instruments TI-36X Solar

Continuing on the Labor Day 2018 of posts, I now present a retro review of a scientific calculator I couldn’t put down as a young student: the TI-36X Solar. For the class of non-programmable solar scientific calculators back in the day, the TI-36X was the one of the high end calculators. 

If you are expecting a review of the multi-line (and current) TI-36X Pro, then please click here:  https://edspi31415.blogspot.com/2011/04/

General Information

Company:  Texas Instruments

Type:  Solar Scientific Algebraic (postfix)

Display:  10 digits with 2 digit exponents (power of 10)

Power:  Solar

Memory:  3 (store, recall, sum (M+), exchange)

Years:  1991 - 2011

Original Cost: $15 - $20

Documentation:  Manual, Quick Reference Card

Texas Instruments no longer manufactures new TI-36X Solar calculators, so if you want one, pawn shops or internet websites such as eBay would be the place to get one.

Features

* Logarithmic and trigonometric calculations

* Base conversion and Boolean logic

* Linear Regression (intercept is labeled ITC, slope labeled SLP, correlation labeled COR)

* Fractions with conversion and improper fraction conversion (automatic simplification)

* Combinations and permutations

* Eight Scientific Constants and Ten Metric-US Conversions

Scientific Constants ( [3rd], (CONST), appropriate key)

C:  speed of light

g:  Earth’s gravitational constant

me: Mass of an electron

e:  electron charge

h:  Plank’s constant

Na:  Avogadro’s constant

R:  Ideal Gas Constant

G:  Universal Gravitational Constant

Metric-US Conversions

Centimeters/Inches  (1 in = 2.54 cm)

Liters/Gallons  (1 gal = 3.785411784 L)

Pounds/Kilograms (1 lb = 0.45359237 kg)

Fahrenheit/Celsius Temperature

Grams/Ounces (1 oz = 28.34952313 g)

The Exchange Key ([x<>y]) key does a lot  

The TI-36X makes good use of the exchange key, and it is a non-shifted key which facilitates easy operation.  In addition in allowing users to interchange arguments (i.e. switching the subtrahend and minuend in a subtraction calculation), the exchange key allows for:

* Entering n and r in combination and permutation calculations

* Entering x and y in Rectangular to Polar conversions

* Entering r and θ in Polar to Rectangular conversions

* Entering x and y in paired-data statistics (linear regression)

Modes

DEC:  Decimal representation, floating decimal mode, normal calculations

BIN, OCT, HEX:  Binary, Octal, Hexadecimal integer modes, respectively.  Boolean functions such as AND, OR, and NOT only work in these integer modes

STAT 1:  1 Variable Statistics

STAT 2:  2 Variable Statistics/Linear Regression

TI-36X Solar:  1991 keyboard, 1996 keyboard, 2004 keyboard

Keyboard Design and History

The design of the keyboards of the TI-36X Solar evolved over the years.  The first major design was rectangular, most of the keys were black with light font.  The arithmetic keys were at first dark gray, then blue.  This was the shortest keyboard in length.  The calculator had a slide hard case.

Then in 1993, the TI-36X Solar had a keyboard with a rounded edge at the bottom of the keyboard.  The 1996 version is slightly longer in size, with the yellow 2nd key and blue-purple 3rd key.  Constants were labeled in green. 

The final design of the TI-36X came in 2004, with an even bigger keyboard.  This time the keyboard was silver with the royal blue 2nd key, lime green 3rd key, and constants marked in maroon.  This is the most readable keyboard, however the 2004 version needs more light to operate than its previous versions.  This version also had a snap on case. 

With the exception of the 1993 revamp, the keys were plastic.  The 1993 version had rubber keyboards.  The keys generally had a good response to them. 

Of the three TI-36X Solar calculators I have now (just purchased one with the 1991 original style keyboard), the manufacturing dates where 12/1990 (1991 keyboard), 6/2000 (1996 keyboard), and 6/2010 (2004 keyboard). 

TI-35X, the battery operated version of the TI-36X Solar, 1993 keyboard with rubber keys

The TI-36X Solar had a battery operated version, named the TI-35 Plus.  I have a TI-35X in a 1993 version keyboard with the rubber keys. 

The TI-36X Solar was itself a successor to the TI-36 Solar.  The TI-36 Solar lacked constants, English-US conversions, fractions, and linear regression, but had complex number arithmetic. 

The successor the TI-36X Solar is the TI-36X Pro, which has a multi-line screen, adds among other things, integration, derivatives, function tables, textbook input/output, matrices, and additional regression types.

My introduction to the TI-36X Solar was when in I was in middle school back in the early 1990s.  I couldn’t put this calculator down.  I wished I remembered to use the hard cover because one day I had the TI-36X Solar in my pocket with a pencil.  The pencil poked through the solar panel and it was the end of it. 

  

Verdict

The TI-36X Solar is a feature rich scientific calculator and as I said before, one of the line non-programming solar scientific calculators at the time.  The keyboard is responsive, you have a nice set of common constants and conversions.  I like the layout of the keyboard.  This is a good calculator to pick up, even if it is just for collection purposes.   For a retro calculator, the TI-36X Solar is a definite recommend from me. 

Source:

Woerner, Jorgen.  “TI-36X Solar” Datamath http://www.datamath.org/Sci/Modern/TI-36XSOLAR.htm 

Eddie

All original content copyright, © 2011-2018.  Edward Shore.   Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited.  This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author.  Please contact the author if you have questions.