Absolute value is a tricky thing. Students love it because it is so easy. They probably come away thinking, "Could I have done this right? That just felt way to easy. And even if I did do it right, it seams pretty pointless." There actually are mathematics applications to absolute value. Here are a few.
Average Deviation – There are a number of formulas that
measure the variability of data. A common one is the standard deviation.
However, average deviation is similar and easier to compute. The average
deviation simply finds the average distance each number is from the mean. To find the average deviation, the
distance from the mean is found for each piece of data in the set. Those distances
are added and then divided by the number of pieces of data. If the mean is 32,
we would want 28 and 36 to both be considered positive 4 units away from the
mean. Absolute value is used so there are no negative values for those
distances.
Example:
- A set of data is {21, 28, 31, 34,
46}. The mean average is 32.The average deviation is 6.4.
Statistical Margin of Error – As mandated by the U.S. Constitution,
every ten years the government is required to take a census counting every
person in the United States. It is a huge undertaking and involves months of
work. So how are national television ratings, movie box office results, and unemployment
rates figured so quickly – often weekly or even daily? Most national statistics
are based on collecting data from a sample. Many statistics that are said to be
national in scope are actually data taken from a sample of a few thousand. Any
statistic that is part of a sample is subject to a margin of error. (In 1998,
President Clinton attempted to incorporate sampling in conducting the 2000
census, but this was ruled as unconstitutional.)
Example:
- On October 3, 2014 the government
released its unemployment numbers for the month. Overall unemployment
was listed at 5.9%. The report also stated that the margin of error was 0.2%.
Government typically uses a level of confidence of 90%. Thus there is a 90% chance
that the actual unemployment rate for the month was x, where |x-5.9| ≤ 0.2.
Richter
Scale Error – The Richter scale is used to measure
the intensity of an earthquake. However, like many measurements, there is a
margin of error that needs to be considered. Scientists figure that the actual
magnitude of an earthquake is likely 0.3 units above or below the reported
value. If an earthquake is reported to have a magnitude of x, the
difference between that and its actual magnitude, y, can be expressed using absolute
value: |x-y| ≤ 0.3.
Body
Temperature – “Normal” body temperature is assumed
to be 98.6° F. For any student that has made the case that anything other than 98.6°
prevents their attendance at school, there is good news. There is a range
surrounding that 98.6 value that is still considered in the normal range and
will allow your attendance at school. Your 99.1° temperature is probably just
fine. Supposing plus or minus one degree is safe, an expression could be
written |x-98.6| ≤ 1.0, which would represent the safe range. Why is the
absolute value a necessary part of this inequality? Without it, a temperature of
50 degrees would be considered within the normal range, since 50-98.6 = -48.6, which
is, in fact, well less than 1.0.
