Just how silly is a 45-day weather forecast? And while we’re at it, just how good is a 2-day forecast?



I’ve got bad news for everyone here in the Fort Collins area: there’s a chance of showers on Labor Day (September 2). It’s a shame because the following Monday (September 9) shows a mostly sunny day, with a high of 77 degrees. (I am writing this on August 8.)

Seriously, AccuWeather? Yes, AccuWeather is releasing a 45-day weather “forecast”, although it’s actually just more of a trendline (it will be colder in October than it is now in August, probably a safe bet). It will predict the highs, lows, and chance of precipitation up to 45 days in advance. Presently, I cannot quantify how terrible these forecasts will be, but they will be terrible.

So that’s the silly portion of this blog post. To me, the more interesting questions are as follows.

Guiding Questions

  • Is a 2-day forecast error prone? What about a 7-day? At what point to weather forecasts become statistically useless?
  • Is there a regional difference in weather forecast error?
  • Which weather forecast models are more accurate for your region?

Suggested Activities

  • Solicit predictions. How accurate do you think weather forecasts are? Do you think a 2-day forecast is appreciably more accurate than a 5-day forecast? Do you think it depends on the region you’re checking (i.e. does, say, a dry climate have more error than a wet climate? do the coasts have more forecast error than the plains?)? Can we quantify any of this?
  • Start collecting data. Keep track of the 2-, 3-, 5-, 7-, and shoot, the ridiculous AccuWeather 45-day forecasts. Compare it to the recorded data (you could stick a max/min recording thermometer out your window, or check archived data). Grab the highs and lows, and possibly recorded rainfall. Yes, the 45-day forecasts won’t start paying dividends for a couple months, but that’ll be some pretty rich data (“rich” has a couple different meanings in this case) to have collected. And thankfully, if you’re teaching Algebra 1, a lot of curriculum maps out there don’t have the “Statistics” unit until late in the year.

How to collect the data? Well, Google Forms makes it nice and easy to record data. See, look, I made a Google Form for you and your students to use. The only tricky part will be some spreadsheet column manipulation, correlating the forecasts with the appropriate weather data collection dates.

  • Develop a method to calculate the error of the forecast. Depending on the grade level, your class could certainly develop an error model based on whatever sophistication you feel appropriate. Maybe an elementary class will just take the differences between the forecasted and collected highs and lows. Maybe a middle or high school class will develop something more sophisticated. Something like Mean Square Error should do nicely.



  • Compare model to model; compare region to region. Which weather model was the most accurate for your region? What if you just took the average of all the forecast models? Would that result in a better prediction? Some forecast sites to check:
  • Toward the end of the year, have students begin making their own Highs, Lows, and Precipitation prediction. By the way, these competitions happen all the time in the atmospheric science community. Maybe after some year-long analysis, your last couple weeks are spent gaming around students making their own predictions. They can look at all the forecasts and make a judgment. There are several ways to “score” it, some are more intense (scroll down to “scoring”) than others. Like collecting the data, Google Forms could be helpful.

If you’re looking for a long-term project that all students can access and be involved in (low barrier of entry, high ceiling), you could do worse than tracking weather forecast accuracy.

Teacher’s Edition

I’ve got some news that will probably shock you: 2- and 3-day forecasts are actually pretty good. And they’re getting better.

But this is me telling you that things are getting better and just citing some peer-reviewed research. That’s not necessarily what we do here. This is where you and your students could potentially come in.

For more on weather and weather forecasting, NCAR/UCAR has some excellent resources.

Update 8/9/2013:

Huge thanks to Frank (@fjvitale) for finding some graphs that better show improvement in TEMPERATURE forecasting. Here’s one that shows the error of MAX temperature forecasting for several different day forecasts since 1972.

You can find more forecasting verification graphs here.

(More of my posts on weather and climate.)

What? How do YOU spend your two-hour school delays?, Water Content in a Snow Cylinder

As anyone in town for NCTM in Denver know, it’s been a bit snowy here this week. In fact, Fort Collins just had its biggest snowfall of the year. But how big?

We had a two hour school delay this morning as my daughter and I were greeted by this on our back doorstep.


“Wow that’s a lot of snow!” she says. But how much snow is it?  Go go gadget EmergentMath!


I got this ridiculously large [cola] mug at a white elephant gift exchange last Christmas. And now I have a chance to use it!


I asked her to make a prediction on how full the mug would be after it melted. We each made a prediction using her hair ties (hers on top, mine on bottom).


We took a couple measurements just for posterity’s sake.



I dunno, we might want them later. For now though, we just stuck with the predictions.

We then watched it melt. Slowly.

Sure enough, we were both way off:

013 015

Wow. All that snow and only that much actual moisture. I have some questions:

  • Is this typical? What if we redid this in the afternoon after the snow had packed a little more? 
  • What if we used different shapes? Could this be a sort of alternative to the how-full-is-the-weirdly-shaped-glass problem?
  • Going back to the original photo, how much water was on that table?

I also have a couple comments:

  • Want an easy way to build buy in? Have kids make predictions on something and make sure it *takes a long time* for them to see if they’re right. Like I said, our delay was a couple hours and this pretty much took up the entire time. This was sort of analogous to Dan Meyer’s now-famous water tank filling task.
  • This seems ripe for Estimations 180.
  • I’m not sure what you could do if you live in a non-snow state. What would Texas use? Sand? Cicadas?

My daughter and I could have gone into the volume of the near-cylinder, which dimensions were useful and that sort of thing. But our two hours were up. It was time to go to school.

Update 4/16: I’ve got my Facebook friends eating out of the palm of my hand. *maniacal laugh*



Sort of related: a couple atmospheric scientist friends of mine started a Facebook page crowdsourcing, archiving, displaying, and discussing clouds: Community Cloud Atlas 

You should join their Facebook page and tell them to get a twitter account.

How does one provide the complex data of global warming to students?

Update (3/12/2013): An atmospheric scientist friend of mine, Katie, suggested a few edits to this post, primarily to clear up a few of the tools listed here. The edits are in bold.

My initial thesis on this post was originally going to be “why don’t teachers let students investigate global warming very often?” While this may not answer it here’s a terrifying google search for any teacher who is interested in having their students do some independent research on climate change. Google: “global warming raw data“.


So the first result is a good one. A legit one. There are lots of links to reputable sites maintained by reputable scientists. Then the second result is a yahoo! answers post. The the third (third!) google result for a simple query on raw data turns up World Net Daily, a website for conspiracy theorists and people that think they’re going to be put in FEMA camps any day now. That is not a reputable site. They provide the opposite of “raw data”.

This is not a post about the messy politics and confusion-campaigns around climate change. But this does point to a particular difficulty that you’d hope would be much simpler: where can we find raw temperature data that we can actually use? For the record, a google search of “raw temperature data” yields much more acceptable initial results. But still, many of those results can be extremely difficult for a secondary math or science teacher to pick up and use, let alone students. For one, climate data is often presented in a file format that requires heavy coding knowledge or special programs to process (such as NetCDF). Second, it’s hard to know where to start with temperature data. Do you start by geographic location? Do you take the annual mean across the globe? How would one do that, exactly?

So this is the problem, and maybe a fundamental problem of teaching science: data are messy. We have to rely on others to package it for us. Scientists are interested in providing the raw data because they want people to have access to true observations, but that raw data is so vast and difficult to process (but not that difficult to interpret!) you have to get at least a Master’s degree before you can even start to decipher it. And often, scientists aren’t interested in culling the data to make it more digestible for the public. They’d prefer to show you the graph. This is great for communication, but not great for independent research. And worse, they’re now fighting on the same plane as disingenuous charlatans who are paid to be as such. So let’s provide students of science the raw data in a way that anyone with Microsoft Excel and a genuine curiosity can begin to explore the very real phenomenon of climate change.

My favorite site that does that is this NASA’s GISS Surface Temperature Analysis. In terms of accurate, raw, commentary-free, accessible, customizable, and processable data, I haven’t found a better place to start. Bookmark that site. Tell your students to go to that site. Start locally.

To find specific historic local weather stations, Katie recommends using the map rather than the search function. The map appears to have better functionality. So click on your favorite vacation spot and go find that precious, precious raw data.



Once you have the ASCII data (shown here), it’s simply a matter of copying and pasting it into Excel, or if you’re incredibly ambitious (or teaching a Stats class perhaps), having students import it into R, one of the industry standards.

For the uninitiated, let me translate a few things: 

D-J-F= December-January-February average

M-A-M=March-April-May average

J-J-A, S-O-N = I think you get the idea….

The last column, metANN = annual mean temperature. This actually might be the best first place to start. 

Berkley also has a nice data set organized by country. However, the accessible to-layperson data is a bit more hidden.


If you’re not careful, you’ll end up downloading intense, non-accessible-to-the-layperson, NetCDF data. Which, again, is fantastic data, but difficult to work with yourself.

But now we’ve got two sites with data that can be tossed into Excel, R, or even those statistics packages designed for secondary students. Now that we have that data, we can do a lot with it.

Suggested Activities

  • Have students investigate the temperature trend in their area.
  • Create a linear model that predicts temperature as a function of year locally.
  • Assign each group or student a different region of the world to investigate and develop a linear model for.
  • Or what about this: develop a sinusoidal equation that describes monthly temperature. Get some trig in there.
  • Ask the question: is our town/state/country/planet heating up or not? Or is it too uncertain to tell?
  • Can you find local stations that DON’T show a warming trend? Katie suggests looking at weather stations closer to the poles to consider the potential impact of polar temperature trends. This might be a bit science-y, but it’s something I’d happily let students explore in a math class.

Once you have actual data, you can start to test it to assess that last, fundamental question (which then spurs thousands of other questions, like “should I have children?”). Is ß>0 under the general linear model? Once we have that answer, even if it’s just locally, we can start to talk about the implications.

Based on this plot, can you tell me when it started snowing? Or, can we fit a sine curve to it?

Now, I’m biased of course since I got a MS degree in Atmospheric Science, but I feel like weather and climate is one of the biggest untapped wells of potential math problems. I think there are two reasons for that (aside from the “lack of expertise” / “it’s not in the textbook” angles):

1) The math involved is often considered “too advanced” and/or involves copious amounts of data.

2) There is significant amounts of uncertainty involved.

If I’m right, then this is troubling. It’s troubling that either of these aspects of weather and climate would make people shy away from using it as a tool to promote mathematical thinking in the classroom. Students should absolutely be exposed to and explore the concept of uncertainty. Ditto for data processing. I’d argue that the ability to assess uncertainty and/or large quantities of data is more relevant than the Pythagorean Theorem. Also, possibly more interesting. Consider this: would your students be more engaged by the good ol’ ladder problem, or by analyzing climate data or incoming cold-fronts?

Now, there are probably a myriad of other reasons that weather and climate, which is such an important part of all peoples’ lives that we constantly engage in it conversationally, isn’t studied in the classrooms (it’s difficult to visualize or represent fluid dynamics on an atmospheric scale; the field is still relatively new and there are many open questions still out there). But don’t let either of the above “problems” with weather and climate science make you shy away from utilizing it, if at all possible. I’d love to see a math curriculum incorporate scores of atmospheric science … or an atmospheric science curriculum incorporating scores of math concepts.

/steps off soap box

OK, sorry about that tangent. Here’s your artifact for today.


This plot of temperature and dewpoint.

And, I dunno, maybe this picture too.

Guiding Questions

  • When did it start snowing?
  • What is dewpoint? And what does it have to do with snow?
  • That temperature plot looks pretty pattern-like until 10/25, what kind of curve could be fitted to that plot?
  • Can we use some sort of pattern-deviation to determine when “weather will happen?”
  • So how much snow is that? Wait, how do they measure snowfall totals?

Suggested activities

  • Based on the data from 10/21 go 10/24, develop an equation that represents the “normal” or “expected” temperature.
  • Use Weather Underground to find your local temperature history for a given time to do the same.
  • By the way, that can also be done to determine monthly average temperatures and dewpoints. Create a sine wave for monthly averages?
  • Convert the dewpoint plot to relative humidity – the preferred measure of atmospheric moisture content by the public.
  • After deciding when the snow began falling, go back and check your solution with actual weather reporting. (group closest to actual time gets free sno-cones?)
  • Based on that photo, estimate how much snowfall was measured (hint: snowfall totals are measured according to the amount that’s left after it’s melted).

Honestly, when it comes to weather and climate in general, I’d be interested to hear what students are interested in. Is it climate change? Tornadoes? Blizzards? All of these are going to incorporate aspects of data collection and uncertainty. That’s probably a good thing.