|WikiProject Statistics||(Rated Start-class, High-importance)|
Ambiguity in 'twice as much'Edit
someone please indicate the meaning of the 1st paragraph hypothesis... if population never changed, let's say 100 people. in the 80's 40% was smoking, in the 90's 30%. that's a reduction of 10 people. so, the change of 10 from 100, is 10%... where does the 25% cames from? — Preceding unsigned comment added by 18.104.22.168 (talk) 07:45, 3 December 2011 (UTC)
- 10 people who quit is 25% of 40 people who smoked. Abolen (talk) 20:01, 3 December 2011 (UTC)
- Still "assuming the same total population in both years" is wrong. That only needs to be true if the 25% should also reflect on the change of the absolute number of people who. A reduction from 40% to 30% is a 25% decrease, no matter how the population changed. The whole reason of using % is to become independent of the absolute numbers. And 30%/40%=0.75=75% and that is always true. Therefore, I will remove the false statement. --22.214.171.124 (talk) 12:36, 21 December 2011 (UTC)
I disagree with this statement in the article: "Statements such as "between 1980 and 1990, the smoking rate decreased twice as much as the lung cancer rate" are ambiguous: it is not clear whether percentages or percentage points are being compared."
I don't see how it is ambiguous. By saying "the smoking rate decreased twice as much as the lung cancer rate" the speaker would merely be saying that the decrease of one was twice the value of the other. I see how a statement could be incorrect when a speaker misuses the term "percent" but that term is not mentioned in this sentence. When no specific unit is mention it is convention to assume the author is referring to absolute value. Therefore, one must also assume that if one were to nominate a unit of measurement to be applied to the value referred to as decreasing, then one would also have to apply the same unit of measurement to the the value with which it is being compared. After all, you don't very often hear a beef farmer say "my herd of cows has decreased by twice as many chickens as last year" do you now?
- Suppose smoking rate decreases from fraction 0.2 to 0.1 of the population, while lung cancer rate decreases from 0.01 to 0.005 of the population per year. Then the decreases here are either
- a fall in both cases by a factor 0.5
- a reduction of the fraction affected by 0.1 of the population in the first case, or 0.005 in the second case.
- So here the "descrease" is either the same or different by a factor of 200 depending on whether change is measured multiplicatively or additively. It might be better to describle this avoiding percentages, but the article is about "percentage points". Melcombe (talk) 18:28, 6 January 2012 (UTC)
- Some of this is discussed further in Relative change and difference, and perhaps all of the "measuring change" stuff could be moved/dealt with there. That article has a few external links/references, but I don't know if they are relevant to this particular point. Melcombe (talk) 13:43, 30 January 2012 (UTC)
- It's pp. But is there supposed to be a space before it? — Preceding unsigned comment added by 126.96.36.199 (talk) 22:45, 31 July 2012 (UTC)
e.g. going from 1% to 9% is an 8 percentage point increase. Who point the citation needed tag in the opening line? Why? Why would a citation be needed for such an obvious fact? Is someone trying to suggest that they believe the increase is actually 8 percent?--XANIA - ЗAНИAWikipedia talk | Wikibooks talk 23:21, 15 December 2014 (UTC)
The definition: "A percentage point or percent point is the unit for the arithmetic difference of two percentages. For example, moving up from 40% to 44% is a 4 percentage point increase, but is a 10 percent increase in what is being measured."
may be interpreted in that way that 44% - 40% = 4% is wrong. This is not true, since 44% is 44/100 and 40% is 40/100. Thus, 44% - 40% = 44/100 – 40/100 = 4/100 = 4%.
I hope that everyone agrees with 50% = 1/2 and 25% = 1/4. Thus 50% - 25% = 1/2 – 1/4 = 2/4 – 1/4 = 1/4. If 25 % = 1/4, then 1/4 = 25%.
One may argue that percentages indicate ratios, not differences. However, the difference of two ratios is a ratio again. Proof: Let’s consider this:
As one can see, we've got the fraction again. Moreover, in our case (when talking about percentage) m = n = 100 since a percentage (from Latin per centum "by a hundred") is a number or ratio expressed as a fraction of 100:
Thus, the result has the same denominator as input numbers (namely x/100 and y/100). Therefore, by the definition of percentage, we are allowed to say that (x-y)/100 is (x-y)%.
The definition of the percentage point states: "40% to 44% is a 4 percentage point increase, but is a 10 percent increase in what is being measured". This is also true, but here, we need to be very sure what is being measured. Let's clarify how this 10% increase is calculated (and let's start with m=n for the simplicity):
Thus, we've got a fraction again, however in this case the denominator has changed. In other words, we are taking the value of x as 100% (or simply rescaling x to 100). Otherwise, by the definition of percentage, we are not allowed to call that fraction as percentage.
We should also note that, the 10 percent increase (from 40% to 44%) could be calculated only when the subtraction operation is involved (minus 1 in the equation above actually corresponds to subtracting 100%). Therefore, the protest that "percentages indicate ratios, not differences" is incorrect.
Final example: 1
There is 1/2 (or 50%) of a very tasty pie on the table. One hour later, there is only 1/4 (or 25%) of pie on the table. We can say this:
- The pie has decreased by 25% (or by 25 percentage points), since 50%-25%=25%. The denominator remains the same.
- The rest of the pie has decreased by 50%, since (50%-25%)/50% = (0.5-0.25)/0.5 = 0.5 = 50%. But here, the denominator has changed. Therefore, we are not talking about the pie, but about the rest of the pie. And we need to say this clearly when talking about 50% decrease. Otherwise, it could be confusing.
Final example: 2
For example, if Bob takes out a $1,000 mortgage from the bank at an interest rate of 10% annually (compound interest). The yearly mortgage payment is $150. The first payment will include an interest charge of $100 ($1,000 x 10%) and a principal repayment of $50 ($150 - $100). The outstanding mortgage balance after this payment is $950 ($1,000 - $50). The next payment will be equal to the first, $150, but it will now have a different proportion of interest to the principal. The interest charge for the second payment will be $95 ($950 x 10%) while the principal prepayment will be $55 ($150 - $95). This means that Bob is ready to pay interest charge of $100 in the first year, $95 in the second year, etc. since the principal sum is decreasing every year.
But suddenly, the bank states new interest rate of 15% for the second year! In this situation, Bob is going to pay the interest charge of $142.5 ($950 x 15%) in the second year. We can say this:
- The interest rate has increased by 5% (or by 5 percentage points), since 15%-10%=5%.
- The interest charge has increased by 50%, since (15%-10%)/10% = (0.15-0.10)/0.10 = 0.5 = 50%. In other words, Bob is going to pay $142.5 instead of $95, what is 142.5/95 = 1.5 = 150%. This is exactly 50% more (150% - 100%) than expected interest charge for the second year. And that is exactly 5% increase of interest rate, since the interest rate is the amount charges expressed as a percentage of the principal (principal for the second year is $950). More precisely, $142.5/$950 = 1.05 = 105%.
In many situations, the increase/decrease of interest rate is misinterpreted as increase/decrease of interest charge and vice-versa. Thus, it is quite common to express the increase/decrease of interest rate in percentage points, thus, avoiding such confusions. However, no one should disagree that the percentage point is equivalent to percentage. 188.8.131.52 (talk) 10:45, 19 January 2021 (UTC)
- Whatever your examples above prove or don't prove, the term "percentage point" was developed to avoid this ambiguity: interest rates rise by 2%, from 1% to 3%, or from 1% to 1.02%? If there were no ambiguity, there would be no need for the term. Tony (talk) 11:04, 21 January 2021 (UTC)
- Dear Tony1, I do understand that need for such term. I am not against it. However, from the mathematical point of view, the arithmetic difference of two percentages is the value in percentages again. And the people use it. Therefore, if wikipedia accepts only "percentage points", then: 1/ it may bring even more misunderstandings, 2/ it is wrong. 184.108.40.206 (talk) 12:24, 26 January 2021 (UTC)