Any measurement made with a measuring
device is approximate.
If you measure the same
object two different times, the two measurements may not
be exactly the same. The difference between two
measurements is called a variation
in the measurements.
Another word for this variation
 or uncertainty in measurement  is "error."
This "error" is not the same as a
"mistake." It does not mean that you got
the wrong answer. The error in measurement is a
mathematical way to show the uncertainty in the
measurement.
It is the difference
between the result of the measurement and the true value
of what you were measuring. 
The
precision
of a measuring instrument
is determined by the smallest
unit to which it can measure. The precision is said
to be the same as the smallest fractional or decimal
division on the scale of the measuring instrument. 

Ways of Expressing
Error in Measurement:
1.
Greatest Possible Error:
Because
no measurement is exact, measurements are always made to
the "nearest something", whether it is stated or not.
The greatest possible error
when measuring is considered to be one half of that
measuring unit. For example, you measure a length
to be 3.4 cm. Since the measurement was made to
the nearest tenth, the greatest possible error
will be half of one tenth, or 0.05.

2. Tolerance intervals:
Error in measurement
may be represented by a tolerance
interval (margin of error). Machines
used in manufacturing often set tolerance intervals, or
ranges in which product measurements will be tolerated
or accepted before they are considered flawed.
To determine the
tolerance interval in a measurement, add and
subtract onehalf of the precision of the measuring instrument to
the measurement.
For example, if a measurement made with a metric ruler is 5.6
cm and the ruler has a precision of 0.1
cm, then the tolerance
interval in this measurement
is 5.6
0.05 cm,
or from 5.55 cm to
5.65 cm. Any measurements within this range are
"tolerated" or perceived as correct.

Accuracy
is a measure of how close the
result of the measurement comes to the "true",
"actual", or
"accepted"
value.
(How close is
your answer to the accepted value?) 
Tolerance
is the greatest range of
variation that can be allowed.
(How much
error in the answer is occurring or is acceptable?) 

3.
Absolute Error and Relative Error:
Error in measurement
may be represented by the actual amount of error, or by
a ratio comparing the error to the size of the
measurement.
The absolute
error of the measurement shows how large the error
actually is, while the relative error
of the measurement shows how large the error is
in relation to the correct value.
Absolute errors do not
always give an indication of how important the error may
be. If you are measuring a football field and the
absolute error is 1 cm, the error is virtually
irrelevant. But, if you are measuring a small
machine part (< 3cm), an absolute error of 1 cm is very
significant. While both situations show an
absolute error of 1 cm., the relevance of the error is
very different. For this reason, it is more useful
to express error as a relative error. We will be
working with relative error. 
Absolute Error:
Absolute error is simply the amount of physical error in
a measurement.
For example, if you know a length is 3.535
m + 0.004 m, then 0.004 m is an absolute error.
Absolute error is positive.
In plain English: The absolute error is the
difference between the measured value and the actual
value.
(The
absolute error will have the same unit label as the measured
quantity.)
Relative Error:
Relative
error is the ratio of the absolute error of the
measurement to the accepted measurement. The relative error
expresses the "relative size of the error" of the
measurement in relation to the measurement itself.
When the
accepted or true measurement is
known, the
relative error is found using
which is considered to be a measure of accuracy.

Should the
accepted or true measurement
NOT be known, the
relative error is found using the measured
value,
which is considered to be a measure of precision.


In plain English:



4.
Percent of
Error:
Error in measurement
may also be expressed as a percent
of error. The percent of error is
found by multiplying the relative error by 100%.
Ways
to Improve Accuracy in Measurement 
1. Make
the measurement with an instrument that has the highest
level of precision. The
smaller the unit, or fraction of a unit, on the measuring
device, the more precisely the device can measure.
The precision of a measuring
instrument is determined by the smallest unit
to which it can measure.
2. Know your
tools! Apply correct techniques when
using the measuring instrument and reading the value
measured. Avoid the error called "parallax"
 always take readings by looking straight down (or
ahead) at the measuring device. Looking at the
measuring device from a left or right angle will give an
incorrect value.
3. Repeat
the same measure several times to get a good average
value.
4. Measure
under controlled conditions. If the object
you are measuring could change size depending upon
climatic conditions (swell or shrink), be sure to measure
it under the same conditions each time. This may
apply to your measuring instruments as well.

Examples:
1. Skeeter,
the dog, weighs exactly 36.5 pounds. When weighed on a
defective scale, he weighed 38 pounds. (a) What is the
percent of error in measurement of the defective scale to
the nearest tenth? (b) If Millie, the cat,
weighs 14 pounds on the same defective scale, what is
Millie's actual weight to the nearest tenth of a
pound 

Answer
(a) 

(b) 
Let x = Millie's actual weight
14 = x + .041x
x = 13.4 pounds 
2.
The actual length of this field is
500 feet. A measuring instrument shows the length to
be 508 feet.
Find:
a.) the absolute error in the measured length of the
field.
b.) the relative error in the measured length of the
field.
c.) the percentage error in the measured length of the
field


Answer:
a.) The absolute error in the length of the field is 8 feet.
b.) The relative error in the
length of the field is
c.) The percentage error in the
length of the field is
3. Find the
absolute error, relative error and percent of error of the approximation 3.14
to the value
,
using the TI83+/84+ entry of pi as the actual value.
