Odontologists--remember we need basic, foundational research in * all* areas of our discipline. Are our procedures scientifically

Critically examine ** everything** we've been taught. Question the scientific basis of

Keep in mind the quote often attributed to W. Edwards Deming: ** "Without data you are just another person with an opinion." **I would add, if you use

We insist on evidence-based treatment in health care, why not in forensics?

Merely saying "we're following the science" without verifying that the "science" being followed is actually true is the same thing religions and cults do.

Roger D Metcalf DDS, JD

PO Box 137442

Fort Worth, TX 76136-1442

ph: +1-817-371-3312

fax: +1-817-378-4882

metcalfd

I have taken a number of statistics classes in graduate school as a foundation for my research about dental age estimation. I am offering my opinions as a current student-user if you are looking for a new graphing calculator to use in your studies.

When you look online for calculators, you will find the ones we are most interested in for stats class are called either *"scientific"* or *"graphing."* At its most elementary, scientific calculators have smaller screens that generally display two or three or four lines of input/output, while graphing calculators have large screens, often in colour, that can display graphs and charts such as histograms and scatterplots.

Scientific models are usually quite a bit less expensive than graphing models, but, while you can get by with a scientific model, a graphing calculator will make things much easier for you on homework and on exams.

If you decide to stick with a scientific calculator, make sure that, at the minimum, it does "one-variable stats" i.e. when you put in a list of numbers, the calculator will give you things like the mean and variance of the list. A step up is the calculators that do "two-variable stats" and with those you can enter two lists of numbers, and the calculator will give you not only means and variances, but correlation and covariance and, if you pick the right one, will do simple linear regression. An example of a scientific, non-graphing calculator is the classic **Hewlett-Packard HP21S** shown in the right column.

An important consideration would also be whether your instructors allow or prohibit certain calculator models on exams, and, if you need to take standardized exams such as the SAT, ACT, or GRE, which calculators are allowed for those.

IMHO, if you want to get the most out of your class, consider the *graphing* calculators presented here.

Texas Instruments has made calculators for a long time and produced a *lot* of different models. TI has been deeply involved in and focused on the educational market for a long time.

The most recent TI models I have used include the **TI-84 Plus CE Python edition**, and the **TI-nspire **series**, **along with the older** TI-92+,** the **TI-89 Titanium**. The** 89 Ti** and, especially, the **92+** have been around for a *long* time, but they are still among the very best, IMHO, and are ** very **useable for stats classes.

First, the **TI-84** is simply an awesome calculator for stats classes. However, in “Distributions” while it has the inverse Normal, t, and Binomial, there is * no* inverse Chi Sq or F function for finding critical values, and that's a fairly big limitation in my eyes. Otherwise, if those inverses are not very important to you, or you don't mind referring to tables in a book, the

Texas Instruments TI-84 Plus CE

Texas Instruments TI-92+.

The **89 Titanium** and **92+** can be considered together, since they are quite similar, if not identical, in function and differ only in form factor. These are very powerful and both have inverse Normal, t, Chi Sq and F functions. Working with lists is easy with both, and plotting distributions is easy —but the plots are not in colour like with the **84**. Probably a minor point. These both do the 2-sample F-test--as does the **TI-84.** Calculating the test F-value itself is only half the battle, getting the critical F-values right can be sort of tricky if you have to do them by hand with the convoluted procedures sometimes presented (you have to switch around the degrees of freedom and invert some things), so it's nice to have the calculator help with that.

The **92+**is one of the primary calculators I use because the screen is larger and the form factor and keyboard make it easier for me to enter stuff quickly, especially lists--the small font on the **89 Ti** is hard to read sometimes. Another point to note is that the **89/92+ **screens are not backlit and in colour like the **84's** and **nspire's **are, if that's important to you. You will also see that the **92+ **is in landscape mode in contrast to all the other calculators here.

Texas Instruments TI-89 Titanium

Next, I like the **TI-nspire** series very much, as well, and these also have inverse Normal, t, Chi Sq, F, and Binomial functions. Again, as with the **92+/89**, working with lists is easy and plotting is easy, though still not quite as easy for me as with the **84**. Further, with the **TI-nspire** one has to learn to work with “Documents” for most of the things you’ll want to do. Not difficult to learn, but just maybe not straightforward and intuitive-- nonetheless, the **nspires** are also great for stats. It appears to me that TI is focusing their corporate energy primarily on the **nspires** and the **TI-84** models.

TI-nspire CX-II CAS

The **TI-84** and **89** have the more traditional type calculator keyboards with just number keys and a lot of function keys. But the **92+** and the **nspires** have, in addition, alphabetic keys on the ABC-style keypad, and that might be important and helpful for your work. (But note that some exams and/or classes don't allow use of calculators that have alphanumeric keypads or wireless capabilities.)

Hewlett-Packard have been making calculators for as long as TI has. While they certainly support the education market, they seem to be a bit more oriented towards the engineering crowd. My favorite all-time calculator is the **HP Prime** and **HP’s** are what I reach for when I have to do a lot of arithmetic—I came up through undergraduate college using **HP** calculators, and so I parse math problems in my mind in RPN (Reverse Polish Notation) style.

The **HP Prime** is awesome for stats. It's pretty easy to plot things--and with the touchscreen, you can "stretch" or "squeeze" the plot with your fingers to resize it so you can see what you want to see. It has a plethora of inverse functions: Normal, t, F, Chi Sq, Binomial, Geometric, and Poisson—very nice. The reason I keep mentioning inverse functions is because that's how you find the critical values for your statistical tests--sure, you could go look them up in a table, but that's sort of the point of having a calculator, so you don't have to carry around a book full of tables!

However, a minor downside for me is that working with lists is not quite as easy as with the **TI’s**, and that can be very important when you’re in the middle of an exam. And plotting things is not always quite as straightforward as with the TI’s--please keep in mind that plotting and working with lists is* not**difficult* on the HP, it's just that the **92+** and** 84** make it *really* easy. But the big selling feature to me will always be the **HP's RPN arithmetic**, and I just can’t get away from doing math in RPN*. The **HP Prime** has an advanced Computer Algebra System (CAS) based on Giac/Xcas.

The **HP’s** have had, for years, an extremely helpful feature I don’t find implemented as well on the other brands—the * SOLVER *(see below). Enter a complicated equation in the Solver, and then the

The **Casio CG-500** has a Solver, but seems to handle only one equation at a time while the **HP** will store 10. Not only that, but with the HP, you can save different versions of the Solver app itself, *each* of them with 10 functions, so you could wind up saving a LOT more Solver functions than just the nominal 10. The **TI'**s also do have a "solve" function, but it does not work quite the same as the **HP Solver **or the** Casio's **version. However, if you don't really need the Solver, it is quite easy to just define a function with the **TI's **or** Casio. NumWorks** have a Solver, too, but I haven't had a chance to try it out, yet. Solvers help prevent one from making silly arithmetic mistakes.

The HP Solver.

Another *very* cool thing the **Prime** has is the "**Probability Wizard,**" which greatly helps visualize and solve critical-value calculations for Normal, t, Chi Sq, F, and Geometric distributions. **To me, this feature alone is almost worth the price of the calculator.**

HP Probability Wizard

** I have no affiliation with any of these products or companies except as a user. However, if someone wants to pay me for referrals, I’m certainly open to that, LOL.**

The **TI-nspires** and **89 **and** 84** have a form-factor that is probably close to what you think about when you think "calculator." The **HP Prime** is similar, but a bit wider and a little, well, "flatter" than one might have thought; additionally, the **Prime** screen is larger than the **TI**'s, and it's a colour touchscreen to boot! The **Casio CG-500** is just *big! *The **TI-92+** is even bigger and shaped differently than any other calculator, and, again, is in horizontal format, but, for me, it is pretty ergonomic. **NumWorks **is slightly more compact than the others, and the default font is sort of small.

If you want to try out the **Hewlett Packard** calculators, but not spend a lot of money, HP offers official emulator apps for several of its models in the Apple App Store for iOS. Keep in mind, though, that standardized exams and most classes don't allow use of cell phones or wireless devices on exams.

**Casio** also has a **ClassPad** emulator for iOS (many functions disabled, though). **NumWorks** provides a very nice emulator, as well.

In fact, there are many 3rd party emulators you can find on the web for several of the **various different HP and TI models**. Check ticalc.org and hpcalc.org and hpmuseum.org for a whole lot of very helpful information as well as for apps you can download to your calculator.

These emulators can be every useful for trying out different calculator models and for doing homework, but, again, you probably won't be able to use your laptop or phone during a high-school or college exam, and you definitely can't use those on the SAT, ACT, or GRE.

If batteries are a concern, the **TI-89** and **92+** and the **Casio CG-500** all use replaceable batteries (some use **AA, **some use **AAA)**, while the **TI-84,** the **nspires**, **NumWorks,** and the **HP Prime** all have fixed rechargeable batteries.

The **Casio CG-500** and **CG-50** both also seem to have a lot of potential, but, at the present, IMHO, that **CG-500** touchscreen keyboard is a disadvantage for me for time-limited exam purposes, especially if I have to keep up with a stylus....but I am *really* going to try to use the **CG-500** more. The **CG-50 **does seem to be comparable to the TI-84 and does pretty much everything one needs in many stats classes.

The **TI-84 Plus CE Python** edition, the **TI-nspire CX-II series,** The **Casio** **CG-50** and **CG-500**, **NumWorks,** and the **HP Prime** all have Python capability, which expands what you can do with them.

* If you're not familiar with RPN (Reverse Polish Notation), or "postfix notation," here's a nice tutorial site:

https://hansklav.home.xs4all.nl/rpn/

Very basically, with an RPN calculator, instead of entering something like "2 + 3 = " you would key in "2 Enter 3 Enter +". Essentially, you put your numbers in first, and then tell the calc what to do with them--pretty much the way we think through a problem. And you might note there is no "precedence of operations" to deal with in this system--that is taken care of automatically as you do your calculations. Once you get the hang of it, it is much easier to work through complicated algebra problems with RPN rather than using algebraic entry and trying to keep up with several sets of nested parentheses--at least, for me it is.

Hewlett Packard HP Prime

The **HP Prime**, like the **TI-84 **and** 89**, has the more traditional type of calculator keypad with number keys and a lot of function keys. These calculators all provide for alphabetic entries using the “Alpha” and/or “Shift” keys. But the best of them all, I think, is still the **92+** with that nice, big QWERTY keyboard.

The **Prime, TI-nspires, 84, 92+, 89,** **NumWorks,** and the **Casio Classpad** all have either arrow keys or a touchpad to move the cursor around on the screen, and further, the **HP Prime** and the **CG-500** **Classpad** both have touchscreens, as well.

Hewlett Packard HP-21S

An honorable mention goes out to the **HP-21S**! This is an older calculator and is one of the few old HP’s that use algebraic entry for math instead of RPN. But it does something I’ve not seen on any other calculator—it has Normal, t, F, and Chi Sq functions *and inverses* on dedicated (shifted) keys right on the keyboard, making calculations of critical values and probabilities *extremely* easy. It has (shifted) keys for sum of *x,* sum of *y,* sum of *x^2*, sum of *y^2*, sum of *xy*, mean of *x* and *y*, and variance of *x* and *y* which can be *very* handy in stats class, but you'll have to find a used one somewhere.

Another company that has been making calculators for as long as TI and HP is **Casio.** Right now I'm working with the **Casio Classpad fx-CG500**. To start with, it has inverse Normal, t, Chi Sq, F, Binomial, Poisson, Geometric and Hypergeometric functions—so that’s pretty awesome. It plots things nicely (but, sometimes, slowly), and working with lists is pretty easy. The screen is *huge* for a calculator, and the calculator itself is big and solid-feeling. It does the 2-sample F-test.

The big downside to the **Casio**, for me, though, is the touchscreen keyboard. The calculator has a very minimalist hardware numeric keypad (no function keys to be found), so the alphabetic and other keyboards (math, special characters) have to be accessed as touchscreens. Additionally the keyboard (as with the **TI-nspires**) is not QWERTY, but is in straight ABC order. Using the stylus is highly recommended, so it seems not to be super convenient to use, IMHO.

My **Casio **actually appears, I think, to have very good capability for some stats classes, but I fear I will never know for sure because the keyboard is such a drawback for my particular uses. On a time-limited exam this could be crucial—but, hey, if you’re not worried about exams and don’t mind using the touchscreen keyboard, then the **Casio **might work very well for you.

Casio fx-CG500

Casio also have the **fx-CG-50** which is presented as being comparable to the **TI-84**. I have a **CG-50 ** and it seems to be a very capable calculator, as well.

**So, my recommendations: **

1) For high-school and college undergraduate stats classes (whether in math department or business** or a science department) and lower-level grad school stats classes (for non-math or stats majors)—the **TI-84**, I think, is the hands-down winner. It seems to me that TI has *tailored the 84 specifically for stats users,* and it does the job very well. Just remember there's no native inverse F or Chi Sq functions--

2) If you’re going on to more advanced stats or maths graduate-level classes, the **TI-84** is still great, but you might want to look at the **TI-89/92+ **or ** TI-nspire** series, which might serve your purposes even better. (Note: the

3) I still love my **HP Prime**, and it probably has more capability than the **TI-84 **for power users, but, in turn, it is a little more difficult to learn—and a lot of folks HATE using RPN math (though the MODE can easily be switched from RPN to Algebraic or Textbook, if desired).

So, if you are a little more technically inclined--i.e. a geek***--or are going on to higher-level math classes such as math stats, or, especially, engieeering classes, the **HP Prime** is probably for you! And it has a fairly advanced CAS based on Xcas/Giac (Xcas/Giac is free and can be installed on your Windows/Mac/LINUX desktop, as well).

** Summary: I think most folks in stats classes would be best served by one of these current models: **

**1) TI-84 **

**2) TI-nspire **

**3) HP Prime**

**4) Casio CG-50**

**5) TI-89**

To get a very powerful calculator at a *very* reasonable price, check for a used **TI-nspire** model on eBay. There are a lot of these sold as surplus models from secondary schools....look for one with CAS (computer algebra system), too. To me, this might be the most cost-effective way to get a really great calculator.

Now, to muddy the waters a bit, however, * f** or me* it's pretty much a toss-up between the

I realize that does not really narrow things down very much at all, and that not everyone wants to have three or four calculators on their desk. If** I** was going to use

So...here's the **NumWorks** calculator. It has never occurred to me before to write the phrase "this is a beautiful calculator," but *this* is a beautiful calculator! It is produced by a smaller company headed by, apparently, an ex-Apple engineer, and it shows! Again, this is a beautiful piece of hardware.

Numworks

* Consumer Reports* did a much more extensive review of this one than I could do...read the review here.

I just got mine and one thing I see already is that it has a "Solver" app, and I love that! I think a goal is for this calculator to be intuitive to use, and I think it is. But what is very cool is that it has all the regular distributions you need to deal with, plus it is **extremely** easy to get the * inverses* and a plot to see which tail you're dealing with--

Some folks don't care so much for the gold/white/black colour scheme, but I think this thing is pretty stylish with a very "clean" layout--maybe even elegant.

One issue I think I have already come across is the absence of ANOVA--that's sort of unexplicable in a high-end calculator, so maybe I overlooked something somewhere. The Regression function has a neat feature of letting you more-or-less work on several different regressions all in the same window at the same time--so that's interesting because this seems it would be a perfect place to implement ANOVA. But I haven't found it, yet, and that seems curious.

All-in-all, the NumWorks seems to have a lot of potential, but it has now been "locked down" due to requirements of standardized testing programs. The features that are implemented are not as sophisticated nor as numerous as on others in this bunch. The NumWorks would be useable in a very basic stats class, but, unfortunately, not so helpful in its current revision, IMHO, as you take more advanced stats.

Still, their Probability applet, I think, is one of the best! Also, they have made it extremely easy to update the OS, and other manufacturers might want to follow suit!

** I know this sounds like a "wishy-washy" cop-out, but** **you can't go really wrong with any of the graphing calculators mentioned here for stats classes. When you learn the math principles, you can figure out how to do the stats with any of these. **

** For business classes, the considerations may be a little different—if you’re doing busines STATS, the the **TI-84** would, of course, be great. But for ECONOMICS or FINANCE calculations, there might be other factors to take into account. All the calculators mentioned above (exc. the **HP-21S**) appear to have FINANCE apps that work quite well--and, in fact, the **TI's, HP Prime, NumWorks, and Casio Classpad** **CG-500** all seem to have very good FINANCE TVM (time-value of money) *solver* apps that are extremely helpful--so you might have to check around and see which of those work best for your classes—and don’t forget the “gold standard” that a lot of folks still use in finance and business is the iconic **HP-12** that's been around pretty much forever.

Hewlett Packard HP-12

WP-34S 3rd-party conversion of HP-30b

***...now if you're* really* a geek and going on to higher-grad-level stats classes designed for stats or math majors, or engineering classes, you might want to take a look at the **WP-34S** calculator (above). This is an esoteric third-party conversion of the **HP-30b** business calculator--a "repurposed" calculator**.**

Warning: This calc is not for the faint-of-heart and the official documentation is less than straightforward, IMHO. There is a ** WP 34S RPN Scientific Calculator Beginners Guide** that might help. You will have to teach yourself a fair amount here.

From some cursory use, I think the WP-34S emulator that runs under iOS is probably worth looking at, if you want to use this model!

© Copyright 2013, 2019 Roger D Metcalf. All worldwide rights reserved. No reproduction without permission. Neither the Tarrant County Medical Examiner's District, Tarrant County, the American Board of Forensic Odontolgy, the American Society of Forensic Odontology, the Royal College of Physicians, Oklahoma State University, nor any other organizaion mentioned here necessarily supports or endorses any information on this website. Any opinions, errors, or omissions are my responsibility, and mine alone. This site DOES NOT REPRESENT the official views of any of these--or any other-- organizations. Similarly, those other organizations may not fully represent my views, either.

Roger D Metcalf DDS, JD

PO Box 137442

Fort Worth, TX 76136-1442

ph: +1-817-371-3312

fax: +1-817-378-4882

metcalfd