## Linear Regression by the Median of Slopes

I have been trying to come up with a lesson about linear regression that involves more than pushing a few buttons, like on the TI-8ish, or using sliders in Desmos. I tried to search the web for lessons of other people but I could not find what I was looking for. Then I came across a method of finding the line of best fit called Theil–Sen estimator. Here is the method.

As defined by Theil (1950), the Theil–Sen estimator of a set of two-dimensional points (

x,_{i}y) is the median_{i}mof the slopes(y−_{j}y)/(_{i}x−_{j}x) determined by all pairs of sample points. Sen (1968) extended this definition to handle the case in which two samples have the same_{i}x-coordinate. In Sen’s definition, one takes the median of the slopes defined only from pairs of points having distinctx-coordinates.Once the slope

mhas been determined, one may determine a line through the sample points by setting they-interceptbto be the median of the valuesy−_{i}mx._{i}^{[8]}As Sen observed, this estimator is the value that makes the Kendall tau rank correlation coefficient comparing the sample data valuesywith their estimated values_{i}mx+_{i}bbecome approximately zero.

I really like this idea because it reinforces a lot of procedures of linear equations. Here is how I might do the lesson. A link to the entire Desmos graph is here.

First give the students data and have them plot it with Desmos. This data is the annual gross ticket sales (in 100’s of millions) where x=0 for 1995. Using the table feature in Desmos is great.

Next I would have students find the First Order Differences and plot these on the same graph. We would have a discussion about what these values mean and also talk about how these are approximately constant so a linear model would be a good fit.

Next we would begin finding the median slopes. We might begin by asking how many different slopes could be found between 17 points. Obviously, we would not find them all so we would assign a certain amount for each student to find. Then we would gather up all of those slopes and plot them in Desmos. This should be a great visual example to see the outliers of slopes within the data. (For this example, I only found 10 different slopes. Also, note that the first oder differences could be used as slope values. Those slope values are for consecutive points.)

Then we can discuss what “average”, (mean, median, mode, midrange), we should use to find the “average slope”. In Desmos, finding the median slope is easy. Click on the top line, then hide it. Click on the bottom line, then hide it. Click on the new top line, hide it. Click on the new bottom line then hide it. Continue doing so and this will result in the median slope. Here is a picture of the final two.

We can also plot that median slope with the first order differences. This could bring up a good discussion about do we really need to find other slopes or could we just use the first order differences to find the “median slope”

Next we can go back to the table and find the median y-intercept. In the Desmos table, we will make a column of values that is the expression y-(median slope)x. We can also plot those points to show what the y-intercept would be for each data point. Here is that graph.

Now that we have all of those different y-intercepts we can use a slider to estimate the median y-intercept value. We could also throw the values into a spread sheet if we wanted, but I think the slider will be good enough. I made the slider have a lower bound of 4 and an upper bound of 5. The b value ended up being 4.554.

Finally, we are ready to plot the line of “median fit.” using the equation y = (median slope)x + (median y-int)

For only using ten different slopes, I would say that the line looks pretty good. However, the data did a have a strong correlation to begin with. I have not compared the “median line” to line of least-squares because I think that would be a good follow up. I think this method goes into the heart of regression. Students get to see how many different lines are used to find the best line. Student review stats concepts and how outliers impact different averages. Students are creating a lot of evidence for their model, instead of just relying on the “r-value”.

One other thought would be to have student’s create an error region for the model. This might help them understand ideas of interpolation and extrapolation. Plus, it might allow us to discuss standard deviation, too. In the graph below I graphed {median slope(x) + 1.15(median y-int)} and {median slope(x) – 1.15(median y-int)} to create a 15% above and 15% region. I could have found the standard deviation of the median b value and done three standard deviations above and below.

The more I explore this concept the more it seems like it turning more into a statistical analysis. I need to determine if that is the route I want to go on since the class I am developing this for is “Math Modeling” course.

I hope all of this gives you some ideas about linear regression. I have not designed the lab sheet that will go with this yet. I would love to hear feed back if you have any.

## Your Homework: Play Temple Run

Summer is here. Yippee for me! This means that I can get back to blogging some of ideas. So, here is my next idea, The Math of Temple Run.

Have your ever played Temple Run? Probably, yes, given the game’s popularity. If not, here is information about the game. http://www.imangistudios.com The game is free to play. Did you catch that – FREE! But what is better than being free is that Temple Run has a lot math problems waiting to be explored. Here are some that I will be trying out.

- How fast is the person running? The game keeps track of how far you run in meters. This means that you can time how long the person runs and then calculate the speed.
- Is the person’s speed possible? I won’t spoil it for you but you need to see how fast the person runs.
- The person runs faster as the game progresses. Students can make a chart of distance and time because the game flashes up the distances as you travel. Does the increase of speed follow a quadratic pattern or is it more like a piece-wise linear?
- Next have students play the game and record 15 rounds of data. Here is a link to the table of data that I made. https://www.desmos.com/calculator/barud9egsn
The data looks to have a strong linear correlation, which allows us to explore rates of change. What does the rate of change mean for the blue dots and what does the rate of change mean for the red dots? Are certain games better than others? Is a better game based on the distance? Is a better game based on the amount of coins? Is a better game related to the number of coins compared to the score?

- In Desmos we can use a slider to create a line of best fit. The ones I made were y=(coins/dist)x and y=(score/dist)x I will probably discuss with students why we can make the initial value of zero. (Actually, one of the goals in the game is to go 1,000 meters and get zero coins.) Next, with the slider values we can either make the numerator or denominator equal to 1 and adjust the other slider accordingly. Once they have the line of best fit, we can talk about what it means for the data points to be above/below the line of best fit.
- Remember that each student will be playing the game, hopefully. So, this means we will have many different graphs. This will allows us to talk about how you can look at a graph and say, “That is good player. This is a so-so player. Etc.” or to be able to look at graph and point out who has had more experience playing the game.
- How is the score calculated? After students record the amount of coins, distance, and score they will have some data to try and figure out how the score is calculated. The game does not tell how the score is calculated. All you see is a running total in the upper right corner. Wikipedia does give a formula for the score. http://en.wikipedia.org/wiki/Temple_Run However, that formula was not working for me. I might doing some type of error. But this is good because it gives students the opportunity to figure it out. Plus, the formula opens up the door to the floor function and ordinal numbers.
- I might throw in Game Theory at the end. Not quite sure.

Plenty of stuff to work on there. I have not made the worksheets yet but I will update the post soon. If you could, please play the game 15 times and record your data into this Desmos graph https://www.desmos.com/calculator/b2bycwrenn. Save your work and then post it here in the comments or on twitter @LukeSelfwalker. I will gather up the data and post it later. Thanks for your help.

## Standard Deviation of Sunset and Sunrise

Thanks to Kate Owens (@katemath) for sharing a great graph and saying the phrase “standard deviation” in a tweet. I was going to teach standard deviation to my Survey of Math class the next day, so this was perfect timing. The following is how I used the graph in class. Note that in description of the graph, the standard deviation is given. This was not shown to the students until later. http://visual.ly/daylight-saving-time-explained

## Dartboard and Equations

While reading Dan Pearcy’s blog entry about Rich Questions I could not agree more about his statement

I love it when people share rich questions. Generally because I find that I can’t keep track of all the resources which come my way but I do seem to remember great questions.

For me, piling up resources is so easy to do, while actually implementing them is the difficult part. So, I am trying a new strategy next semester. Instead of having the “oh this would be a cool problem and oh this would be a cool problem too, oh-oh-M-G and I have to do this”, which ends up in only doing a few problems, I will focus on one problem and see how much I can get out it. Can a whole chapter be focused around one problem? I don’t know but if I find one, it will be placed in the “Rich Question” Category.

After seeing Dan’s post about linear equations and shapes being shaded, I wondered if the two could be combined. Here is my idea. First, I would have students look at this graph made in Desmos and ask the basic question, “What part is shaded in each square?”

(Click on the above picture to see graph at desmos.com.) After we agree that 50% is shaded in each region, I would ask the main question of the project.

If each square is an individual dartboard, then which square is the easiest to hit a shaded region? Which square is the hardest to hit a shaded region?

Hopefully, this would hook their attention to the project. Next, each student would create four different dartboards. One dartboard they think is too easy, one dartboard that is too hard, and one dartboard that is just right (fair). The fourth would be one that is a cheat – that is it looks like 50% shaded but actually it is less. Once each student completes their four graphs they can go around assessing each other graphs with HARD, EASY, FAIR, and CHEAT. Side note: The student that gets aways the least amount of shaded region should earn extra points. This would give more of an incentive to create unique ways of shading 50% of a square. Finally, the general case be examined. How should one shade 50% of a square so that it is the easiest/hardest dartboard?

Where are the linear equations? The students would create the dartboards in Desmos by inputing linear equations. For example would graph the linear equation x+9 with a domain restriction -9 to -1. I think this could provide some good opportunity to solve equations, create equations, and practice domain restrictions. Of course the shading part, which uses a dampening function, would have to be explained. But once that is shown, that would create a bridge of transformations, a concept that seems only start with quadratics and not linear. Also, students might find a reason to know how to find the distance between two points.

Overall, this is still really fresh in my head. There seems to be lots potential I have heard that less is more but with so much great stuff on the internet, focusing on less is tough. But, I think finding a Rich Question can help me do more with less. If you land a bullseye with a Rich Question then let me know. Thanks!