Many people roll their eyes or their brain freezes whenever a mathematical calculation is written about or discussed in conversation. This article uses some simple arithmetic that shouldn’t cause that reaction.
This is a part of a planned series to discuss how numbers can be manipulated or misused to portray favorable or unfavorable viewpoints. Specifically, the series will address percentages, probabilities (or chance), and statistics.
Numbers and Percentages
There are instances when numbers are misrepresented on purpose.
- I say weigh 185 lbs (84 kg) when I know my weight is 192 lbs (87 kg).
- I claim I have $250,000 in my checking account when I know that I only have $37.52.
These are obvious deliberate misrepresentations that we aren’t considering here. We’re going to look at the more subtle (sometimes not so subtle) misrepresentations. Percentage, probabilities, chance, and statistics involve numbers that we encounter in everyday life. These can be deliberately misstated but sometimes their misuse is caused by lack of understanding. Let’s start with being clear about using numbers and percentages.
Losing Weight
If I weigh 200 lbs and need to lose weight, I may be content with losing 5 lbs. If, however, my doctor tells me that I need to reduce my weight to 95% of my current weight, how do I know what that means?
As a number, 95% is the same as 95/100 or 0.95. Converting it to a number makes the arithmetic simpler. So, if I need to go from 200 lbs to 95% of 200 lbs and want to know what that target number is, I can do a simple arithmetic calculation. (NOTE: I will use “(” and “)” for calculation separators below and will use “[” and “]” for parenthetical information within the calculation – for those who are experience with applying parentheses for arithmetic calculations):
1.a.: My Desired Weight = 95% [0.95] X 200 lbs
1.b.: My Desired Weight = 0.95 X 200 lbs = 190 lbs
Now I know what my new target weight should be.
The sometimes trickier question involves calculating the percentage when you know a set of numbers. We can use the same example above (1.b.) in reverse to understand this. If I now weigh 190 lbs and previously weighed 200 lbs, my new weight is what percentage of my old weight? We already know it’s 95% but what if we don’t know?
2.a.: Percentage = (new weight (190) / old weight (200)) X 100%
2.b.: Percentage = 190 / 200 X 100% = 95%
Once you understand percentages you will mentally understand the multiplication by 100%. That additional multiplier (100% = 1) is used to convert the answer into percentages. But what if I want to know how much weight I lost as a percentage. I weighed 200 lbs but now weigh 190 lbs. I lost 10 lbs. Since the 10 lbs once was part of my 200 lb weight, I need to compare the 10 lbs to my previous 200 lbs.
3.a.: Percentage = (amount lost (10) / old weight (200)) X 100%
3.b.: Percentage = 10 / 200 X 100% = 5% (0.05)
You can now check that this answer makes sense by doing the reverse calculation as we did in the first example:
4.a.: Lbs Lost = 200 lbs X 5% (0.05) = 10 lbs
Ok. So far it seems simple. Now, what about other percentages with these same numbers? I now weigh 190 lbs but previously weighed 200 lbs. How much larger, in percentage, was my old weight compared to my new weight?
5.a.: Old weight as % of new weight = (200 lbs / 190 lbs) X 100% = 105.3%
Notice that we didn’t get exactly 5% for that calculation. That can seem confusing. More about that later.
But, sadly, not long after I lost that 10 lbs, I gained 14 lbs. I now weigh 204 lbs. If I want to know the percentage that I gained, how much is that?
6.a.: Percentage gained = (14 lbs /190 lbs) X 100% = 7.4%
Using Percentages
This may seem extremely simple. However, for people who don’t use these types of calculations frequently, the calculations in 3.b., 5.a., and 6.a. can become confused. Example 5.a. shows a slight difference in percentage (if we expected the answer to be exactly 105%). Let’s look at different numbers to see how this variation is more obvious. Using a larger change in the numbers shows more clearly why 3.b. and 5.a. give different results. We will consider 200 reduced by 50 (200 – 50 = 150) in this example. (The “old number” is 200. The “new number” is 150.)
7.a.: Percentage reduction = (50 / 200) X 100% = 25%
7.b.: Percentage of old number to new number = (200 / 150) X 100% = 133.3%
7.c.: Percentage of new number to old = (150 / 200) X 100% = 75%
This demonstrates the change in percentage numbers more visibly than in 5.a. In words, what 7.a. and 7.b. tell us is that:
(a) If we subtract 50 from 200 (leaving us with 150) and want to know the percentage reduction that represents, the answer if found by 50 divided by 200. We use 200 and not 150 because we are comparing 50 with 200. The calculation in 7.a. shows that subtracting 50 is 25% reduction of 200. That makes since because 50 + 50 + 50 + 50 = 200. So 50 is one fourth (25%) of 200. Example 7.c. is a similar view. Since 50 + 50 + 50 = 150, it is three fourths of 200.
(b) For some people, it seems that 200 should also be 25% larger than 150. But in 7.b., we see that 200 is one hundred thirty-three percent of 150 or one third larger than 150. Another way to consider it is to start at 150 and added 50. What part or percentage of 150 are we adding? Of course, 50 + 50 + 50 = 150. So, 50 is one third (33%) of 150. Then, 200, in 7.b., is larger than 150 by one third (i.e., 50 or 33%) of 150.
We encounter numbers like these in the news or in reading. It helps to understand what they mean. “Unemployment is currently 3.8%.” So the number of unemployed is 3.8% of the total “workforce”. If we hear “The Dow increased by 200 points from 30,291” or “The Dow rose by 0.7%”. We understand that 200/30291 = 0.007 (0.7%).
Percentages are not too difficult. Some usages can be confusing to understand. By thinking through these examples, hopefully this is clearer. When we move into discussion of misuse of probability and statistics, an understanding of using percentages will become critical.