Science Buddies: "Ask an Expert"

Hi,

I am doing a science fair project on how accurate are scientific calculators versus their prices? Currently I am having trouble finding a method to test the accuracy of calculators. I have the following calculators: Casio FX-500MS, TI-83+, TI-84+, TI-89. If anybody can show me how to test the accuracy of any of those calculators, that will be much appreciated.

In the fields of science, engineering, industry and statistics, accuracy is the degree of conformity of a measured or calculated quantity to its actual (true) value.

So, to determine if a calculator is accurate, you simply need to know the true value of a calculation, then compare that to the answer of the same calculation that the calculator makes . Put simply, we all know that the true answer to 2+2 is equal to 4. If a calculator, doing that calculation, returns an answer of 4, it is accurate. If it returns some other answer, it's accuracy is less than perfect.

Are you expecting calculators to make mistakes? There might very well be rounding errors in the last digit SOMETIMES, but I would think that most calculators are accurate.

What is your hypothesis?

Hello Holsetymoon!

I hate to sound negative here, but I'm not sure that this is a project that will yield the sort of results that you would like. I think most people would say that barring any physical damage to the calculator (dropping it, exposing it to extreme heat or cold, etc.), all calculators are accurate 99.99% of the time.

However, one route you may explore is how many decimal places a calculator shows. For some numbers (such as Pi) a calculator's accuracy is based on how many decimal places are shown. In the case of Pi, 3.14 is more accurate than just 3, and 3.14159265 is far more accurate than 3.14. Obviously the higher-end Texas Instruments (TI-83+, TI-84+) are going to show more decimal places because they have larger screens, but this is one route that you may want to consider.

Like zzzzdoc asked, what is your hypothesis? I hope this helps!

My hypothesis is:
Does the price of a calculator affects its accuracy?
Well, I know that price (money) is not a very objective factor in science, but I cannot think of anything else. If you can think of a better hypothesis, tell me and I will consider changing it. By accuracy, I mean "Accuracy is the degree of conformity of a measured or calculated quantity to its actual (true) value". I know that calculators are accurate most of the time, but I am testing the degree of accuracy of calculators, like "low accuracy", "normal accuracy", "good accuracy", "very good accuracy", "extremely good accuracy", etc. I am thinking about the number of digits that can be shown in a scientific calculation of a certain calculator and from there determining the degree of accuracy of that specific calculator. I think a calculator that can display 15 digits (including guard digits) after the decimal point in a calculation is more accurate than a calculator which can only display 10 digits after the decimal point. The problem is finding a method to test it. Please help me!

holsetymoon wrote:My hypothesis is:
Does the price of a calculator affects its accuracy?
I am thinking about the number of digits that can be shown in a scientific calculation of a certain calculator and from there determining the degree of accuracy of that specific calculator. I think a calculator that can display 15 digits (including guard digits) after the decimal point in a calculation is more accurate than a calculator which can only display 10 digits after the decimal point. The problem is finding a method to test it. Please help me!

The problem is with the true answers. They would need to be chosen with care. Let me give you an example. Let's go back to the 2+2=4 problem. The correct answer is 4.

A cheap calculator with one decimal place will return the value of 4.0. It is 100% accurate.

A slightly better calculator with 4 decimal places will return a value of 4.0000. It is also 100% accurate.

A really big calculator with 16 decimal places will return a value of 4.0000000000000000. It is also 100% accurate.

Now, if you use an infinitely repeating answer (e.g. PI), you can find an inherent accuracy difference which is solely based on the number of digits reported. But what about going to Wal-Mart and buying a discount calculator for $10.00 that has 8 decimal places, and then going to Sharper Image and buying one for $20.00 that has only 4 decimal places (but has a neat bottle opener, laser pointer, corkscrew, and emergency beacon.) This would be comparing apples and oranges. Clearly cost has multiple independent variables that would act as confouding variables for your problem.

Hopefully this will give you some hints into the challenges that you will face.

holsetymoon wrote:My hypothesis is:
Does the price of a calculator affects its accuracy?
Well, I know that price (money) is not a very objective factor in science, but I cannot think of anything else. If you can think of a better hypothesis, tell me and I will consider changing it. By accuracy, I mean "Accuracy is the degree of conformity of a measured or calculated quantity to its actual (true) value". I know that calculators are accurate most of the time, but I am testing the degree of accuracy of calculators, like "low accuracy", "normal accuracy", "good accuracy", "very good accuracy", "extremely good accuracy", etc. I am thinking about the number of digits that can be shown in a scientific calculation of a certain calculator and from there determining the degree of accuracy of that specific calculator. I think a calculator that can display 15 digits (including guard digits) after the decimal point in a calculation is more accurate than a calculator which can only display 10 digits after the decimal point. The problem is finding a method to test it. Please help me!

Another thing to consider is most calculators carry more digits than they display. So, for example, all calculators you test may be carrying 15 digits, but one may show 4 digits (rounded) and the other 10 digits. However, the actual number they calculated may be identical. As long as the rounding is correct, I don't think you can argue one is more accurate than the other, since the display precision is different.

I agree with the other posters that this is unlikely to be a good experiment. I believe you should consider a project that does not look at the accuracy of calculators.

Louise

Louise wrote:

Another thing to consider is most calculators carry more digits than they display. So, for example, all calculators you test may be carrying 15 digits, but one may show 4 digits (rounded) and the other 10 digits. However, the actual number they calculated may be identical. As long as the rounding is correct, I don't think you can argue one is more accurate than the other, since the display precision is different.

I agree with the other posters that this is unlikely to be a good experiment. I believe you should consider a project that does not look at the accuracy of calculators.

Louise

Okay, I found some links about accuracy of calculators which provides some leads...
http://www.calculator.org/weblog/PermaL ... b05c5.aspx

http://www.rskey.org/~mwsebastian/miscprj/forensics.htm

http://www.hpmuseum.org/cgi-sys/cgiwrap ... read=91383

Note that there are apparently some issues with the testing... these are intended to provide a starting point for you, not just a procedure for you to follow. You need to check and make sure this is an appropriate method.

Louis

Hi holsetymoon!

Another webpage is:

http://www.madsci.org/posts/archives/de ... .Cs.q.html

which describes an 11th grade science fair project in Alabama in 2001 to test calculator accuracy.

There is an interesting footnote on one of the references that David posted that shows how to find out what the internal (not diplayed) number that the calculator has calculated (a point that Louise brought up.)

http://www.geocities.com/SiliconValley/ ... eostus.htm

I still think that the experiment will prove to be too simple (it gets hard to justify an science fair experiment that only takes 5 minutes to do), and cost really can't be one of the variables studied, but as these references show, there are ways to look at the accuracy of scientific calculators.

Hi experts,

I am changing my hypothesis to:
What brands of scientific calculators are most accurate?. Comment if you think this is still not a good idea. This site, http://www.geocities.com/SiliconValley/ ... eostus.htm , as zzzzdoc posted, has some interesting methods of testing the accuracy of calculators. But I still do not understand how the first method works :
"One way to test the accuracy of a scientific calculator is to perform the following formula and see when the answer turns out to be ONE (1).

Y =lim ( X / sin X) where X descends to zero.

If Y is 1 upon X = 0.01 then the accuracy will be classified ´low´.

Simular with X = 0.001, accuracy will be ´below normal´.

With X = 0.0001, accuracy ´normal´. X = 0.00001, accuracy ´good´. X = 0.000001, accuracy ´very good´. X = 0.0000001 or smaller, accuracy ´extremely good´. "

So if anybody can clarify for me how to test that method with TI-83, TI-89, Casio FX-500MS , or any TI or Casio calculators, that will be much appreciated. Thanks again. By the way, happy new year!!!

holsetymoon wrote:Hi experts,

I am changing my hypothesis to:
What brands of scientific calculators are most accurate?. Comment if you think this is still not a good idea. This site, http://www.geocities.com/SiliconValley/ ... eostus.htm , as zzzzdoc posted, has some interesting methods of testing the accuracy of calculators. But I still do not understand how the first method works :
"One way to test the accuracy of a scientific calculator is to perform the following formula and see when the answer turns out to be ONE (1).

Y =lim ( X / sin X) where X descends to zero.

If Y is 1 upon X = 0.01 then the accuracy will be classified ´low´.

Simular with X = 0.001, accuracy will be ´below normal´.

With X = 0.0001, accuracy ´normal´. X = 0.00001, accuracy ´good´. X = 0.000001, accuracy ´very good´. X = 0.0000001 or smaller, accuracy ´extremely good´. "

So if anybody can clarify for me how to test that method with TI-83, TI-89, Casio FX-500MS , or any TI or Casio calculators, that will be much appreciated. Thanks again. By the way, happy new year!!!

I still agree with zzdoc that this is not a good experiement. The "measurement" will take about 30 seconds. Your question is not a hypothesis, and there isn't really much science involved in this. I suggest you re-read the Science Buddies information about "How to Do a Science Fair" starting at:
https://www.sciencebuddies.org/mentorin ... thod.shtml

I'm not sure what you mean by "testing the method" in the procedure above. There isn't anything to test... just plug some numbers in the calculator. I am assuming you are confused by the notation lim(x/sinx) just means you are looking at the limit as x goes to zero. There is an analytic solution to this equation that you can calculate just from the formula given- this is from calculus, so don't worry about that. [If you are interested in this part tho, feel free to ask here again.] Basically, the calculators do these things numerically, not analytically, so they can get the incorrect answer, which is what this page says happens.
If you read below the formula (and ignore this equation) you see the procedure is simple- All you do is calculate 0.01/sin(0.01). If the answer is 1, the calculator has "low" accuracy. You continue through the values of X (0.001, etc) until you find where it equals 1. As I said, this will take about 30 seconds to do, and is not a very scientific project. You are just measuring a property of the calculator, not testing the effect of something on some other property. You may want to explore the topic selector or the project ideas page (see the links at the top of this forum) and see if you can find a project that might be a little more involved.

Happy New Year to you too.


Louise