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Apologies, my mathematics is poor but I saw an advert recently which said:

Up to $350\%$ more likely to quit

How can something be greater than $100\%$? It can't be can it?

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    $\begingroup$ $100 \%$ of $x$ is $x$. $350 \%$ of $x$ is $3,5x$. $\endgroup$
    – Red Banana
    Dec 22, 2016 at 0:08
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    $\begingroup$ @OppaHilbertStyle it seems $\LaTeX$ (MathJax at least) doesn't like European decimal separator... $\endgroup$
    – Ruslan
    Dec 22, 2016 at 4:58
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    $\begingroup$ @Ruslan Indeed, and it can be fixed by placing the comma in braces; 3{,}5x produces the desired $3{,}5x$ (which I'm sure is more annoying than being able to use .!) $\endgroup$
    – pjs36
    Dec 22, 2016 at 5:56
  • $\begingroup$ If a big cup holds three and half times as much as a small cup, it holds 350% of what the small cup holds. If you have 10 employees, adding 5 more would be a 50% increase. Adding 35 more would be a 350% increase. $\endgroup$ Dec 22, 2016 at 10:05

8 Answers 8

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Say 1 out of 10 people quit in year 1. Now if 4.5 out of 10 people quit in year 2 it is up 350%. Does this make sense? The percentages compare numbers relative to each other.

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    $\begingroup$ This would be better out of a hundred so we don't deal with half people. Also, it should be 4.5. $\endgroup$ Dec 22, 2016 at 0:09
  • $\begingroup$ Thanks, I made the correction. $\endgroup$
    – David
    Dec 22, 2016 at 0:11
  • $\begingroup$ Thanks. Well explained indeed. :-) $\endgroup$
    – user401126
    Dec 22, 2016 at 1:32
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    $\begingroup$ It's not that simple. Another (often more useful) way of thinking of "increasing a percentage by a percentage" is that "1 out of 10" is the same as 1:9 ratio, so 350% more becomes 4.5:9 = 3.33 out of 10. If we took your method instead and tried to increase it by 350% again, we'd get "20.25 out of 10 people" which is nonsense. $\endgroup$ Dec 22, 2016 at 8:11
  • $\begingroup$ Agree to disagree on which is more useful. For the context of this question a standard definition of "percent difference" seems to be at least as didactic. What you're describing is an ambiguity in the language, which is often the case when mathematical statements are made without formal definitions. $\endgroup$
    – David
    Dec 22, 2016 at 18:55
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$100$% of something means all of it.

$200$% of the same thing means twice as much.

$350$% means $3.5$ times the original amount.

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    $\begingroup$ "$350$% more" should mean $4.5$ times the original amount. $\endgroup$
    – Henry
    Dec 22, 2016 at 0:15
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    $\begingroup$ @Henry I think the word more is used as "more likely to quit", not "350% more". $\endgroup$
    – BigbearZzz
    Dec 22, 2016 at 0:17
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    $\begingroup$ It is ambiguous, I think either interpretation is satisfactory to answer the OPs question. $\endgroup$
    – David
    Dec 22, 2016 at 0:19
  • $\begingroup$ I agree with that. $\endgroup$
    – BigbearZzz
    Dec 22, 2016 at 0:21
  • $\begingroup$ @David If $10$ in $1000$ quit anyway and $15$ in $1000$ quit with this alternative, then you might reasonably say quit rates are $0.5$ percentage points higher, or $50\%$ more quit, but not $150\%$ more. It would be better to say $1.5$ times as many quit. Similarly $35$ in $1000$ quit with this alternative, then you might reasonably say quit rates are $2.5$ percentage points higher, or $250\%$ more quit, but not $350\%$ more. It would be better to say $4.5$ times as many quit. $\endgroup$
    – Henry
    Dec 22, 2016 at 0:21
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$\%$ (percent) literally means "over 100".

So when people say "$350\%$ more likely to quit", what they loosely mean is that, if $x$ is how many people (usually) quit, then $x + 3.5x$ represents how many people will quit in this new scenario.

Your question "how can it be bigger than 100%" suggests that you are thinking about probabilities which never go higher than 1. But that's because of its definition and the fact that people frequently use percentages to represent probabilities.

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    $\begingroup$ I suspect is should mean $4.5x$ to cover the word more $\endgroup$
    – Henry
    Dec 22, 2016 at 0:13
  • $\begingroup$ Yes. Thank you @Henry $\endgroup$ Dec 22, 2016 at 0:15
  • $\begingroup$ @Henry how is it 4.5 exactly? If that is not another question in itself. $\endgroup$
    – user401126
    Dec 22, 2016 at 0:31
  • $\begingroup$ @cmp: An increase from $10$ is $15$ is a $50\%$ increase. An increase from $10$ to $45$ is a $350\%$ increase $\endgroup$
    – Henry
    Dec 22, 2016 at 0:55
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What you likely have in mind is the fact that the probability of something happening cannot be more than 100%, that is true. However, in this context, 350% more likely means that the current probability is 4.5 times that which it used to be, so it is still alright.

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  • $\begingroup$ I suspect is should mean $4.5$ times to cover the word more $\endgroup$
    – Henry
    Dec 22, 2016 at 0:14
  • $\begingroup$ Ah, that's a fair point. Will edit. $\endgroup$ Dec 22, 2016 at 0:15
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How can something be greater than 100%?

Nothing can be greater than itself. However, a value can be greater than a reference value.

Consider the sentence

$x\%$ of something.

If something is an abstract value, then the sentence makes sense with $x$ greater than $100$. In your particular example, something refers to probability which is a abstract value. So, makes sense to take $x$ greater than $100$ and this corresponds to multiply by a number greater than one.

If something is a concrete object, then the sentence makes sense only with $x$ less than $100$, or equal $100$.

Examples:

  • Makes sense: I will eat $50\%$ of this pizza.
  • Makes no sense: I will eat $150\%$ of this pizza.
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A percentage $x\%$ is literally nothing more than shorthand for $x/100$. It doesn't quite make sense to talk about, say, eating $150\%$ of a candy bar, but that's not a problem mathematically. I can't easily talk about drinking $-2$ or $\pi$ or $i$ candy bars, but those are all perfectly reasonable numbers.

In this particular case, the advertisement is saying that if (say) people had a $10\%$ chance of quitting without the product, then they'd have a $(3.5 + 1) * 10\% = 45\%$ chance of quitting with the product. It would be odd to say that $350\%$ of the specific people who quit without the product later quit with it, but the message is clear as it stands.

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A percentage is just a fraction - say: $25\% = \frac{25}{100}=\frac{1}{4}$ - that is used as a multiplicative factor of some quantity.

Hence we speak of "$25\%$ of the people present" and we have 80 people, the we are speaking of $\frac{25}{100} \times 80=20$ people.

So, if a percentage is used to denote a part of some total, yes, the fraction must be between 0 and 1 (nothing and all), and the percentage must be between $0\%$ and $100\%$.

But we can also use the fraction to denote some arbitrary change or proportional difference. Suppose the probability of some events $A,B$ are respectively $p_A=0.1$, $p_B=0.3$. We see that the latter is three times more probable n the former: $p_B = 3 \times p_A$ ; hence we can say that the event $B$ is "$300\%$ more likely" than $A$.

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Here is a very simple way to calculate a % greater than 100%. I'll give you an example:

Let's say you were trying to figure out what percentage of an increase is there from 30_Mb to 300_Mb of memory usage.

Take the greater value (in this case, it is 300), divide it by the lesser value (which would be 30) and multiply the result by 100.

Example: $\frac{300}{30} \cdot 100$ = a $1000\%$ increase in memory usage.

or you can calculate the multiples with one less calculation by simply taking the greater value (in this case, it is 300), divide it by the lesser value (which would be 30).

Example: $\frac{300}{30} = 10$ times as much memory usage.

Of course, there are different algebraic formulas which can calculate this, but as you said you're not overly versed in mathematics, so I presented a simple basic-math approach to the calculations.

A side note on perceptions, and how it relates to the "350% more likely to quit" scenario.

This is the problem with the perception of a percentage increase; 1000% sounds much greater than 10 times as much, even though they are equal in terms of numeric representation, especially to those that only use the imperial system.

Often, when one "chooses" to use the an expression that something is 1000% more likely, rather than 10 times more likely, there is a false narrative being projected to add a more excessive "sounding" value.

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