## Why is "a 100% increase" the same amount as "a two-fold increase"?

People prefer to avoid the "%" increase for anything more than a few percent, due to confusion it creates: lots of readers fail to realize the distinction between "increase by" and "increase to", and even these who do, make a double take to spot which one was used, especially with values exceeding 100 by not much.

So, is increase of production by 120% better or worse than making it 180% of the previous output? How much is 3000% above norm? Is it 30 or 31 times the norm?

And when you start adding confusion of percent relating to which value they talk about, this becomes a total horror: The production first grew by 50%, then dropped by 50%. Oh, no, it did not return to original value. Currently it's at 75% of the original. Five increases by 10% each are totally not equivalent to increase by 50%.

You are correct in your usage, but it may be preferable to avoid percent if you can use plain fractions and multipliers instead. And on top of that, ALWAYS make sure you give the reference point and scale whenever not obvious, if using multiples and not direct values.

Process this: Today the weather is 15% colder than yesterday.

answered Nov 15 '12 at 9:19

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In almost every specialty area within biology it is necessary to express some things in **quantitative** ways: in numbers, fractions, percentages, averages, ranges, tables, graphs, equations, formulae, and so on. And often the points of interest involve comparisons: one thing is larger than another, one group has fewer diseases than another or lives longer, something is faster than something else or more dense or narrower or cooler and so on. The ability to deal with quantitative aspects of biology is as important as mastery of a working vocabulary and understanding of the scientific method, among other things.

**1**. Quantitative expressions of comparison may appear in many forms. For example, suppose that a person weighed 120 pounds in 1995 and 240 pounds in 1999. How might that change or difference be expressed?

a. Obviously, he gained 120 pounds.

b. You could say his weight doubled during the period of time or that he weighed twice as much in 1999 as in 1995. Both statements are correct and probably obvious to you.

c. It is also correct to say that his weight **increased** by 100% (not 200%). If we take his 1995 weight as the point of reference and ask about the percentage of **change**, then we have **(**(240 lb. - 120 lb.) ÷ 120 lb.**) **X 100 = 100%.

Note well that his weight did not **increase** by 200%. However, it *is* correct to say that his 1999 weight is 200% of his 1995 weight. That is, (240 lb. ÷ 120 lb.) X 100 = 200%. It is important to realize that in both of these equations the lb. units cancel when you divide; so, the percent answer has no units. Here the point of interest is his total weight in 1999 relative to his total weight in 1995. In the previous comparison the focus was on only the part of his weight that was gained during the 4 years. His 1995 weight (120 lb.) was 100% of his weight at that time. So, if he **added** that much again, he added 100% of his starting weight; that is, he **doubled** it. Looking at these two equations, you see that the difference lies in whether you subtract the starting value from the end value before you divide.

d. It is also correct to say that during the 4 year period his weight increased **1-fold** (one-fold). The common "fold" expression means "100%", and it is very often misused in expressing comparisons. If something increases by 100% (that means it **doubles**), it increases one-fold. An increase of 180% is an increase of 1.8-fold. An increase from 200 to 1000 is a 400% **increase**, which means a 4-fold increase. Here, again, the focus is on the amount of change that has occurred relative to the starting value. If you compare the end value (1000) to the starting value (200), then it is correct to say that the end value is 5-fold of the starting value. Similarly, if Bill's salary were $50,000/year and Sam's salary were $12,500/year, then Bill's salary is 4-fold Sam's. And if Sam's salary rose to $50,000, we would say that:

(i) his salary increased by 300%, (ii) his salary increased 3-fold, (iii) his salary quadrupled, (iv) his current salary is 4-fold his former salary, (v) his current salary is 400% of his former salary, (vi) his current salary is 4 times his former salary.

**2**. The expression "orders of magnitude" sometimes appears in comparisons. One **order of magnitude** means one power of ten. So, the numbers 100 and 10,000 differ by two orders of magnitude; or we can say that 10,000 is two orders of magnitude greater than 100. Similarly, a meter and a micrometer differ by six orders of magnitude: **3**. Suppose that a car's speed decreased from 100 mph to 75 mph. That is a decrease of 25%. We could also say that the speed decreased **by 1/4** or that the speed decreased **to 3/4** of the original speed.

**4**. Suppose that in a study of the squirrel population in a large city park we found that the number of squirrels was 165 in 1987. If the number fell by 40% in the next 10 years, how many squirrels were there in 1997? If the number fell by 2/3 in that period, how many were there in 1997? Suppose that in this squirrel study the number was 248 in 1990 and 171 in 2000. What is the percentage change? Obviously the change is a decrease; so, we write: **5**. You may see expressions of change or difference in other formats, e.g.:

Suppose that Marsha's blood cholesterol level is 180 mg/dL (that's milligrams per deciliter of blood) and that this is a 20% reduction from what it used to be before she altered her diet and exercise habits. What was the cholesterol value previously?

One approach: If 180 mg/dL is a 20% reduction from what it used to be (call the former value "x"), then 180 mg/dL is 80% of "x", that is: Another approach: Start with the former value (call it "x") and subtract from that 20% of itself to get the new value of 180 mg/dL. **HINT: (a small point but a useful one sometimes)**. If this is the correct answer, you should be able to check it by working backwards to the given value. That is, if Marsha had a 225 mg/dL value and reduced that by 20%, what's her new value?

225 mg/dL - (0.2 X 225 mg/dL) = 225 mg/dL - 45 mg/dL = 180 mg/dL. So, our solution must be right. Got that? Do the units come out right here?

**6**. In using percentage comparisons some people become confused when they lose sight of what the reference point is. Here are two examples.

a. Bob managed to reduce his weight from 200 lb. to 150 lb. That's a 25% decrease. Later, though, he lost his self control and his weight went back up by 25%, which put him at 187.5 lb.

Some people would say that's incorrect. If he lost 25% and then gained 25%, his weight must be back at 200 lb, where he started. **Not so!** The 25% loss was relative to the 200 lb. weight. However, the 25% gain was relative to the 150 lb. weight.

b. Mary and Sarah both entered a physics competition that involved taking a difficult exam. Neither of them won an award. The next year both women entered the competition again, and each improved her score by 20% over the previous year. But only Sarah won an award, even though the minimum score for an award (75) was the same as the year before. How can that be? Some people will assume that the women's scores in the second year were the same because the degree of improvement was the same. Not necessarily! If the first scores were 60 (Mary) and 70 (Sarah), the equal percentage improvement would still leave Mary with a score (72), below the required minimum, while Sarah's score (84) was above that minimum value.

**7**. Here's one last example of expressing magnitude of change; and in this sort of instance the meaning is not just wrong but silly. Suppose you hear someone talking about his pulse (heartbeat rate) before, during, and after strenuous exercise. He says that his pulse increases 100% during maximum physical exertion and that after resting for a while it decreases 100%. A 100% decrease means that it has dropped to zero; a few minutes of that and he's dead! Whatever the pulse was, 100% is all there is.

## “Times” and “folds”

*[This post has been updated from its original form to make it better organized, clearer, and more direct and succinct.]*

Bill Walsh is absolutely, completely, 100%, unequivocally, and in all other ways right in his position on the meanings of “times” and “fold” in his recent disagreement that is as much mathematical as it is semantic.

If I start with $100 and end up with $250, did that money grow 2 1/2 times?

A reporter and I are having a good-natured disagreement: He says yes, and I say no.

The confusion between growing, say, 1 1/2 times and 2 1/2 times comes from the fact that some people use *grow* to mean either multiplying or adding, depending on the situation or, I guess, their whim. But *grow* should not mean to multiply. It should only mean to add a number to a starting value. For example, when you say a child grows by an inch, you don’t mean that their previous height was multiplied by 1 inch. The same logic applies when we say something grows by a percentage. For example, say the height of something grew by 1%. That means that 1% of its previous height was *added onto* that previous height, not that its height was multiplied by 1%. Importantly, that percentage is not just an abstract number but refers to a specific, concrete, physical measurement; a raw quantity, say in inches or meters.

The same applies when we convert percentages into *times*. When Bill Walsh and his friend refer to money growing by 1 1/2 or 2 1/2 times, they are really referring to percentages: *percentages of the starting value*. The only way that multiplication comes into a “growing” calculation is when we use a percentage (or “times” or “factor” or “fold”) to calculate how much gets added. When we say something grows by a certain factor of a starting value, what we mean is that we multiply the starting value by that factor and then add that product onto the starting value.

These concepts can be easily seen with a simple linear transformation. Here is the operation as Bill Walsh and I see it:

Let the function “grows

Ntimes” be the linear transformation such that ifxgrowsNtimes, then

x↦x+Nx(xbecomesNtimeslarger thanx).

With this vocabulary, we see that

“$100 grows 1 1/2 times” means $100 ↦ $100 + (1.5)($100) = $250

“$100 grows 2 1/2 times” means $100 ↦ $100 + (2.5)($100) = $350

“100 grows .5 times (grows by half)” means $100 ↦ $100 + (.5)($100) = $150

Alternatively, let’s define a linear transformation according to Bill Walsh’s friend’s use of the English language:

Let the function “grows

Ntimes” be the linear transformation such that ifxgrowsNtimes, then

x↦Nx(xbecomesNtimesx).“$100 grows 2 1/2 times” means $100 ↦ (2.5)($100) = $250

“$100 grows 1 1/2 times” means $100 ↦ (1.5)($100) = $150

“$100 grows .5 times (grows by half)” means $100 ↦ (.5)($100) = $50

In case you aren’t convinced yet, let’s consider the opposite of growing: shrinking. If we use standard English and refer to shrinking as identical to growing except in the opposite direction, then we can easily define an analogous but opposite operation. But first, note that nothing can shrink by more than 100% (1 time); if something shrinks 100%, there is none left.

Here is how I and, presumably, Bill Walsh would define the shrinking transformation:

Let the function “shrinks by a factor of

N” be the linear transformation such that ifxshrinks by a factor ofN, then

x↦x–Nx(xbecomesNtimessmaller thanx).

Note the replacement of “*N* times” from the original formula with “a factor of *N*” in the new formula. This is partly because of how English works and partly because of what I said above about the impossibility of shrinking more than 100%; it can sound awkward to use the plural “times” if the factor is less than 1, and if a value shrinks by 30%, we don’t say it shrank by .3 times but rather that it shrank by a factor of .3, or alternatively by 30%. My “shrinking” transformation thus produces:

“$100 shrinks by a factor of .75 (by 75%)” means $100 ↦ $100 – (.75)($100) = $25

“$100 shrinks by a factor of .25 (by 25%)” means $100 ↦ $100 – (.25)($100) = $75

“$100 shrinks by a factor of .5 (by half)” means $100 ↦ $100 – (.5)($100) = $50

Now let’s define a different “shrinking” transformation analogously to Bill Walsh’s friend’s “growing” transformation:

Let the function “shrinks by a factor of

Nbe the linear transformation such that ifxshrinks by a factor ofN, then

x↦Nx(xbecomesNtimesx)“$100 shrinks by a factor of .75 (by 75%)” means $100 ↦ (.75)($100) = $75

“$100 shrinks by a factor of .25 (by 25%)” means $100 ↦ (.25)($100) = $25

“$100 shrinks by a factor of .5 (by half)” means $100 ↦ (.5)($100) = $50

Not only is that usage of English incoherent, but it requires that growing by half be identical to shrinking by half!

“Now, John, you’re setting up a straw man,” you say. “No one uses *shrink* like that.” Then why do they use *grow* like that?

We can conclude that *grows by* and *shrinks by* are equal but opposite operations and that they each require multiplying the starting value by the growing or shrinking factor and then *adding that product* to the starting value. (For the “shrinking” operation, we can use the phrase “adding that product” to mean adding a negative number, which is the same as subtracting a positive number.)

Now, the word *fold*. As far as the word *fold* goes, I thought its meaning seemed clear to me, but its meaning *as used* seems different, and I would typically avoid using it if I were writing a scientific paper and not just editing others’ papers. In my job as a scientific editor, I think every time I’ve ever seen the word *fold*, it has meant “entailing multiplication by a factor of [the number that comes before it]”. In other words, a 2.5-fold increase always is used to mean “multiplied by 2.5 times”. Therefore, people would say both that $250 is 2.5-fold greater than $100 and that $250 is 2.5-fold (of) $100. That makes no sense to me. Well, no, it does make some kind of sense, but it is inconsistent sense. Well, no, it’s consistent mathematically, because regardless of the construction of the sentence, you just always multiply the original number by the fold factor. But it is inconsistent semantically.

Folds are also frustrating when referring to decreases, but I’ve come to accept formerly non-sensical fold decreases and not care anymore. For example, in the real, physical world, nothing can decrease more than 100%. If some quantity decreases 100%, none of it is left, and there is no such thing as negative matter or energy, so it is non-sensical and meaningless to say something decreased by more than 100%. If something decreases by half, 50% of it is left. If something decreases by two-thirds, one-third of it is left.

Well, if you took *fold* to mean “percent of” or “fraction of”, then the most anything could ever decrease would be 1-fold. If something decreased 0.5-fold, that would be decreasing 50%. Etc. That, however, is not how any biomedical research scientist has ever used *fold* that I’ve seen. They say something “decreased 7-fold”, meaning the final value was 1/7th of the original. If something decreased 150-fold, the final value was 1/150th of the original. That’s stupid, but I guess everyone’s consistent, so now *fold* means “involving a factor or ratio of the original value”.

Bill Walsh has experienced similar frustration with *fold*:

My friendly adversary pointed me to a dictionary that defines the verb

tripleas meaning “to increase three times in size or amount.” And there is the-foldmodel. Atwofold increaseis doubling, athreefold increaseis tripling, and so on. To which I respond: None of the dictionaries on my shelves are that sloppy, and those shelves also hold an otherwise wonderful usage book in which the author is tripped up by-fold, insisting that tripling would be a twofold increase. (It’s a special case,-fold, because “a onefold increase” is not only never used but also impossible. You can fold something in two or three or more, but you can’t fold it in one.)

[I would love to know what wonderful usage book that is. —JTP]

His friendly adversary’s dictionary would, unfortunately, agree with most biomedical scientists on the use of *-fold*: they use a twofold increase to mean multiplying by a factor of 2 (doubling), even though multiplying by and increasing (growing) by ought to mean different things. I suppose you could argue that increasing (growing) by can mean adding to *or* multiplying by according to the whims of the author and following no consistent or pre-defined rule, but that is illogical to me and goes against the meanings of the words “increase” and “grow” as I understand them, especially when the word “by” is added after them.

This is a perfect example of why I am largely prescriptivist: so that meanings can stay as consistent as possible and people separated by time and space (and mathematical ability) will mean the same thing when they use the same words. I will stop being prescriptivist when I gain the ability to understand how people can not care that the same words mean substantially, crucially different things to different people.

If *shrink* is not the perfect opposite of *grow*, *less than* is not the opposite of *more than*, and *grow (by)* can mean two mathematically distinct things, then, well, I don’t know anything and it’s pointless to write or talk about anything.

This entry was posted in Grammar, Language. Bookmark the permalink.

Sours: https://www.jpetrie.net/2012/01/15/times-and-folds/Enter the original number and the final number into the calculator to determine the total fold increase.

### Fold Increase Formula

The fold increase formula is as follows:

F-A:B = B/A

- Where F-A:B is the fold increase from A to B
- B is the final number
- A is the original number

### Fold Increase Definition

A fold increase is defined as the ratio of an increased number to the original number. For example, an original number of 15 and the final number of 30 would be a 2 fold increase (30/15=2).

### What is a 20 fold increase?

A 20 fold increase is defined as an increase in a number of 20 times or in other words a %2,000 increase. A 20 fold increase of the number 2 would be 20*2 = 40.

### What does 2 fold increase mean?

A 2 fold increase would mean that the final number is 2 times the original number. A 2 fold increase of the number 5 would be 10, so 5*2=10.

### How to calculate a fold increase?

To calculate a fold increase, first, determine the original number.

For this example, we will say that number is 5.

Next, determine the final number.

For this example, we will say this is 15.

Finally, using the formula above, calculate the fold increase.

F-A:B = B/A

F-A:B=15/5

**F-A:B = 3**

## Fold increase 1

## Fold change

**Fold change** is a measure describing how much a quantity changes between an original and a subsequent measurement. It is defined as the ratio between the two quantities; for quantities *A* and *B the fold change of *B* with respect to *A* is *B*/*A*. In other words, a change from 30 to 60 is defined as a fold-change of 2. This is also referred to as a "one fold increase". Similarly, a change from 30 to 15 is referred to as a "0.5-fold decrease". Fold change is often used when analysing multiple measurements of a biological system taken at different times as the change described by the ratio between the time points is easier to interpret than the difference.*

Fold change is so called because it is common to describe an increase of multiple *X* as an "*X*-fold increase". As such, several dictionaries, including the Oxford English Dictionary^{[1]} and Merriam-Webster Dictionary,^{[2]} as well as Collins's Dictionary of Mathematics, define "-fold" to mean "times", as in "2-fold" = "2 times" = "double". Likely because of this definition, many scientists use not only "fold", but also "fold change" to be synonymous with "times", as in "3-fold larger" = "3 times larger".^{[3]}^{[4]}^{[5]}

Fold change is often used in analysis of gene expression data from microarray and RNA-Seq experiments for measuring change in the expression level of a gene.^{[6]} A disadvantage and serious risk of using fold change in this setting is that it is biased^{[7]} and may misclassify differentially expressed genes with large differences (*B* − *A*) but small ratios (*B*/*A*), leading to poor identification of changes at high expression levels. Furthermore, when the denominator is close to zero, the ratio is not stable, and the fold change value can be disproportionately affected by measurement noise.

### Alternative definition[edit]

There is an alternative definition of fold change,^{[citation needed]} although this has generally fallen out of use. Here, fold change is defined as the ratio of the difference between final value and the initial value divided by the initial value. For quantities *A* and *B*, the fold change is given as (*B* − *A*)/*A*, or equivalently *B*/*A* − 1. This formulation has appealing properties such as no change being equal to zero, a 100% increase is equal to 1, and a 100% decrease is equal to −1. However, verbally referring to a doubling as a one-fold change and tripling as a two-fold change is counter-intuitive, and so this formulation is rarely used.

This formulation is sometimes called the relative change and is labeled as *fractional difference* in the software package Prism.^{[8]}

### Fold changes in genomics and bioinformatics[edit]

In the field of genomics (and more generally in bioinformatics), the modern usage is to define fold change in terms of ratios, and not by the alternative definition.^{[9]}^{[10]}

However, log-ratios are often used for analysis and visualization of fold changes. The logarithm to base 2 is most commonly used,^{[9]}^{[10]} as it is easy to interpret, e.g. a doubling in the original scaling is equal to a log_{2} fold change of 1, a quadrupling is equal to a log_{2} fold change of 2 and so on. Conversely, the measure is symmetric when the change decreases by an equivalent amount e.g. a halving is equal to a log_{2} fold change of −1, a quartering is equal to a log_{2} fold change of −2 and so on. This leads to more aesthetically pleasing plots, as exponential changes are displayed as linear and so the dynamic range is increased. For example, on a plot axis showing log_{2} fold changes, an 8-fold increase will be displayed at an axis value of 3 (since 2^{3} = 8). However, there is no mathematical reason to only use logarithm to base 2, and due to many discrepancies in describing the log_{2} fold changes in gene/protein expression, a new term "loget" has been proposed.^{[11]}

### See also[edit]

### Notes[edit]

**^**"Free OED – Oxford English Dictionary".**^**"Definition of TWOFOLD".**^**Cieńska, M.; Labus, K.; Lewańczuk, M.; Koźlecki, T.; Liesiene, J.; Bryjak, J. (2016). "Effective L-Tyrosine Hydroxylation by Native and Immobilized Tyrosinase".*PLOS ONE*.**11**(10): e0164213. doi:10.1371/journal.pone.0164213. PMC 5053437. PMID 27711193.**^**Cunningham, M. W. Jr.; Williams, J. M.; Amaral, L.; Usry, N.; Wallukat, G.; Dechend, R.; LaMarca, B. (2016). "Agonistic Autoantibodies to the Angiotensin II Type 1 Receptor Enhance Angiotensin II–Induced Renal Vascular Sensitivity and Reduce Renal Function During Pregnancy".*Hypertension*.**68**(5): 1308–1313. doi:10.1161/HYPERTENSIONAHA.116.07971. PMC 5142826. PMID 27698062.**^**Li, B.; Li, Y. Y.; Wu, H. M.; Zhang, F. F.; Li, C. J.; Li, X. X.; Lambers, H.; Li, L. (2015). "Root exudates drive interspecific facilitation by enhancing nodulation and N_{2}fixation".*PNAS*.**113**(23): 6496–6501. doi:10.1073/pnas.1523580113. PMC 4988560. PMID 27217575.**^**Tusher, Virginia Goss; Tibshirani, Robert; Chu, Gilbert (2001). "Significance analysis of microarrays applied to the ionizing radiation response".*Proceedings of the National Academy of Sciences of the United States of America*.**98**(18): 5116–5121. doi:10.1073/pnas.091062498. PMC 33173. PMID 11309499.**^**Mariani, T. J.; Budhraja V.; Mecham B. H.; Gu C. C.; Watson M. A.; Sadovsky Y. (2003). "A variable fold change threshold determines significance for expression microarrays".*FASEB J*.**17**(2): 321–323. doi:10.1096/fj.02-0351fje. PMID 12475896. S2CID 16668234.**^**"Prism".*www.graphpad.com*. Retrieved 2018-06-07.- ^
^{a}^{b}Robinson, M. D.; Smyth, G. K. (2008). "Small-sample estimation of negative binomial dispersion, with applications to SAGE data".*Biostatistics*.**9**(2): 321–332. doi:10.1093/biostatistics/kxm030. PMID 17728317. - ^
^{a}^{b}Love, M. I.; Huber, W.; Anders, S. (2014). "Moderated estimation of fold change and dispersion for RNA-seq data with DESeq2".*Genome Biology*.**15**(12): 550. doi:10.1186/s13059-014-0550-8. PMC 4302049. PMID 25516281. **^**Pacholewska, Alicja (2017). "'Loget' – a Uniform Differential Expression Unit to Replace 'logFC' and 'log2FC'".*Matters*. doi:10.19185/matters.201706000011. ISSN 2297-8240.

### External links[edit]

## Bioinformatic Fold Change Analysis Service

**Fold change (FC)** is a measure describing the degree of quantity change between final and original value. For instance, for a data set with an original value of 20 and a final value of 80, the corresponding fold change is 3, or in common terms, a three-fold increase. Fold change is computed simply as the ratio of the changes between final value and the original value over the initial value. Thus, if the original value is X and final value is Y, the fold change is (Y - X)/X or equivalently Y/X - 1. As another example, a change from 60 to 30 would be a fold change of -0.5, while a change from 30 to 60 would be a fold change of 1 (a change of 2 times the original).

However, confusion and ambiguity can arise from this use. For example, although 2 fold is 200%, or 2x, a 2-fold increase, is, as noted above, understood by some to mean an increase of 300% as in "90 is 2 times greater than 30." Yet, several dictionaries, including the Oxford English Dictionary and Merriam-Webster Dictionary, as well as Collins's Dictionary of Mathematics, define "-fold" to mean "times," as in "3-fold" = "4 times." Likely because of this definition, many researchers use both“fold”and“fold change” to be synonymous with "times," as in "2-fold larger" = "2 times larger." Among some experts in this field use persists of fold change as in "40 is 1-fold greater than 20." Therefore, one could argue that the use of fold change, as in "X is 3-fold greater than 15" should be avoided altogether, since some will interpret this to mean X is 45 whereas others will understand this to mean that A is 60.

Fold change method is often used in the analysis of gene expression data in transcriptomics, proteomics and metabolomics, for measuring the changes in different conditions. The obvious advantage of fold change analysis is that it makes sense to biologists. A disadvantage to and severe risk of using fold change method in this setting is that it is biased and may miss differentially expressed genes with large differences (Y-X) but small ratios (X/Y), leading to a high miss rate at high intensities.

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Well, I think you dont mind, and Ill turn off the light to attract less attention. And so I was left in complete darkness in a dirty basement, with my bitter thoughts. An absolutely naked and completely defenseless woman, waiting for no one knows what.

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