Category Archives: Asked & Answered

Asked & Answered: 13

When I became a Damn Yankee [1] back in the aughts, one of the first things I noticed was the outsize presence of Christian radio here, both AM and FM, and both music and talk.  There are more religious radio stations in the Asheville area (15 within a 30-mile radius) than NPR stations in the entire state of South Carolina (nine) or Tennessee (also nine).

I soon found an easy way to locate (and bypass) Christian stations — just set the car radio to “scan” and listen for giveaway words like righteous, almighty, rejoice, praise, miracle, glory and power, along with the instant-bingos Jesus, Savior and God.  At least one of the above is sure to be heard in any six-second fragment of any Christian radio broadcast here.

The abundance of God’s Radio-Spoken Word in these parts led me to wonder: are there so many religious radio stations here because there is so much sin to stamp out, or because there is so much righteousness to self-congratulate?  Is there any correlation at all between the reach of religious radio in a region and the size of its cesspool of sin?

Before we get to the answer (after all, this is an Asked & Answered article), let’s talk about the possible types of correlation between one thing and another [see illustrations below].

As shown in the first diagram, direct correlation is when one factor increases along with the other.  If sin and religious radio were directly correlated, then one would tend to find more sin in areas that have more religious radio stations.  There would be several ways to explain such a correlation:

(a) Religious broadcasters might be drawn to set up shop in high-sin communities, because that is where The Word is needed the most (cf. the Willie Sutton Rule).

(b) Or it might be that sinners hear the Sunday sermons, feel the full power of the Lord’s forgiveness, and then figure, “Hey, the slate is clean, this gives me a whole new week of sinning to do.”

(c) Or maybe religious broadcasts actually encourage sin!  But that could only be true if they made listeners believe things like vaccinations are deadly, children need to be “delivered” from same-sex households, and if you convert to Islam you may beat your wife.  Since these are all unthinkable, this hypothesis may be a bit shaky.

This brings us to inverse correlation (second diagram) where as one factor rises, the other tends to fall.  If that were the case here, then we could say: the more religious stations, the fewer the citations.  What might explain such a catchy correlation?

(a) Perhaps religious radio broadcasts reach into the hearts of hardened men and subdue their wayward impulses.  Listeners might say to themselves, “Nah, I don’t need to rob that bank today.  It’s Sunday, the Lord’s day, and they’re closed.”

(b) Or it could be that religious radio strengthens the wills of the righteous, who then elect law-and-order politicians, who then pass punitive laws, which mandate long prison terms for sinners, who decide to flee to other states.  Could be that.

But it is also possible there is no correlation at all (third diagram) between the number of religious radio stations and the amount of sinning in a given area.  How would one know?  Asked — and now answered.


I started by selecting fifteen cities at random from five US regions (Northeast, Southeast, Central, Texas-Oklahoma, California) and three population ranges (Large = 3-6 million, Medium = 1-2 million, and Small = 0.5-1.0 million).  The population figures for each area (see table below) were based on the estimated number of residents within a 30-mile radius of city center.  [Sources:  Free Map Tools and MCDC (Missouri Census Data Center).]

Now, I needed a proxy for the amount of sin in a given area, since the true definition of sin is known only to God.  (As we will all find out on Judgment Day, won’t we!)  For my proxy I chose the crimes homicide, robbery and assault — purposely ignoring property crime — and then totaled the respective rates for each city, with each crime given equal weighting.  [My main source here was the FBI although all my other sources cited FBI statistics.]

Finally, I counted the number of religious radio stations, AM and FM, within 30 miles of the center of each city.  [Source:]  I included stations whose format was listed as religious, gospel, Christian contemporary or Spanish Christian (found primarily in California and Texas).  The bar chart below shows the number of religious radio stations (and the percentage of the total they represent) for each metro area, grouped by region.


What surprised me about the radio station data was that most of the selected metro areas had about the same number (12.5 ± 1.3) of religious stations, regardless of population or region.  Cities with significantly more religious stations were Austin (22), Kansas City (23) and Houston (31).  Spanish Christian was by far the dominant religious format in Houston but represented only a handful of the religious stations in Austin and Kansas City.

The percentage of religious-format stations in a given metro area was typically 18-22%, again irrespective of population or region.  The notable outliers were Kansas City (40%), Houston (39%), Austin (33%) and Asheville (33%).  By comparison, religious radio in Chicago (13%) was a bit light, at least on a percentage basis.

Let’s now consider the original question — does religious radio have sin-fighting power?  To get a visual sense of the correlation, if any, I plotted each city’s rate of violent crime (selected crimes per million residents) vs. the prevalence of religious radio (stations per million residents) within 30 miles of city center.  I then calculated the best-fit curve [2] through these data points:

Hallelujah! as Leonard Cohen might say if he were alive today and subscribed to this blog.  It looks like the more religious radio stations per capita, the lower the crime rate, right?  Go tell it on the mountain, people — mount a transmitter on every steeple!

Or maybe not.  On closer inspection, this graph reveals a somewhat different correlation: the large cities are all on the left side of the graph, the medium cities are all in the middle, and the small cities skew to the right.  It is almost as if the rate of violent crime and the concentration of religious radio stations both depend on population density. 

In fact, we already noted that the raw number of religious stations within a 30-mile radius of a city does not depend on its population.  (Perhaps it has more to do with the available frequencies on the radio dial.)  That being the case, small cities will naturally tend to have the greatest concentration of religious radio stations, when expressed in terms of stations per million residents.  So this graph, in many ways, says nothing at all!  Or more precisely, while there are differences in crime rates, there is not a strong correlation between crime and religious radio, at least for this sample of cities.

To be able to say the same thing mathematically, I decided to analyze this data set with a statistical tool called multiple linear regression (MLR).  This tool helps identify which factors are strongly correlated and which are not, so that one can select a model that best fits the data.  I evaluated many models with various combinations and powers of the factors, but none of them fit the data better than the following two-factor model:

C = 64.5 + 1.22 P 2 – 1.04 S

where C is the crime rate (crimes per year per million people), P is the local population (millions) and S is the local concentration of religious radio (stations per million people).  This model suggests that crime rises with the square of the population but falls as the concentration of religious radio increases.

To illustrate the workings of this model, let’s pretend that Chicago doubled the number of religious radio stations in the area, from 12 to 24.  (This means S for Chicago would rise from 2.0 to 4.0.)  The model predicts that this would lead to a 2-point drop in the rate of violent crime, or twelve people a year who are not murdered, robbed or assaulted.  If you lived in Chicago, wouldn’t you want to be one of those twelve?

But before we start building a bunch of radio towers, we need to check the goodness-of-fit of the model to see what its predictions are worth.  According to the MLR analysis, the correlation coefficient for this model is only 0.5 (1.0 would be a perfect fit).  This means there is a weak-to-moderate correlation between the crime rate and the factors proposed to explain it.

Finally, the MLR analysis reports a 91% chance that P is a significant variable, but only a 69% chance that S is significant.  In other words, the religious radio effect is probably just a lot of noise, as you likely suspected before you even read this article.

I hope you at least learned something today that you wouldn’t normally hear on a Sunday.  Now, go thy way, and from henceforth sin no more.


[1]  I learned the local definition of Damn Yankee — a Northener who moves to the South and stays there — when I was only half-jokingly called one in a 2007 job interview here.
[2]  Here I need to insert a note about the crime rate data for Scranton, PA, the town where Joe Biden lived until he was 10.  Whereas the other crime rates in this study were based on the most recently available data (2018 and 2019), the 2018 crime rates in Scranton took an incredible leap — I figured that this must reflect some kind of change in reporting methods or coverage area.  So, even though my graph shows 2018-2019 crime data for Scranton, I used an average of Scranton’s 2016 and 2017 violent crime rates for the purpose of calculating the best-fit curve.
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Asked and Answered 3.4

Hello, and here we are again.  I thought I was done with this series on hanging pictures, but it seems physics never dies — it just gets more complicated.

Some commenters on my previous articles (Why Frames Tilt Forward, The “Hang It with Two Hooks” Calculator, and The Physics of Hanging Pictures) asked how they could hang items on wall studs, if the studs are off-center from the desired hanging spot.  This seemed to be a rather specialized topic, and beyond the scope of my series, so I deferred until now.  But a recent commenter rekindled my interest and finally inspired me to take a look.

Before I proceed, however, I must mention that I’m not the first to address this problem.  The number-one result (as of now) I found in searches for “hang item on off-center studs” is this article on by an author named MolecularD.  The author describes the principles involved and offers a set of equations (minus the math) that are meant to show the reader where to place the wall hooks.  Unfortunately, some readers commented that they did not get the desired result when they followed the author’s instructions.

The solution provided in the article is such a complicated equation that there is no way for me to verify it without essentially solving the problem myself.  Which is what I will do now, taking a somewhat simpler, more intuitive approach.

The four consecutive views in Figure 1 demonstrate the concept:

Concept of Hanging a Frame on Two Studs

View (A) depicts a frame hanging on a wall, centered at our desired position (dotted line), using a wire on a single hook.  Because of the symmetry of the system, there is no tendency for the frame to rotate one way or the other.  Ignore for now the fact that the wire extends above the top of the frame.

View (B) shows the studs in the wall behind the frame (we use a stud-finder to spot them).  The two studs are different distances from the center of the frame.  We drive a nail into the center of each stud, just touching the underside of the wire.  This does not cause the frame to rotate.

In View (C), we attach a piece of wire (blue) to the original wire, from the point where the first nail touches the wire to where the second nail touches the wire, without any slack.  The load is now shared between the central hook and the nails in the studs.  But this still does not cause the frame to rotate.

In View (D), we snip away the original wire where it touched the nails, leaving our new wire in place.  The nails in the studs now assume all the load, with the higher nail bearing more than the lower.  Still the frame does not rotate, so we have found the solution.

Obviously, I don’t expect readers to repeat these steps to hang their pictures — this was just a demonstration of concept.  Instead I will offer a calculator, with instructions for taking measurements, placing the hooks and cutting the wire, to help the reader achieve the final result.

That is, if you really insist on using studs.  Personally, I think it would be easier in most cases to forget about the studs and use the Hang-It-With-Two-Hooks calculator that I presented in my earlier article.  You would fasten the hooks to the wall with toggle bolts, which can hold a significant amount of weight when paired with the appropriate hooks.  (This video shows how to install them.)  But in the end, it’s your call.

The Setup

Oh, you’re still here!  This must mean that you really, really want to use two studs to hang your item.  Okay then, onto the intricate details.  Please consult Figure 2 (below) to get a sense of the important lengths and measures:

Diagram of Frame Hung on Two Studs

Start by measuring the height H and the weight of the item you want to hang.  Then mark the spot 0n the wall corresponding to the top-center of the item.  All other measurements will refer to this point.

Next, use your stud-finder to measure XA, the distance from top-center to the center of the closest stud, and XB, the distance from top-center to the center of the next-closest stud.

Now inspect your hanging hardware.  You want to (ideally) hide all your hardware behind the item you are hanging, which means the higher hook (A) should not show.  Therefore, you should choose a value for ZA, the distance from the top of the frame to the bottom of Hook A, that is slightly greater than the length of the hook.

While you are it, measure the length (D) of the D-rings attached to the item.  If you plan to attach the wire directly to the item, then this length is zero.

Your next measurement is WD, the distance between the D-ring attachment points.  If you have not yet attached the D-rings to your item, then mark the spots where you think they should be attached, and measure the distance between those marks.

Note that I have not asked you to specify Y, the distance from the top of the frame to the D-ring attachment point, or ZB, the distance from the top of the frame to the bottom of Hook B, or S, the length of wire to cut.  These values will be returned by the calculator.

There is one last thing you may have noticed on the diagram: to make the item hang true, you need to install a guide hook below Hook A to equalize the slack in the wire — and the forward tilt — on the left and right sides.  More on this later.

The Math and The Calculator

I provide geometric and algebraic solutions in this attachment.  The result we are most interested in is:

ZB = ZA+ (XBXA) tan θ

where θ is the wire angle, tan θ = (Y ZA)/(WC XA) and WC = ½ WD.

The formula for ZB assumes that Y, the D-ring attachment point, is a given.  But I don’t ask you to specify Y directly, as this involves a judgment call.  Ideally, the ratio Y/H would be about 1/5 (the “one-fifth rule”) to minimize forward tilt of the item.  But this might call for too small a wire angle and create too much tension in the wire.  On the other hand, if the  wire angle is too large, and Y/H is greater than 1/3, then the forward tilt could be excessive.

So what I did in the calculator is ask you to specify the wire angle, with 30° as the default. (The minimum entry is 20° and the maximum is the angle corresponding to Y/H = 1/3.) The wire angle is used to back-calculate Y as described in the attachment.

If the default angle seems to provide a reasonable value for Y/H, then go ahead with it, assuming the wire tension is not too high.

If the calculator flags one of your entries as out-of-bounds, don’t ignore it.  The calculator will not report any results if Y/H is greater than 1/3, and it will warn you if the estimated wire tension exceeds 25 lbs.  (You are responsible for selecting the appropriate hardware.)

Results are reported to the nearest one-eighth-inch.   The calculator provides guidance on positioning the guide hook and attaching the wire to the D-rings.  The suggested length of wire S includes 6 extra inches (3 inches per side) for tying the loose ends to the D-rings.

Final Notes

It may be a challenge to hang your item on three hooks.  I suggest you find a helper, if only for you to have someone to complain to while attempting it.  (Still, watch your language.)  You might start by feeding the slack of the wire through the guide hook and onto Hook A.  Then slide the item toward Hook B and feed the wire over Hook B.

You ask, do I have to use the guide hook?  If your item weighs much of anything, then yes.  The farther that Hook B is from the center, the more the item will tilt forward at Hook A,  since there is more slack in the wire on that side.  And the more front-heavy the item, the more uneven the forward tilt will be.  The guide hook helps keep the wire close to the wall on the Hook A side.

I end with my usual disclaimer.  My calculator makes it easier for a person to hang an item on two off-center studs using hooks and wire.  But whether this method is suitable in your situation is a judgment only you can make.  You assume full responsibility for your project. I offer this calculator as a convenience, but I accept no liability for damage of any kind, even if the suggestions offered in this post are followed exactly.

If you’re not confident how things are going to work out, you can always do a mock-up in your garage before you mark up your walls.

With that out of the way, good luck.  I would be interested to hear about any successes, failures or problems.  As always, your suggestions and feedback are welcome.

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Asked and Answered: 12

Last year, I told you about Spelling Bee, a word game that appears daily in the New York Times.  The point of the game is to form as many words as possible, using any of the seven letters provided, any number of times, as long as the center letter is used at least once.  It is a fairly undemanding diversion which, along with my cup of coffee, helps get me going in the morning.  Plus, it gives me something to kvetch about to my friend Eric, who also plays.

Eric and I often compare notes and complaints about the Bee of the Day.  My usual beef is about the exotic foodstuffs (e.g., BOBA, CALLALOO, GHEE) included in the answer list, and Eric (former chemistry professor) and I both gripe about the chemical words, such as NICOTINIC and PROPANOL, that of course should be accepted by the Bee but are not.

Nonetheless, we were pleasantly surprised by a recent Bee in which BORON (Element 5) and CARBON (Element 6) were among the answers.  This got me wondering: what is the greatest number of element names one can generate from a set of seven different letters?

My first step toward an answer was to create a spreadsheet to count the number of times each letter of the alphabet appears in the list of element names.  Note: I decided to limit the number of elements in my list to the first 100, i.e., from HYDROGEN to FERMIUM.  Elements 101 and beyond — all man-made — are unfamiliar even to Eric and me.  One of those is darmstadtium (Element 110) of which only a few atoms have ever been produced. So it’s not like I’m disrespecting Nature by excluding darmstadtium and its ilk.

Anyway, back to my spreadsheet.  I found that the consonants appearing most often in my 100-element list were M (50), N (36), R (33), L (22) and T (22).  And as you might guess, the most common vowels were I (56) and U (50).  After spending about 15 minutes playing around with frequently-appearing letters, I was able to find two different seven-letter sets which “contain” the names of four elements.  (Before I reveal, would you like to try?)

The first set I found was EIO/BMNR. (I’ll refer to these sets by their vowels/consonants). This set spells the elements BORON, BROMINE, IRON and NEON.  And my second set was AEIO/DNR, which spells IODINE, IRON, NEON and RADON.  Interesting, but…

I was, of course, not satisfied.  Humankind needed to know: are there other seven-letter sets that spell out four element names?  And more importantly, are there seven-letter sets that spell out five (or more) element names?  I did not yet have these important answers.

So, for humankind’s sake, I was obliged to resort to brute-force computation, employing the only modern programming language I know — PHP.  I am familiar with PHP because it is the language used by WordPress, the platform for this and millions of other blogs.  And though I have a PHP reference manual, most times when I want to write code for a new task, I just do an internet search — 99 percent of the time someone has already done the thing that I want to do and has provided functions and/or code for it.

And that is (mostly) how I wrote a PHP program to print out all the seven-letter sets that spell out four or more element names.  I’ll spare you the details, but suffice it to say that before now I had no idea there were PHP functions like count_chars (finds the number of unique letters in a word), array_intersect (lists the items that two sets have in common) and implode (combines a set of individual letters into a single word).  Those functions served me well here, but they (like many other PHP functions) are so special-purpose that I can’t imagine any programmer having good command of them all.

In any case, I ultimately wrote a program that evaluated all 213,333 of the seven-letter sets containing one or more vowels and one or more consonants found in the element list. Without further delay, here are the results.




None.  Zero.  Not-a-single-one-ium.

So there you have it.  There are 15 different seven-letter sets which can be arranged to spell four element names, but there are no seven-letter sets that will spell five element names, at least not with respect to the first one hundred elements.

If some nerd ever uses this edition of Asked and Answered to win a bar bet, I will expect due credit, if not a beer.

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