Percentage Question
December 31, 2013 4:35 AM   Subscribe

Help me understand percentages.

Quick question about "percentages".

I know that a percentage is a number expressed as a fraction of 100.

I also know that a number can be more than 100%. For example, an item can be marked up more than 100%.

However, a lot of people also get upset when people say that you give something 110%, as it impossible to have more than 100%.

Can you help me get my head wrapped around this? In fairly basic terms?

posted by dbirchum to Education (17 answers total) 2 users marked this as a favorite
100% is the entire original amount. This means that, if you have a price of $5, that's the whole original amount so going up 100% means going up another $5. This works because there is not a mathematical upper limit to the amount of money someone can charge.

In some cases, like effort, 100% (the entire amount) can't be increased. Thus, you can't GIVE more than 100% because there's nothing left to give after 100% has been spent.
posted by Mrs. Pterodactyl at 4:44 AM on December 31, 2013 [1 favorite]

In the case of an item being marked up, the original cost is treated as '100%'. This means that if an item costs $100 to produce, and it's sold at $300, the item has been marked up 200% (twice the original $100 of the cost price). You can always increase dollar amounts, so it's possible to do this pretty much infinitely.

In the case of 'giving 100%', the 100% refers to everything that's available. Unlike the previous example, where you can infinitely increase a price (selling a $100 item at $1,000,000, for example) it's impossible to give more of yourself than 100%, because more than that doesn't exist.
posted by littlegreen at 4:45 AM on December 31, 2013

You're right on all counts. You can mark up an item in a store more than 100%, but logically you can't give more than 100% of yourself to something. That is, if you've given all you have (in terms of effort or time or whatever), there is no more to give. But it's a common expression suggesting that someone is giving above and beyond the norm, or above and beyond what we think they have to give.
posted by wisekaren at 4:45 AM on December 31, 2013

Consider whether it makes sense for an item to have a 110% discount. If you've already given 100%, there's nothing left to give.
posted by empath at 4:47 AM on December 31, 2013 [1 favorite]

I used to have a pet peeve about giving 110%. Then I stopped and learned to love the bomb. When someone is being asked to give 110%, they're being asked to give more effort than they think they are capable of giving. They are not being asked to run from Marathon to Athens and actually die; Pheidippides gave 100%.
posted by Nanukthedog at 4:53 AM on December 31, 2013 [1 favorite]

"Giving 110%" is a turn of phrase, not a mathematical concept. George McGovern was running for president some time back when it was learned that his running mate, Thomas Eagleton, had had treatment for a mental illness in his past. He infamously declared that he supported Eagleton "one thousand percent," then replaced him just a short time later. So maybe something under 1,000% is the top limit of the turn of phrase.
posted by yclipse at 5:00 AM on December 31, 2013

There's a difference between talking about what percentage of the total something is, and talking about what percentage the increase or decrease of the total amount is. So, while it's impossible to have a glass of water that is 110% full (because the glass would overflow when you tried to pour the last 10% in), it would be completely possible to say that you took a glass of water that was 10% full and increased the amount of water in the glass by 110% (which would mean the glass is now 21% full, because 10+(110%*10)=21).

That's why people get annoyed when people say they've given 110% of their total available effort (because they would have run out of available effort at 100%), but understand when someone complains that a store has raised its price on peanut butter by 110%. They're being a bit pedantic, because the former is clearly just hyperbole, not an actual statement of math, but that's where the difference lies.
posted by decathecting at 5:00 AM on December 31, 2013 [2 favorites]

Everyone is right. Another way to look at the impossibility of 110% in some circumstances is to look at possible outcomes. The likelihood of something happening can only go up to 100%, no more.

For example, what is the likelihood that it rains tomorrow? 50% means 1 in 2 chance. 100% means it's absolutely going to rain. 110% means . . . nothing; that's just nonsense. If I said that there was a 110% chance of rain tomorrow, I'd be talking figuratively, not literally.

Or, what is the percent likelihood that the Colts will win their playoff their game this weekend? Again, anything from 0% to 100% makes sense, but over 100% means they have a better chance than a guaranteed win, which is impossible.

Now, it's true that when people talk about giving 110% percent effort, they are talking about an amount, not a likelihood, but there are some things of which you just can't give more than 100%. I can't give you 110% of my right leg, for instance. As others have mentioned, talking about effort in this way is just poetic.
posted by (Arsenio) Hall and (Warren) Oates at 5:08 AM on December 31, 2013

There are some things I've heard of that give over 100%, but it's shorthand for "100% of normal/safe operation level." The example that comes to mind is, I believe, the climactic battle of "The Hunt for Red October," though I know from my dad's Navy experience it was reflecting a real practice by the Navy where some power systems were involved. At some point, they set a level for the sub's nuclear that was 100%, but when the captain needed more power to save the ship, so he asked for 105%, then 110%. Sure, the reactor's now running hot with a greater propensity for failure, but what's worse-- that risk, or the risk of getting hit by a torpedo?

Now that I think about it, they did something similar in "Captain Phillips," also with the ship's power system. In this sense, and in the human sense, it means "push the limits."

The trick with percentages is to identify whether the percentage is describing the amount of change in something (discounts, markups, etc.) or whether it's the amount of a whole that's being described (in which case 100% is usually the limit.
posted by Sunburnt at 5:13 AM on December 31, 2013 [1 favorite]

A slight niggle on "markup" and similar terms like "appreciation". Generally speaking, one is often referring to the amount above 100% that has been added to something's price/value. So if I buy, say, pants at $20 and sell them at $24, my markup isn't 120% (note that 24/20=1.20-120%), but is a mere 20%. Likewise, if my $1000 investment over the course of a year becomes $1025, that's an appreciation of 2.5%, not of 102.5%.

The same thing happens with discount, of course. If I cut the price on a $30 gadget to $21, it's become 70% of what it was, which reflects a discount of 30% (i.e. how much it has changed from its original value).

However, percentages greater than 100 are often useful. For one thing, they're a good way to reflect how a value has changed after an increase, so one could equivalently say "these pants are marked up 20% from wholesale" or "the retail price of these pants is 120% of their wholesale cost.". This somewhat less digestible phrasing is important when, for instance, calculating investment value over time, where it might be more computationally useful to say "the investment becomes 105% of its original value over the course of a year" than to say "the investment increases in value by 5% over the course of a year".

As for where percentages greater than 100 might reasonably come up: if a price is more than twice its wholesale/production cost, then that constitutes a markup greater than 100%. For instance, if I buy wholesale chocolate at $12/pound and sell it at $30/pound (which is not actually an absurd price for well-manufactured chocolate packaged into, say, 1oz bars), my markup is 150% (because $30 is 250% of $12).

There are a lot of scenarios, however, where a percentage greater than 100 simply doesn't make much sense. If you're speaking of a part of a whole, greater than 100% doesn't make sense --- you couldn't leave 120% of your assets to someone in a will, say. This is the underlying principle which irks a lot of people told to "give it 110%". Probabilities fall into this class too, since a probability represents a "slice" of possible outcomes, and there's no probability larger than "always occurring", which is to say, 100% of the time. Likewise, a discount of 100% would compute to a cost of $0 or free, so unless you're paying someone to take something away (which you might!) you couldn't really speak of a discount of greater than 100%.
posted by jackbishop at 5:19 AM on December 31, 2013 [1 favorite]

It's hyperbole.
posted by Sys Rq at 5:46 AM on December 31, 2013 [2 favorites]

[please stop the Spinal Tap derail]
posted by jessamyn (staff) at 6:43 AM on December 31, 2013

I agree with Sys Rq. It's 123% hyperbole for effect.
posted by Doohickie at 9:02 AM on December 31, 2013 [1 favorite]

I remember once a sports talk show host quoting an athlete who was recovering from injury, and mocking him for saying he felt "100% better," because that meant 1x what you were before, so it was the same thing! But there's a difference between feeling 100% of where you were you were at (that would be equal), and feeling a 100% increase of where you were at (which would be double). As another example, 10 increased by 50% is 15, not 5.

Also, people will sometimes use percentages when they could just use multiples ("X is up 300%!" instead of just saying three times).
posted by TheSecretDecoderRing at 9:31 AM on December 31, 2013

I've done statistical analysis which is rife with percentages. In formal statistical analysis especially reviewing or peer reviewing scientific papers, you try to look at the language and fundamental data that scientists and statisticians use along with the percentage figure they give, because unfortunately we can't stop using language to frame the context and the figures we talk about.

In formal statistical analysis, if you're forced to include language into the analysis you do, you have to pay close attention to the use of (in my case English language) comparatives and articles used in the language itself to describe the effect of the percentages.

For example:
"of" usually means that 100% is identical to 1 or 1x the figure being discussed.
So "50% of an apple" is the same as 1/2 of an apple.
Or "100% of a pie" is the same as the whole pie, or 1 pie.
So if you said you wanted 50% of 8, you would get 4. Or 100% of 8 would be 8.

"better" or "better than" or "more than" usually means that you take the original amount as an implied 100%, but not always. It's slippery language.
The usual expectation is that "30% better than before" means that something is 130% as good or as accomplished or performative as it was last time it was measured. So if the original number was 10, then 30% better than 10 is 13.
Or "150% more than last year" again usually means that the current amount is now 250% or 2.5x the original amount. Again, if the original number is 10, then 150% more than 10 is 25 because it's 150% + the implied 100% of 10, so it's 250% of 10, which is 25.

The problem is that sometimes the folks who are writing about things with English language articles and comparators surrounding percentages are indeed sloppy and instead of factoring in the implied 100%, they don't, so they add it or add it erroneously, so when they mean to say that it's actually 2.5x of (or 150% more than or 250% of) the original amount, what they actually say is that it's 350% (and sometimes even neglect to specify "of" or "better than" so you have to dig down into the weeds and figure out what they meant FOR THEM.

And if they don't publish their original data or their math, then you mail them or their publisher and wait for weeks for a response (in scientific papers and publishing, etc.) if you ever get one. Why yes, I have done this and sometimes never gotten a response.

Another problem with percentage use in modern parlance as well as in scientific reporting (both by scientists themselves and the press) is that we've sort of slowly shifted away from absolute values and are often talking about relative values. I can go into this further but it's probably not totally salient to this discussion. But if you consider that often funding for scientific research is awarded by folks who are not really solid on the science being done, you can see why a scientist might report big punchy numbers. A big indicator lots of folks use to evaluate the worth of scientific results is high numbers. So there's often an economic driver for folks to speak in relative risks and risk ratios rather than in absolute numbers.

As an example:
Let's consider that an important metric is reported as an absolute risk of prevalence (total number of cases in the world versus world population) as 5%. So this means that 4 out of 100 people in the world have this thing.
A scientist does some work that results in a verifiable, multiply reproduced drop in prevalence risk from 5% to 2% in treated populations. The work is supported by double-blind studies so the drop of 3% in risk among the population is well established, verifiable and widely understood as causal. But the drop in risk is only 3%.
It looks MUCH BETTER to funders of future and related work if this drop is reported as a relative risk, and unfortunately there's no set standard of which number to put in the denominator or the numerator of the fraction the calculate it, but let's go with the most pop: big number on top, little number on bottom. So 5/2. That's 250%! So the RR dropped by 250%! Right? Right? Or maybe it dropped by 5/3? 166%? Or 2/5? 40%? Or wait, do I subtract 100%? I can't subtract it from 40%, but what about 66% or 150%? Which one would you report?
There are many problems here, but one of the ones most applicable to this question is that each ratio, each relative risk, assumes that the reporter and the audience recognize which number is used where and what it represents.
The best way to report this would probably be to report the change in absolute risk and stay away from using unclear relative risks altogether, but if you did report relative risks, you'd want to articulate what changed and how, something like "After treatment with Substance A, the relative risk dropped by 150%, from an original absolute risk of 5% to an improved risk of 2%." But usually what we get in headlines is RISK DROPS BY 150%!
And you're left with the question, "From what to what?"

This is why it's an important question to clarify the intent and meaning of percentages. Percentages can easily be used to cloud and distort reported statistical findings. Certainly not everyone does this, but it does happen and it can be used to make a scientific finding seem more important than it is.
posted by kalessin at 11:19 AM on December 31, 2013

oop, typo. In the first part of the sciency example, the risk of prevalance is 5% so that should be 5 out of 100 people in the world have this thing.
posted by kalessin at 11:25 AM on December 31, 2013

Giving 110% is like the Beatles saying I need your love 8 days a week. It makes a better song than if it were 7 days.
posted by Obscure Reference at 5:19 PM on December 31, 2013 [2 favorites]

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