# Archive for category Maths

### Littlejohn and renewable energy – both fueled by wind

I don’t normally blog about stupid Littlejohn columns – it’s a bit “dog bites man” – but today he’s written a piece about wind farms, and he’s really outdone himself on this one.

Here’s a quick rundown of the highlights.

At midday yesterday, wind power was contributing just 2.2 per cent of all the electricity in the National Grid. You might think that’s a pretty poor return on the billions of pounds spent already on Britain’s standing army of windmills.

In fact, for the amount of energy produced, onshore wind power is only slightly more expensive than coal, and less expensive than nuclear. Offshore wind is quite a bit more expensive, but hopefully this will come down as production gets more organised.

### The Sun lies about disability benefits, what a shock

GOT hay fever? Why not throw a sickie?

Even better, take the rest of your working life off.

Well, this is going to go well.

New figures show that under Labour the state was happy to pay your way, no questions asked.

Those claiming Disability Living Allowance soared from 2.1million in 2000 to 3.1million last year. The annual cost is now £12billion.

So, 3.1 million have “taken the rest of their working lives off” on Disability Living Allowance, and the state is “paying their way”? Well, no.

Disability Living Allowance is a supplementary payment, given to people with disabilities, which helps cover their care and mobility costs – in The Sun‘s case, they seem to be talking solely about the part of the DLA that covers care, since that’s where the 3.1 million figure comes from. There are different levels of DLA, depending on how severe the disability is, but even in the most severe case – someone who requires 24 hour care – the recipient would only get £73.60 a week, or about £3,800 a year, and on average, people only receive about £46.30 a week, or £2,400 a year (and 500,000 of that 3.1 million get nothing at all). No-one has “taken the rest of their working lives off” to live on £2,400 a year.

Incidentally, that part about the annual cost being £12 billion does seem to be including the cost of mobility allowance as well – the cost of the care part of the DLA is only £6.4 billion a year. It sounds like a lot, but like I say, it only actually works out at about £46 per person per week – not very much at all when you think about the cost of a private carer, or the earnings lost by a friend or family member who takes time off work to provide care.

Clearly The Sun must realise this – they complain that “Many of those handed up to £73.60 a week are laid low with ailments such as “alcohol abuse” or allergies“, clearly hoping that we won’t realise that £73.60 is not all that much money. There maybe people on DLA because of alcohol abuse or allergies, but in that case, it will be because their condition is so serious that they need part-or-full-time care. To qualify for even the lowest rate, you need to be either physically unable to cook for yourself or require care for part of the day. That’s more than just “someone who cannot get out of bed because their hangover is so bad“.

The Sun also says that “The vast majority of claimants have never been medically assessed“, which also isn’t true. Most people aren’t assessed by the Department of Work and Pensions, true, but in order to qualify for DLA, you need to have been diagnosed by your doctor. Everyone who is on DLA was assessed by their doctor.

Now at last the Government plans to order regular assessments to weed out the workshy.

It should make the economy look healthier by a few billion pounds a year.

That’s something not to be sneezed at.

Unpaid carers are worth about £87 billion to the economy per year, by reducing the strain on the NHS. Making it even harder for them is hardly going to make the economy any healthier.

Edit: The Express’s coverage is more or less the same, but with TPA quotes and the added bonus that they express incredulity that people with back pain might have trouble moving around. WHO’D HAVE THOUGHT?

(The Sun discards its “Sun Says” columns each day. I’ve preserved this one beneath the fold)

### A bumper crop of nonsense equations in the Mirror

This piece is only in the print version of the Mirror as far as I can tell, but it’s so daft I can’t let it go. Equations! Loads of them! All bollocks!

First:

Tractors:

q = (vcvs)ks

This formula, produced by University College London and the Green Flag breakdown company, tells you how long you spend stuck behind a tractor, where q is the time spent, vc is the speed of your car, vs is the speed of the tractor and ks is the number of tractors you meet per journey.

The trick to explaining why this is bollocks is something called dimensional analysis. Basically, since we’re trying to work out how long we spend stuck in traffic, the result should be a time. Instead however, we have speed, measured in miles per hour – ie, distance per time – times number of tractors. This is not a time – we’ve still got that pesky distance in there which we haven’t got rid of. Therefore, this formula is nonsense.

The perfect bacon sandwich

N = C + {fb(cm) . fb(tc)} + fb (Ts) + fc . ta

This came from Leeds University, who reportedly spent 1000 hours tasting 700 variations of bacon sandwiches to make this formula. All I’ll say is that that works out at 1 hour 25 minutes per sandwich. They must be very slow eaters.

The formula itself, as described by the Mirror, are meaningless. Those fbs and fcs are “functions of the bacon type”, which are never explained. cm represents condiments (if you can tell me what “7 + ketchup” equals, you’re a better mathematician than me), tc is cooking time, Ts is temperature, ta is the time taken to fill the sandwich and C is the force (in Newtons) required to break the bacon.

Since we don’t know what fb and fc is, I haven’t a clue how the units work, but if fb can be a function of time, temperate or condiment, this must be a very versatile function!

The perfect cheese sandwich

W = [ 1 + ((bd)/6.5) – s + ((m – 2c)/2) + ((v + p)/7t)] (100 + l/100)

Now, at least the dimensions more or less work in this one, from Bristol University and, apparently, a cheese company. This tells you, supposedly, the thickness of cheese in the perfect sandwich (in millimetres). b is the bread thickness, d is the “dough modifier”, s is the thickness of the of the margarine, m is the thickness of the mayonaise (and c is the creaminess of the mayonaise), v is the thickness of the tomato, p is the thickness of the pickle, t is the tanginess of the filling and l is the lettuce thickness.

If you look at this one, at least all the things that are being added together are thicknesses (except for the “creaminess”), so our end product is also a thickness!

Of course, it doesn’t say how you assess “dough modifier”, “creaminess”, or “tanginess”. The company’s website has an online calculator (it’s viral marketing, so I’m not sute I want to link it, but, err, here it is, just don’t buy cheese from it!) which does the equation for you, but you never actually have to use the creaminess modifier, so that’s no help.

The best way to open champagne

P = T/4.5 +1

This is actually a good one. P is the pressure in the bottle in atmospheres, T is the temperature of the champagne in centigrade. It’s only an approximation, so it falls apart at very high or low temperatures, but around room temperature, it actually works alright. At 4.5 degrees, the pressure is 2 atmospheres, at 9 degrees it’s 3 atmospheres, and at 18 degrees it’s 5 atmospheres. This tallies pretty well with the figures in the Mirror and on this site. One point for the Mirror!

The perfect joke

x = (fl + no)/p

Ah, back to form. Supposedly, x is the funniness of the joke, f is the funniness of the punchline, l is the length of build up, n is the number of times the comedian falls over, o is the “ouch factor” and p is the number of puns.

One obvious problem: if your joke has no puns, p = 0, so you’re dividing by zero and your joke is infinitely funny!

One other obvious problem: according to this, longer jokes are ALWAYS funnier than shorter ones. So according to this formula:

“A man walks into a bar. Ow.”

Is nowhere near as funny as.

“A man walks down the street. It’s a bright early summer’s day, and a few clouds scud across the afternoon sun. Thirsty, he decides to go for a drink and walks into a bar. Ow.”

One more obvious problem: increasing the ouch factor supposedly ALWAYS makes the joke funnier. So if the comedian trips over, that’s slightly funny. If he gets hit with a frying pan and keels over, that’s funnier still. And if he falls out of a third story window, lands on the concrete below and breaks both his legs, his pelvis and three vertebrae, that should be utterly hilarious!

So in other words, if you were to use this to produce the perfect joke: a man tells a longwinded monologue – but with no puns – while repeatedly suffering debilitating injuries. Since delaying the punchline increases funniness too, the punchline should NEVER come. Monologue, no puns, no punchline, repeated torture. Sounds like my sort of thing!

The perfect amount of gravy for Sunday lunch

(W – (D/S))/D x 100

Where W is the weight of the uncooked food, D is the weight of the cooked food, S is the amount your food shrinks.

I can kind of see where they’re coming from on this, but what the formula is trying to do, basically, is tell us how much water the food has lost during cooking (though I can’t make sense of the dimensions here – I don’t know whether it’s telling us the weight of the gravy, the amount of gravy, or what). This isn’t the same as the perfect amount of gravy. For example, chips lose very little weight during cooking, but if anyone tells me that chips shouldn’t be absolutely smothered in gravy then that is CRAZY TALK.

Pulling a cracker

O = 11 x C/L + 5 x Q

Where O is the angle(?) C is the is the circumference of the cracker, L is the length, and Q is the quality of the cracker (1, 2, 3).

The dimensions work in this one, but I can’t really make any sense of it. The better quality your cracker is, the lower the angle you pull it at? I guess it’s meant to say that thicker crackers need a stronger pull. I’ll give this one a maybe.

Anyway, everyone knows the secret is to twist the opponent’s end of the cracker before you give it to them.

The perfect holiday beach

(A x W) + 2[E x C x V] + v[F x S] / (T + TC)

Sigh.

A is the quality of the beach, W is the weather, E is the entertainment, C is the culture, V is the activities, F is the familiarity, S is the shopping, T is the time to destination and TC is the cost of the holiday as a percent of your income.

Great. And if anyone can tell me what entertainment times culture times activities is, or why, if all of these are so subjective, the Mirror can just crown the Tuscan Rivera the best, I’d be delighted.

(And if familiarity makes a beach better, are you not allowed to go to new beaches?)

The perfect sandcastle

OW = 0.125 x S

OW is the amount of water, S is the amount of sand.

This barely counts as equation. It literally just says “use 8 parts sand to one part water”. Moving on.

The perfect neckline

O = NP(20C+B)/75

O is the “naughtiness”, N is the fraction of nipple showing, P is the percent of exposed frontal area, C is the cup size and B is the bust measurement in inches. If O is greater than 100 then the neckline is “obscene”.

Well, the most obvious problem with this is simple: have you got big breasts? Congratulations, you’re automatically more “obscene” than a smaller breasted woman wearing the exact same outfit. That’s a great message to be sending out, Mirror!

Secondly, there’s that “nipple fraction” parameter. If the nipple fraction is 0, the naughtiness is also 0. So if you walked around completely topless, save for a pair of pasties, that would be completely prim and proper.

In fact, even if you walked around topless, so N = 1, P = 100, you would need to be at least a 36 B or 34 C for that to count as “obscene”! (and that’s ignoring the fact that band size over-estimates the underbust measurement by several inches)

Conclusion

Sometimes I wonder if it’s worth getting worked up about these things. After all, it’s all a bit of fun (and advertising. Always advertising).

The thing is, though, this is why people hate maths. When people think of maths, they don’t think of the really interesting stuff – like the beauty of fractals or using the power of statistics to win thousands on the lottery. They think of dull formulas, spending hours in school plugging numbers into them and rearranging them without ever really understanding them.

When done properly, a formula shouldn’t just be a bunch of unintelligable letters, without any deeper meaning. A formula should be able to speak just as loudly as words do.

E=mc2, for example, isn’t just a formula to plug numbers into. It also tells us that energy, E, and mass, m, are one and the same. You can destroy mass to create energy – as happens in a nuclear bomb – and you can use energy to create mass – so as you go faster, you get heavier (though sadly this is only noticeable when you get close to the speed of light, c).

That’s a lot of information packed in three letters!

### Hangovers, statistics and a dodgy hookup

It’s one of the oldest clichés in the book. You go to a party, get completely hammered, and wake up in bed with a dodgy PR firm.

Today’s ill-advised hookup is a threesome between The Express, The Mirror and a non-alcoholic drinks company called Sweet Lady Beverages, who claim that “the average Briton will spend five years of their life with a hangover“.

Before we look at the article itself, a quick sanity check. Life expectancy in the UK is roughly 80 years, and it’s unlikely people are going to experience hangovers before the age of about 15 or so. So, at maximum, that gives the average Brit about 65 drinking years. If the Express‘s statistics are true, we spend 8% of our adult lives hung over – we would spend more time hungover than we would eating. It’s amazing anyone gets anything done.

The article goes on to say that:

[Britons] will suffer the ill effects for a whole day – usually a Sunday – at least once a week between the ages of 21 and 38.

Bear in mind that this an average. According to Sweet Lady Beverages, the average person is hung over every week until the age of 40, and those hangovers last all day. That sounds a tiny bit excessive. After all, one – much more scientific – study found that having even just one hangover per month over an extended period is linked to a major increase (around 2.36 times) in heart attack risk.* And yet somehow, we’re not dropping like flies.

As far as I can tell – there’s no information about this survey available on the web outside these two articles – Sweet Lady Beverages simply asked visitors to its site to answer some questions about hangovers. There’s no published methodology; in other words, they don’t say what questions were asked or what precautions they made to make sure they had a fair sample.

For instance, they could have asked

It would certainly explain the odd results they got.

The Sweet Lady Beverage company is quoted by the Express as saying

The message we can take from this is simple – by reducing our alcohol intake we can reduce the amount of time feeling wretched.

Oddly on-message for a company selling alcohol-free drinks, wouldn’t you say?

* I can’t find many good scientific studies of hangovers. A lot of them are rather hamstrung by the fact that surveys usually take place in university, and therefore involve university students – not very representative of the drinking habits of the wider population! Nevertheless, this paper suggests that only 15% of the population have more than hangover per month.

Edit: The Daily Mail has now picked up the story too.

### Evidence? Where Cameron’s going, he doesn’t need evidence!

David Cameron is going to give a speech today with Lord Reid today about the AV referendum. In it, he will say:

“Too often debates about AV are less like political arguments, and more like scientific discussions, where people get lost in a language of proportionality and preferences, probabilities and possibilities.

“Of course, some of these things are important. But for me, politics shouldn’t be some mind-bending exercise. It’s about what you feel in your gut – about the values you hold dear and the beliefs you instinctively have. And I just feel it, in my gut, that AV is wrong.”

Just a reminder, this came from the Prime Minister of the United Kingdom, not a Richard Littlejohn column or some particularly stupid Comment is Free post. The man who directs a nation of 62 million people and controls the sixth largest economy in the world doesn’t think politics should be about thinking and weighing up different options. Oh no, that’s just “some mind-bending exercise”! No, politics is about what you feel in your gut* and your blind instincts.

Voting systems are, at heart, all about mathematics – each voting system is just a different way of counting people’s opinions. You cannot discuss any voting system without taking into account the way it behaves mathematically. The language of “proportionality and preferences, probabilities and possibilities” is not swamping the debate, it is the debate.

Imagine if this was any other debate. Imagine if David Cameron said we should ignore the clinical trials when deciding whether to fund a particular drug, or the climate models when considering pollution controls, on the grounds that all this scientific evidence was “mind-bending” and got in the way of his “instinctive beliefs”. No-one would think that was an appropriate way for an elected official to make decisions.

If you push the evidence out of the debate, all you’re left with is empty sloganising, blatant untruths and tribal party politics. Without a proper debate on the pros and cons of each voting system, the AV referendum just becomes a Cameron/Clegg popularity contest. That’s a terrible way to decide an issue that will shape government in the UK for decades to come.

* Most of what’s in your gut is digested food, so perhaps Cameron is just saying that politics should be full of shit?

### Another day, another stupid asteroid scare

The road I live on has a railway running across the end of it. Every day, hundreds of tons of metal speeds along the line just a hundred metres or so from my house. Yet I don’t live in fear of waking up one morning and finding a train’s crashed into my house, because of course the trains are restricted to the railway tracks.

Space is much the same. Asteroids are whizzing around over our heads every day, but they follow precisely defined orbits through the sky. An asteroid passing close to the Earth is no more a “near miss” than a train passing my house without hitting it is a “lucky escape”.

It’s a simple enough idea, you’d think, and yet…

“Phew, that was a near miss: Cigar-shaped asteroid stronger than ’15 atomic bombs’ whizzes by earth”

Thankfully, the 50m long rock that could have destroyed a small country went barely noticed as it passed earth at a distance of some 2,085,321 miles.

Yes, the asteroid 2011 GP59 could have destroyed a small country. If it was two million miles closer.

Once again, this article has been taken from the Australian news site news.com.au, who seem to have a thing for scaremongering stories about space; they also started the rumour that Betelgeuese would go supernova in 2012 and gave credence to the shameful “supermoon” story. At least the Mail‘s headline is less awful than news.com.au’s, who’ve gone with “Scientists find asteroid with potential power of 15 atomic bombs. Heading this way. Tonight.” which surely has to rival shouting “fire” in a crowded theatre in terms of unethical scaremongering stupidity.

Incidentally, the claim that it’s “as powerful as 15 atomic bombs” doesn’t come from any scientific authority. It comes from the news.com.au journalist – who doesn’t appear to be a science journalist at all, but a technology journalist – digging up an old New Scientist article about an asteroid that exploded with the energy of three nuclear bombs (three of the very small Hiroshima bombs, I should point out, not a modern nuclear bomb), and then scaling it up. This is a stupid calculation for a number of reasons:

1. You can’t just say “this asteroid is 10 metres long, this asteroid is 50 metres long, therefore it’s 5 times bigger”. It’s the volume which is important – the length times the width times the height. Assuming the asteroid is 5 times bigger in each direction, then it’s 5 x 5 x 5 times bigger, which is 125 times the size. If the journalist hadn’t cocked up his maths, he could have made this asteroid sound EVEN SCARIER. Except…
2. The amount of energy an asteroid has depends on its speed. A fast moving asteroid carries far more energy than a slow moving one, and a small increase in speed causes a much larger increase in energy.** The gravitational pull of the Earth as the asteroid approaches plays a large role in determining its speed, so the energy it would have would depend on the route it took to Earth. Since this asteroid is not heading for Earth, it’s meaningless to ask how much energy it would have if it hit Earth.
3. It also depends what the asteroid is made of. Most asteroids are made of dust and ice, and burn up harmlessly in the atmosphere. A few – maybe one per year – explode high up in the atmosphere. And a very, very few – mostly large metallic asteroids that don’t burn as well –  hit the ground. Again, we don’t know what this asteroid is made of.
4. THE ASTEROID IS TWO MILLION MILES AWAY AND WILL NOT HIT EARTH AT ANY TIME IN THE FORESEEABLE FUTURE.

At any rate, there are literally thousands of asteroids this big – or indeed much bigger – rattling around near-Earth space, and there must be thousands more we haven’t detected yet. It’s worth being sensibly worried about the risk of currently undiscovered asteroid hitting us, but getting worked up about an asteroid that we know can’t hit us is just stupid.

Now, if you’ll excuse me, I’m off to dig a train-proof bunker in my garden.

* Psst, Daily Mail Reporter. “Earth” has a capital “e”.

** In fact, energy is proportional to speed squared – if you double your speed, your energy goes up fourfold.

### Channel 5 and the Express – some unsurprising statistics

Inspired by my last post, I’ve decided to have a quick look at the Express‘s TV review archive.

Channel 5 was bought by Richard Desmond, owner of Express Newspapers, on 23 July 2010.

So far this year, a Channel 5 show has been a “Pick of the Day” 38 times (warning: all links are direct to the Express site)

There have been 35 weekdays so far this year. On average, that means a Channel 5 show is a pick of the day slightly more than once a day – 1.09 times per day to be precise.

Over this time last year, a Channel 5 show (back then, the channel was called “FIVE”) was a pick of the day 3 times:

That works out at roughly one pick every twelve days, or 0.085 picks per day.

CSI and Law and Order were both on FIVE at the time, but funnily enough, neither of them were mentioned at all over this period. CSI was hardly ever an Express pick of the day before the takeover – from the looks of it, it got four mentions between 2007 and 2009, or about twice a year. Since October 2010, however, it’s been pick of the day 10 times in just 16 weeks.

I know it’s hardly surprising that the Express and Channel 5 are a bit chummy, but that doesn’t make it any less of a shock when you see it in terms of raw numbers.

Edit: Thanks to Lazer Guided Melody in the comments for pointing out that the statistics need more context. Here’s a very quick count of the number of recommendations over the same period for each of the major channels:

BBC1: This year, 25. Last year, 48
BBC2: This year, 21. Last year, 13
ITV1: This year, 31. Last year, 42
Channel 4: This year, 8. Last year, 37
Channel 5: This year, 38. Last year, 3

### A survey? How romantic!

Blah blah romance is dead, say The Express and The Mirror. Why?

Only nine per cent of those surveyed had ever sent a letter – and most of those were over 50 – while more than two-thirds prefer to say “I love you” by text.

An additional 24 per cent would rather send an email to express their feelings, while 14 per cent said they would post a message on their lover’s Facebook wall.

In further shocking news, very few couples surveyed courted via telegraph, and fewer still stood on hilltops furiously waving sweet nothings in semaphore. The Express quotes relationship expert Jo Barnett saying this shows that “[people] want an instant relationship with instant physical contact, they feel they’ve not got enough time to romance their partner” rather than the more immediately obvious conclusion that people are just sending fewer letters in general.

If the love letter is dead, then it seems male chivalry is also on its last legs as just four per cent of men said they would send flowers to their partner’s place of work. Only five per cent stand up when their partner stands from the dinner table, while only 12 per cent have booked a surprise weekend away.

This is the same rubbish that comes out every few months; pearl-clutching panic about how “chivalry is dead” (a phrase which appears 49 times on The Mail‘s site, incidentally) when it would be more accurate to say “society’s norms about how men should treat women are (slowly) changing to be less infantilising” (a less snappy headline, I’ll admit). After all, I’m pretty sure if I stood every time my girlfriend left the table, she wouldn’t find that chivalrous but instead rather creepy.

Anyway, I’m sure there was a perfectly valid, academic reason this survey was done, right?

The study was to mark the DVD release of romantic comedy Going the Distance.

Going the Distance is released on Blu-ray Triple Play and DVD today, from Warner Home Video.

Oh.

### Short and royally silly statistics

Being a monarch is four times more dangerous than being a soldier fighting on the front line! Charles and William are doomed!

Well… if you look at every European king and queen since the year 600, and assume that the regicide rate today is exactly the same as it was a thousand years ago, that is…

The Guardian‘s coverage is a bit better – at least they seem to know the difference between past and present!

### News of the World shows us how not to use statistics

Thank god for churnalism. The News of the World yesterday published an article claiming to have found the most “workshy” neighbourhood in Britain, but of course that ended up locked behind its paywall. Luckily, The Daily Express and The Daily Mail have both churned out articles based on NotW‘s, so I don’t have to pay Murdoch to read it.

The community that they claim is most “workshy” is a small area called Cottsmeadow Estate in Birmingham. Before I even go into the statistics of this, I’d like to point out that having a lot of people on benefits does not mean an area is “workshy”. Perhaps it’s an area where a major employer recently went bust. Perhaps it’s an area with a lot of affordable, accessible housing perfect for disabled people. Or perhaps, as seems to be the case here, it’s an especially deprived area, which has been very hard hit by the recession. After all, the majority of people in the area receiving benefits are getting Jobseekers Allowance (and quite a few more people are receiving income support, which means they work part-time).

Anyway, the newspapers claim that 106 people of working age live on the estate, of whom 105 are on benefits. Population statistics for individual “census output areas” are only available by request, annoyingly, so I’ll have to take that on faith for now. However, the population data is just an estimate, not a robust census, and when you’re dealing with areas as small as 100 people (out of a population of 60 million), you’re bound to have quite a bit of error in there.

The newspapers claim to have tracked down the lone worker – the mind boggles over how they could possibly have gone about this (did they go from door to door asking people “do you have a job?”), especially since the data in question dates from June of last year – in employment terms, that’s rather stale. Now, if they had found the only person on the estate who wasn’t receiving some sort of benefit, they’d almost certainly have breached the Data Protection Act – giving the name of the only person who does not receive benefits is, in effect, exactly the same as revealing everyone else does. This is precisely why the DWP anonymises their data – they randomly round each figure up or down to a multiple of 5, so you can’t work out who is or isn’t on benefits by taking advantage of small numbers.

In this case, it looks like they’ve probably underestimated the population of the area. After all, according to the statistics, three months earlier there were 110 people receiving benefits in the area (code 00CNGP0059) – more than the estimated population! This seems to be the only reason to focus on such ridiculously small areas. The data is divided into “census output areas” – the smallest division that the Office for National Statistics uses, and therefore most error prone too. Looking at the ward Cottsmeadow Estate is in, Washwood Heath, there appears to be about 4,950 people receiving benefits out of a working age population of around 15,000. This is a sample almost 150 times larger than just Cottsmeadow Estate and a much fairer way to gauge the number of people receiving benefits in the area.

These articles, had they been written properly, could have carried an important message – some areas are more deprived than others, and we need to make sure that everyone has access to work. The way the Mail and the Express (and presumably NotW, but alas I don’t have a copy of the article) cover it however completely destroys any attempt at nuance, tarring whole neighbourhoods as being full of “workshy” “scroungers”, regardless of what the statistics and basic common sense say.