Explaining Fitts Law: doing the math behind it.

Kevin Lee
Kevin Lee
Sep 5, 2018 · 5 min read

I remember once for a test, I was asked to explain Fitts law once as a test question. This was my explanation:

“Fitt’s Law can predict the time it takes to move to a point based upon the size and distance to the target.”

So what does that all mean?

Let me explain.

So back in 1954, this guy named Paul Morris Fitts did an experiment to explain speed accuracy trade offs. he was interested at looking at creating a metric that would explain the difficulty of pressing a target.

but he basically made something that looked like this:

Source: https://www.yorku.ca/mack/hci1992.html

Basically, you had to move the pen back and forth as fast as you could between the two black lines. and vary the width of the lines or the distance. this is supposed to predict how hard and how long it would take for you to click an object.

So I tried the same experiment out myself that Fitts did those 60 something years ago. If you want to try it out too, hop on the dork express!

You will need:

a pointer (any tool that doesn’t change size from person to person cause fingers are different sizes so that doesn’t work as well)

Targets(paper works)

A stopwatch (your phone has one probably)

Math skills (Yes finally using them outside of school)

Excel (or any tool that can graph numbers)

My article (duh)

So here are two formulas that you need to use for this:

ID= Log 2 (A/W+1)

Where ID= Index of Difficulty, A= Amplitude, and W= Width.

And this Other formula to predict movement time: MT=a+b(ID)

MT= Movement time a= intercept from the experiment, and b=slope from the experiment. and ID is once again Index of Difficulty.

(Fitts came up with these formulas from experimenting, but we get to skip that hard part)

If none of this makes any sense, that’s okay, we are going to go through the experiment and learn together!


Figuring out the ID Index of Difficulty for your experiment

So the first thing you do is you gotta figure out how big or far your targets are gonna be. The Index of Difficulty is supposed to calculate for you how hard it is to move between your targets quickly. You do it by using their formula I chose to do the increase in difficulty from 1–5 for my experiment:

So ID= Log2 (A/W +1) rearranged = A/W = (2^ ID) — 1

so if you go from 1–5 (5 being harder) you calculate the width or distance of each thing by plugging those into ID and then keeping something constant so you only have to solve for 1 variable.

EX: 1= Log2(A/.5 +1) So A=1.5

I chose to keep the width in this case constant so I wouldn’t have to make multiple sheets of paper and I could just move them further apart. you can actually do this either way you want, These are the numbers I got from keeping my width constant and calculating A (or distance between targets):

1 stays at 1 because it was easier for calculating 1 difficulty. log 2 (2) = 1 or 2¹=2

Doing the Experiment

When you have those numbers, you can try out the experiment yourself! Get that pointer (mine was a paperclip) and set up your targets at those distances you just calculated! ( I just taped mine down and moved it every time we did a task) get your stop watch ready, and have your participants (at least 2) move between both targets a couple of times ( this is because you aren’t going to be too accurate when starting and stopping so it helps you average the time for one movement) I had mine go 15 times between the two and then just divided my time by 15 after they did it. and repeat until all of the difficulties have been completed.

These are my times for my subjects:

Just looking at the numbers you can see that as the Index of Difficulty goes up, so does the Movement Time.

So now what?

We do what a lot of researchers do with this kind of data, we slap it into a graph, and then make a formula from it. So I plugged my numbers into excel and graphed them out, and found a best fit line on it. this is what should come out of it.

Now the more people you test, the more accurate it will get so if you’re testing a new pointer device, this would actually be a helpful exercise.

So using that formula from before:

MT=a+b(ID)

Expand that out by plugging in the formula for ID:

MT= a+b(Log 2 (A/W+1))

And plug in the numbers from that graph y=mx+b (from algebra) in this case the m= b and the b=a so you end up with this:

MT=.1919s + .0625s(log 2(A/W +1)).

This serves as a prediction of the time it takes to move when using a paper clip.

Now you can predict how long it will take you to move from one point to another! WOW so interesting! You must be feeling pretty intelligent now huh?


Take away

So what was the point of this whole exercise?

Fitts law in a nutshell is that the bigger and closer your target is, the easier it will be for you to press, so if you have a big button on your screen, it’s easy to click.(and vice versa) “WHY DIDN’T YOU JUST SAY SO AT THE START KEVIN?”

I wanted you to see and appreciate the beauty of the math and science that goes into some of the design principles we take for granted every day.

This was used for things like designing those buttons that you and I use on apps all the time to help my fat fingers press. Now you have another hammer in your toolkit of design to support your choices! also if the math behind this was simple enough for you, it will make those 8th graders swoon over your algebra skills as well.

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