Some Notes on Precision

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Some Notes on Precision
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How long is the green line?
green line
purple line
The very best one can say is that it is about 5
and one half units or a little less than 5 and
one half units. The measurement tool and our
eyes limit the accuracy of a measurement.
Since the measurement tool only shows the nearest
one unit, normally one would round the
measurement to the nearest one unit. Thus, the
green line is 5 units.
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How long is the purple line?
It is not appropriate the say that the line is
about 4 and three fourths units. This would
suggest a precision that you cannot determine
using this measurement tool. The line is
actually just as close to 4 and two thirds. One
could say that the purple line is a little less
than 5 units.
Again, however, since the measurement tool only
shows the nearest one unit, normally one would
round the measurement to the nearest one unit.
Thus, the purple line is 5 units.
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A note on rounding
When you round to the nearest one unit, you are
seeing which ones unit the line is closest to.

The green line is between 5 units and 6 units.
It is closer to 5 units.
The purple line is between 4 units and 5 units.
It is closer to 5 units.
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How long is the purple line?
one unit
one tenth unit
Now the grid provides tic marks to indicate
tenths of a unit. We can now round to the
nearest tenth of a unit. In doing this, you are
seeing which tenths unit the line is closest to.
This is a little harder on the eye.
The green line is between 5 and four tenths and 5
and five tenths units. It is clearly closer to 5
and four tenths units. We can round it to 5.4
units.
The purple line appears to be just at 4 and seven
tenths units. It may not be exactly at this
point, however it is 7/10 is clearly the closest
tenth. We can round it to 4.7 units.
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• A more precise measurement of the green and
  purple lines would require instruments more
  precise than the naked eye.
• Because of the limitations of measurement
  instruments and our eyes, measurements are never
  absolutely precise. The true measurement can
  always be a bit different than the recorded
  measurement.
• We will now consider two concepts
• the greatest possible error, and
• the relative error.

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When we obtained a measurement of 5 units for
both lines, we were measuring their lengths to
the nearest unit. Any line whose actual length
was between 4.5 and 5.5 units would be measured
as 5 units. The greatest possible error in this
measurement would be 0.5 units or one half unit.
The actual length might be 0.5 units less or 0.5
units more.
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What is the Greatest Possible Error in these
measurements?
one unit
one tenth unit

• Measurements
• The green line is 5.4 units.
• The purple line is 4.7 units.
• When we obtained a measurement of 5.4 units for
  the green line, we were measuring its length to
  the nearest tenth of a unit. Any line whose
  actual length was between 5.35 and 5.45 units
  would be measured as 5.4 units.
• The greatest possible error in this measurement
  would be five hundredths (0.05) units. The
  actual length might be 0.05 units less or 0.05
  units more.
• What about the purple line?

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one unit
one tenth unit

• Measurements
• The green line is 5.4 units.
• The purple line is 4.7 units.
• The greatest possible error for the purple line
  would be the same, 0.05 units. When we obtained
  a measurement of 4.3 units for the purple line,
  we were measuring its length to the nearest tenth
  of a unit. Any line whose actual length was
  between 4.65 and 4.75 units would be measured as
  4.7 units.
• The greatest possible error in this measurement
  would be five hundredths (0.05) units. The
  actual length might be 0.05 units less or 0.05
  units more.

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green line
The new orange line is measured as 0.3 units.
Since we are measuring to the nearest tenth of a
unit, the greatest possible error is again 0.05
units. Its actual length could be from 0.25
units to 0.35 units.
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• Measurements
• The green line is 5.4 units.
• The orange line is 0.3 units.
• The significance of an error of five hundredths
  of a unit will depend upon what one is measuring.
  If you are measuring a piece for the Space
  Shuttle the error may be more important than if
  you are measuring a piece for a house.
• Another consideration is the size of what you are
  measuring. For our lines, an error of five
  hundredths of a unit could be considered more
  significant in the measurement of the shorter
  orange line than in the measurement of longer
  green line.

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Relative Error

• Measurements
• The green line is 5.4 units.
• The orange line is 0.3 units.
• A percent used to express the relative error of a
  measurement based on the size of what is being
  measured. Relative error is a percent formed
  from the ratio of the greatest possible error to
  the measured length of an object.

x 100
Calculate the greatest possible error for both
lines.
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Relative Error
x 100
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The possible error in the measurement of the
orange line is relatively greater than the
possible error in the measurement of the purple
line. The measurement could be off by nearly
17 for the orange line and only by about 1 for
the green line.
Make sure that you can calculate the greatest
possible error and the relative error of a given
measurement.
Note that proportional reasoning comes up
everywhere!
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