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
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).