Monthly Archives: November 2009

How to lie with your graphs

Stumbled across an interesting little post about 5 ways that you can lie with your graphs.  Actually, a better lesson to take from the post is “5 ways other people lie with their graphs and now you can call them on it.”

The post was on Talking Squid, entitled “Five Easy Lies”: Two of the most common ones I’ve seen in talks are below:

  • Choose your cutoffs
    image 
  • “The trend shows no increase for the last [n] days/months/years.”

    Don’t mention the previous [20 n] data points.

  • Talk about the trend of the trend
    image
  • “Sure the graph is going up for now, but the rate of increase is going down.”

    If this fails, talk about the rate of increase of the rate of increase. Keep on differentiating until you find a curve that matches your needs. If all else fails, try logarithms.

Read the rest…

The last comment reminds me of Mar’s Law that stumbled upon a couple of days ago:

Everything is linear if plotted log-log with a fat magic marker.

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How do you measure two-thirds?

imageFrom an article by Mary Ann Bragg which appeared on CapeCodeOnline and was also printed in this month’s College Mathematics Journal:

TRURO — Voters narrowly approved one of four zoning amendments late Tuesday night at the annual town meeting. But town officials were still looking at the exact vote count on that article yesterday.

In a vote of 136 to 70, voters passed a new time limit on how quickly a cottage colony, cabin colony, motel or hotel can be converted to condominiums. The new limit requires that those properties be in operation for three years before being converted to condominiums.

The idea behind the zoning amendment is to slow the pace of condominium development in Truro and preserve more affordable accommodations for tourists, according to citizens proposing the warrant article.

Currently Truro does not allow condominiums complexes to be built outright in its zoning bylaws. Instead, property owners must build a cottage colony, cabins, motel or hotel first and then covert it to condominiums through a special permit.

The exact count of the vote — 136 to 70 —had town officials hitting their calculators yesterday. The zoning measure needed a two-thirds vote to pass. A calculation by town accountant Trudy Brazil indicated that 136 votes are two-thirds of 206 total votes, said Town Clerk Cynthia Slade.

But is it?  Is 136 a sufficient number of votes to be considered two-thirds of the total 206 votes?  Let’s check:

If you use the fact that [tex]frac{2}{3} approx 0.66[/tex] and then proceed to multiply 206 by 0.66 you get 135.96.  There were 136 votes in favor which is  more than 135.96 so that means it passes, right?  If you think so, then you’d be WRONG!!! 

The main problem is the rounding.  In fact, [tex]frac{2}{3} = 0.666666ldots[/tex] or using repeated decimal notation, [tex]frac{2}{3} = 0.bar{6}[/tex].  When you round, you are actually creating an error that, in this case, makes a pretty significant difference.

Think of it another way, lets compare 136 / 206 to 2 / 3.  First, just do it by decimal approximation:

[tex]frac{136}{206} approx 0.660194174757 < 0.6666666667 approx frac{2}{3}[/tex]

My calculator cannot exactly represent either of these fractions but its accurate to 12 decimal places and I can clearly see that 136/206 < 2/3 so the vote should not pass.

Do you remember another way you can compare fractions?  Find a common denominator and convert each fraction, then compare. 

[tex]frac{136}{206} cdot frac{3}{3} = frac{408}{618}[/tex]

[tex]frac{2}{3} cdot frac{206}{206} = frac{412}{618}[/tex]

So, here we see that, again,

[tex]frac{136}{206} = frac{408}{618} < frac{412}{618} = frac{2}{3} [/tex]

This second method of checking is even better than the first because there are no approximations involved.  We’ve confirmed, absolutely, that 136 votes out of a total of 206 does NOT constitute two-thirds.

Fortunately, a good citizen made an anonymous call in Truro, MA, to clear this up.  What perplexes me is that they decided they needed to let the State Attorney General’s office decide on the correct count. The mathematical explanation wasn’t good enough. Can you say quantitative illiteracy?

Read the entire story here.

Interesting facts about Euler

image I’ve not studied much of the history of mathematics but occasionally I read from a few books I have on my shelf on the subject.  When my mind is bogged down and I am unmotivated on my current projects, I pick up, say, Makers of Mathematics, by Stuart Hollingdale.

Today, I flipped open to the chapter on Leonhard Euler (one of my mathematical heroes) and learned (or re-learned) a few interesting facts about the man.

  • Entered the University to study theology and Hebrew but his mathematical abilities attracted the attention of Johann Bernoulli who gave him a private lesson once a week.  He received his master’s at 17.
  • His father greatly desired him to pursue his theological ambition’s but was convinced by Bernoulli that his son was destined to be a great mathematician.  Leonhard Euler remained a devout Calvinist all his life.
  • At the age of 26, Euler took on the leading mathematical position at St. Petersburg Academy
  • He and his wife had 13 children, only 5 of whom survived to adulthood.
  • He lost sight in his right eye fairly early in his career, probably due to overwork.
  • He spent 25 years at Berlin Academy and then returned to St. Petersburg at the age of 59 about which time he lost sight in his other eye.  The blindness didn’t stop him.  In fact, he completed a comprehensive analysis on the theory of the Moon’s motion.  All the complicated analysis was done entirely in his head.
  • In 1771, his house burned down.  In 1776, his wife passed away.  He died in 1783 at the age of 76 still active to the end.
  • All told, he published more than 500 books and papers during his lifetime, while a further 400 appeared post-humously.  It has been computed that his publications during his working life averaged about 800 pages a year.

For the record, the correct pronunciation of Euler is “oiler” not “yuler”.  That’s a minor pet peeve of mine.  It ranks right up there with folks that write my name as Scoot instead of Scott.  ;) 

Learn more about Leonhard Euler.