Wednesday, June 20, 2012

Percent and other scales based on powers of 10

Scales based on powers of 10: The most famous scale base on powers of ten in percentage, which really means "per 100". It is much more common to see "53% of the people agree with the president's plan" than ".53 of the people..." or "53 out of every 100 people...". Technically, all those phrases are saying the same thing, but percentage is the most popular.

To get a number based on a power of 10 scale, you take the small number, divide it by the big number and multiply by the power of ten, so it is  small/big*scale. Sometimes we need greater precision because the proportions are so small, the small number is tiny in comparison to the big.

When I ask a class what is the legal limit for blood alcohol while driving, invariably someone will say "point oh eight" and most people will agree. But .08 is wrong; .08 = 8%, and the correct answer is .08% = .0008. I don't blame the students. The number is badly represented and it is an easy mistake to make. Let's take a look at the number on other scales of 10.

.08 out of 100 is the same as
.8 out of 1,000 or
8 out of 10,000 or
80 out of 100,000

80 parts out of 100,000 is a tiny proportion. To give an idea, ounce of pure alcohol mixed into ten gallons of blood would give you 78 parts out of 100,000, and most people have between a half gallon and a gallon and a half of blood in their body, between 4 and 12 pints. The amount of alcohol in a person's blood stream that is over the legal limit is about the same amount of alcohol as found in a capful of mouthwash used after brushing your teeth.

We will look at the per 100,000 scale for another type of statistic, measurements of mortality rates.

Here are the number of homicides in some local cities in 2007.

Oakland: 124 homicides
Richmond: 28 homicides
San Francisco: 98 homicides

Clearly, comparing these numbers is misleading, because we know these cities have very different numbers of citizens, so the standard way to measure these statistics is the per 100,000 population scale, which we find by the formula

small/big x scale

which in this case is

(# of homicides)/(city population) x 100,000

Oakland's population in 2007 is estimated at 415,000, Richmond at 106,000 and San Francisco at 825,000, so the murder rates on this standard scale are as follows

Oakland: 124/415000 * 100000 = 29.9
Richmond: 28/106000 * 100000 = 26.4
San Francisco: 98/825000 * 100000 = 11.9

So even though more people were murdered in San Francisco than in Richmond in 2007, the murder rate in Richmond was over twice as high, because Richmond has barely 1/8 of the population of San Francisco. (note: The trends for the three cities this decade are going in different directions. Oakland's murder rate is on the rise, while Richmond's is falling and San Francisco's has stayed about the same.)

Practice problems: (answers given in comments)
1) Here are the homicide numbers for Oakland, Richmond and San Francisco from 2004.

Oakland: 96 homicides, 399,000 population
Richmond: 40 homicides, 99,000 population
San Francisco: 96 homicides, 775,000 population

Find the murder rates from these years, rounded to the nearest tenth per 100,000 population and rank them from lowest (1st) to highest (3rd).

1 comment:

Prof. Hubbard said...

Oakland: 96 homicides, 399,000 population

96/399000 * 100000 = 24.06...

24.1 per 100,000 population

Richmond: 40 homicides, 99,000 population

40/99000 * 100000 = 40.4040...

40.4 per 100,000 population

San Francisco: 96 homicides, 775,000 population

96/775000 * 100000 = 12.387...

12.4 per 100,000 population

Lowest: San Francisco 12.4 per 100,000
Second: Oakland 24.1 per 100,000
Highest: Richmond 40.4 per 100,000