Category Archives: Statistics

10,000 step challenge

This year I asked for a FitBit for my birthday.  (For those of you that don’t know a FitBit is a pedometer, counting your steps, flights of stairs, daily active minutes, and approximate number of calories burned).  I was excited and curious to clip on my FitBit and see just how far I was walking every day!

But, after a few, short days I was a little confused.  I thought that about 2,000 walking steps = 1 mile, but I was getting FitBit read-outs on my phone that looked like this:

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So, if you’ve been reading this blog for any amount of time, you can probably guess what I did next . . .Yep, I Googled the length of a walking step and discovered that this website (which seems legit to me) estimates that the average length of a person’s walking step is about 2.5 ft, which means that in order for a person with average walking steps to walk 5 miles, they’d have to take 10,560 steps . . . not 10,000.

Then, I started wondering how long my steps were (on average of course); compared to the published average of 2.5 feet/step.  I used my FitBit output for 3 different days

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and discovered that, despite my relatively short legs my walking stride length was pretty average!

Then, I started thinking about a project I used to have some of my students work one, which is now an activity on the NCTM Illuminations Site, called Walking to Class.

This summer, make a walking strides chart of your day (or a trip to and from the park, pool, etc.) but instead of measuring distance in steps, change the units from steps to miles using the average 2.5 foot length, or you could dust off a pedometer and calculate the length of your actual stride!

 

In like a Lion; Out like a Lamb?

This morning as I was walking to work through whipping wind and freezing cold temps, I thought to myself “March, you’re supposed to be going out like a lamb.”  Goodness knows it sure came in like a lion!  Then, I started thinking about Groundhog’s Day and how unreliable that good old Phil actually is!  That made me wonder . . . is there really any truth to this whole lion/lamb thing?

I spent the better part of my morning trying to track down an answer!

This is what I did:

I’m only concerned with how March comes in and out, and not what happens in the middle of the month, so I thought I’d look at the first 7 days of March and the last 7 days of March.  Then, I did a quick search and found that the average March temperature in Iowa for the past 150 years is 34.5 degrees Fahrenheit.

Given this information I decided I would define “Lion” to be a 7-day span in which 4 or more of the days had an average temperature that was less than the average monthly temperature.  Then, a “Lamb” was a 7-day span in which 4 or more of the days had an average temperature that was greater than or equal to the average monthly temperature.

I obtained daily average temperature data for Des Moines from The University of Dayton Average Daily Temperature Archive dating back to 1995.  Because the month of March isn’t over yet for 2014, I didn’t use the temperature data for any of the days in March 2014.  This is what I came up with:

Screen Shot 2014-03-26 at 12.05.03 PM

In the past 19 years March has followed the “In like a Lion; Out like a Lamb,” pattern in 13 different years (or 68% of the time).  It has followed an “In like a Lamb; Out like a Lamb” pattern 5 different years (or 26% of the time).  And once in the last 19 years March has come in like a Lion and gone out like a Lion . . . according to the average daily temperature in Des Moines.

This got me thinking about a few things:

1. This saying seems to be a little more accurate then the Groundhog shadow thing.

2. There weren’t any “In like a Lamb; Out like a Lion” years . . . I wonder how many times that has happened in the past 150 years (if at all)?

3. What do you think of my definition for Lion-like weather and Lamb-like weather?  Would you define it another way?  If so, how?

Don’t worry the groundhog isn’t right anyway . . .

This year my two oldest children really got in to the whole Groundhog Day thing . . . so much so that when they heard on the Today Show that Punxsutawney Phil saw his shadow, signaling 6 more weeks of winter the oldest insisted that there be a “redo,” because he was not going to endure 6 more weeks of winter.

See?  They still have fun in the winter!

See? They still have fun in the winter!

“You guys,” I said.  “Just relax.  The groundhog can’t really predict the weather.  It’s just something fun to do on February 2 every year.  It’s just folklore.”  (My son is currently learning about folk stories in 2nd grade, so I thought this might sell my explanation!)

“How do you know mom?”  (that was my kindergartener, not convinced and not happy about the groundhog’s prediction).  “Maybe he’s right and you’re just saying its folklore so we won’t be sad.”

I don’t really remember how the rest of the conversation went, mostly because there wasn’t any use in arguing with either of them and also because by then, they had completely lost interest in the conversation.  But that kindergartener, she got me thinking.  How often is the groundhog right?  Does he predict the end of winter enough that there might actually be something to this whole “groundhog sees his shadow” thing?

U.S.A. Today wrote an article about the groundhog’s predictions and the National Weather Center also has a thing or two to say about how accurate the groundhog actually is; they report the groundhog has been wrong 15 times and right 10 times.  But here’s the thing . . . according to both websites the groundhog either predicts 6 more weeks of winter or an “early spring.”

Who decides if spring has come early?  How do we know if winter has lasted for 6 weeks?  Is this measured by the air temperature?  The amount of snow on the ground?  The vernal equinox is the “official” start of spring, but clearly that’s not what people are talking about when they refer to the groundhog’s prediction . . . otherwise, no prediction would be needed!

I suggest we get a little more precise about what we mean by the groundhog’s prediction, before we decide whether his predictions are accurate or not.  I propose that if the temperature for half or more than half of 6 weeks after February 2 (so that’s the week of 2/2, 2/9, 2/16, 2/23, 3/2, 3/9, 3/16) has an air temperature at or below the national average for the last 100 years that counts as 6 more weeks of winter; otherwise, spring.  According to this definition of an “early spring” Phil has correctly predicted the weather for the next 6 weeks, 11 times over the course of the last 26 years.

But, I agree spring is much, much better!

But, I agree spring is much, much better!

If we recognize this as an example of binomial probably, we know that about 30% of the time we would have been able to be just as accurate as Phil by flipping a coin where “heads” is early spring and “tails” is 6 more weeks of winter.  I’d say that qualifies as a folktale to me, wouldn’t you?

Math Dice Game

My mom bought me this little game called “Math Dice” for Christmas this year.  Have you heard of it?  I hadn’t, and truth be told, I’m not sure my mom realized it was a game she was buying it for me.

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When I opened the package she said “I thought you could do something creative with those dice and your math blog.”  In the days after Christmas, I scooped the unopened box of dice into our “junk drawer” (sorry mom) and rediscovered them this weekend while cleaning.  On the back the of the box were the directions to the “Math Dice Game.”

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This morning I had a little extra time, so I thought I’d give the game try!

Step One: Roll the 12-sided target dice

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Step Two: Roll the three 6-sided scoring dice.  Combine the three scoring dice in anyway to match or come closest to the Target Number.

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Ummm, really?  1, 1, 2?  The closest I could get to 80 was 24.  This is what I did:

(1+1+2)! = 4*3*2*1.  Can you get closer?

My second roll of the Target Dice was 36

My Scoring Dice roll was 6, 4, 2

Super easy: (6*(4+2))=36.  Did you get 36 another way?

My last roll was 20

And my scoring dice were 6, 3, 1

I couldn’t get 20.  I could get 18 and 21, but 20 right on the money was a little tricky.  Can you do it?

Much, much more to come about this fun Math Game, with my new Math Dice . . . I’m working on a table of possible Target Number combinations as we speak!

M&M’s Revisited (for the Last Time!)

If you haven’t been here before, then you don’t know that we’ve already talked about M&M’s twice (here and here) and you don’t know that we’ve talked a little bit about the colors of M&M’s in the bags.

Well, today I want to keep talking about the different colors of M&M’s in the bags.  Except today I want to talk about the percent of M&M’s that are red, orange, yellow, green, blue, brown.  Before we continue our M&M discussion, do you have a guess?   That is, what percent of the M&M’s manufactured are red, orange, yellow, green, blue, brown?

Hmmm . . . Let’s pretend that we don’t know (maybe you really don’t!).  I think a pretty educated guess would be that 16.67% of the M&M’s are red, 16.67% of them are orange, 16.67% yellow, etc., etc.  Can you live with that guess?

I’m going to use the data I collected in my last M&M post, except instead of individual bags I’m going to look at my entire sample of M&M’s.

Here’s the percentage breakdown of M&M’s:

Screen Shot 2013-11-26 at 9.52.31 AM

Let’s make a nice table, based on what I would expect to get, given my educated guess of 16.67% of each color and what I actually got:

Screen Shot 2013-11-26 at 9.52.41 AM

So, I wonder if the distribution of colors I got in my sample would be likely, if the colors of M&M’s really were distributed evenly at the manufacturer?

Luckily for us there’s a statistical test we can use to answer that exact question.  And, luckily for us its a pretty straightforward test to understand!  It’s called the Chi-Square Goodness of Fit Test.  The Chi-Square Goodness of Fit test compares the observed values (in our case my M&M colors) to the expected values (if our initial assumption was true).  In our case we would subtract the expected value from the observed value and square the difference.  Then, we would divide by the expected value.  We’d do this for each color of M&M and add up the results.  Don’t worry, I’ll do it (actually, I did it with the help of this website). . .

Based on the Chi-Square Goodness of Fit Test it’s fairly reasonable to assume that I could have gotten this distribution of M&M colors given the fact that M&M Mars makes 16.67% of each color of M&M’s.

Screen Shot 2013-11-26 at 10.10.29 AM

So, here’s my next question?  Do they?

(So here’s the thing, about 5 years ago the M&M Mars website used to answer this exact question, but in 2008 they stopped.  This person wrote to M&M’s and posted the response)

Use the distribution for Milk Chocolate M&Ms detailed by M&M Mars and run another Chi Square Goodness of Fit Test with my data (or your own, if you collected any).  How does this compare to the 16.67% guess?

 

Assigning ZIP Codes

Last week I was working on getting a document together that involved typing many, many ZIP codes from across the United States.  This particular document involved looking up addresses for approximately 350 locations and after a while I realized that I was getting pretty darn good at accurately predicting what the first digit of the ZIP code was going to be and vise versa (i.e. if I looked at the first digit of the ZIP code I could guess the location within a few states).

As I was collecting this data into my spreadsheet, I was developing a hypothesis . . . the first digit of the ZIP code is directly related to the year a state joined the union.

Remember, directly related means as the year the state joined the union increases the first digit of the ZIP code also increases.  In other words, the first digit of the ZIP code depends on the year the state joined the union.  To test my hypothesis I used a map of the U.S. and wrote in the first digits of the ZIP codes I knew.

And then, I created a table of values with the same information (X means I didn’t have the ZIP for any location in that particular state, not that a quick Google search couldn’t have helped me find it, but I just didn’t have it in the document I was working from–also, if my hypothesis proved correct I likely wouldn’t need it!):

State ZIP Year of Statehood
Delaware X 1787
Pennsylvania 1 1787
New Jersey 0 1787
Georgia X 1788
Connecticut 0 1788
Massachusetts 0 1788
Maryland 2 1788
South Carolina 2 1788
New Hampshire X 1788
Virginia 2 1788
New York 1 1788
North Carolina 2 1789
Rhode Island X 1790
Vermont 0 1791
Kentucky X 1792
Tennessee 3 1796
Ohio X 1803
Louisiana 7 1812
Indiana X 1816
Mississippi 4 1817
Illinois 6 1818
Alabama 3 1819
Maine 0 1820
Missouri 6 1821
Arkansas 7 1836
Michigan 4 1837
Florida 3 1845
Texas 7 1845
Iowa 5 1846
Wisconsin X 1848
California 9 1850
Minnesota X 1858
Oregon X 1859
Kansas X 1861
West Virginia X 1863
Nevada X 1864
Nebraska X 1867
Colorado X 1876
North Dakota X 1889
South Dakota X 1889
Montana X 1889
Washington X 1889
Idaho X 1890
Wyoming X 1890
Utah X 1896
Oklahoma 7 1907
New Mexico X 1912
Arizona 8 1912
Alaska X 1959
Hawaii X 1959

And I made a scatterplot:

11.19.11

So, I’m going to go ahead and say that my hypothesis was not overwhelmingly correct.  It looks like the year the state joined the union may be related to the first digit of the ZIP code, but clearly my theory has some flaws.  For example, look at the first few entries in the table.  States joining the union after the first few states have ZIP codes of 0, 1, 2!

Ugh.  Then, you know what I wondered about.  Would there have been a need for ZIP codes (i.e. a post office) when the first 13 colonies became states?  In fact, when did the post office start using ZIP codes anyway?  Well, I found my answer . . . 1963.  Yes, really.  1963.

Goodness Gracious.  All 50 states had joined The Union by the time the use of ZIP codes was implemented.

This experience made me think about two things:

1. Just because two variables are correlated, doesn’t mean that one causes the other.

2. I wonder what a better predictor of ZIP codes would be?

Reblogged from The Atlantic: The Myth of I’m Bad at Math

Today I’m reblogging a post I saw in The Atlantic this morning.  I’m reblogging it because I think the authors, Miles Kimball and Noah Smith, do a great job of articulating a sentiment that I think many, many mathematics teachers share.

(And because they talk about Terence Tao, and any true Math Warriors fan knows his importance in the world of mathematics!)

Enjoy.