Download Accuracy, Precision, Errors, Interpolation, and Significant Figures

ACCURACY
• Accuracy is the closeness of a measured value to the true
value.
• For example, the measured density of water has become more
accurate with improved experimental design, technique, and
equipment.
Density of H2O at 20° C
(g/cm3)
1
1.0
1.00
0.998
0.9982
0.99820
0.998203
ACCURACY
• Percent error is used to estimate the accuracy of a
measurement.
• Percent error will always be a positive.
• What is the percent error if the measured density of titanium (Ti)
is 4.45 g/cm3 and the accepted density of Ti is 4.50 g/cm3?
PRECISION
• Precision is the agreement between repeated measurements of the
same sample. Precision is usually expressed as a standard deviation.
• For example, the precision of a method for measuring arsenic (As) was
determined by measuring 7 different solutions each containing 14.3 μg/L of
As.
Measured
Concentration
(μg/L)
18.4
13.6
13.6
14.2
16.0
13.6
17.8
Average = 15.3 μg/L
Standard Deviation = 2.1 μg/L
What is the true concentration of As in this experiment?
14.3 μg/L
Estimate the accuracy of this method.
How precise is this method?
2.1 μg/L
ACCURACY AND PRECISION
• Describe the accuracy and precision of these 4 targets.
Accurate, and
precise
Precise, but
not accurate
Accurate, but
not precise
Not accurate,
and not
precise
ERRORS
• Systematic (or determinate) errors are reproducible and
cause a bias in the same direction for each measurement.
• For example, a poorly trained operator that consistently makes
the same mistake will cause systematic error. Systematic error
can be corrected.
• Random (or indeterminate) errors are caused by the natural
uncertainty that occurs with any measurement.
• Random errors obey the laws of probability. That is, random
error might cause a value to be over predicted during its first
measurement and under predicted during its second
measurement. Random error cannot be corrected.
INTERPOLATION AND SIGNIFICANT FIGURES
• By convention, a measurement is recorded by writing all exactly
known numbers and 1 number which is uncertain, together with
a unit label.
• All numbers written in this way, including the uncertain digit, are
called significant figures.
• For example, the blue line is 2.73 cm long. This measurement
has 3 significant figures. The first 2 digits (2.7 cm) are exactly
known. The third digit (0.03 cm) is uncertain because it was
interpolated or estimated 1 digit beyond the smallest
graduation.
INTERPOLATION AND SIGNIFICANT FIGURES
• What is the volume of water in this graduated cylinder? Always
measure the volume of a liquid at the bottom of the meniscus.
The units are mL.
• The volume of water is 52.8 mL. The 52 mL are exactly known,
and the 0.8 mL is uncertain because it was interpolated or
estimated 1 digit beyond the smallest graduation.
SIGNIFICANT FIGURES AND ZEROS
• Zeros between nonzero digits are significant. That is, 508 cm
has 3 significant figures.
• Leading zeroes merely locate the decimal point and are never
significant. That is, 0.0497 cm equals 4.97 x 10-2 cm and has 3
significant figures.
• Trailing zeros are significant as follows: 50.0 mL has 3
significant figures, 50. mL has 2 significant figures, and 50 mL
has 1 significant figure.
Datum
(grams)
10,034
1.908
0.32
0.00046
150
0.0000160
Number of
Significant
Figures
5
4
2
2
2
3
Datum
(milliliters)
150.
0.705
0.054
5.86 x 10-7
3040
0.0000730
Number of
Significant
Figures
3
3
2
3
3
3
SIGNIFICANT FIGURES, ADDITION, AND SUBTRACTION
• When adding or subtracting do NOT extend the result beyond
the first column with a doubtful figure. For example, …
SIGNIFICANT FIGURES, ADDITION, AND SUBTRACTION
• What is 16.874 + 2.6?
• What is 16.874 - 2.6?
SIGNIFICANT FIGURES, MULTIPLICATION, AND DIVISION
• When multiplying or dividing the answer will have the same
number of significant digits as the least accurate number used
to get the answer. For example, …
2.005 g / 4.95 mL = 0.405 g/mL
• What is 16.874 x 2.6?
• What is 16.874 / 2.6?
SIGNIFICANT FIGURES AND CALCULATIONS THAT REQUIRE
MULTIPLE STEPS
• An average is the best estimate of the true value of a
parameter.
• A standard deviation is a measure of precision.
• Averages and standard deviations require several steps to
calculate. You must keep track of the number of significant
figures during each step. Do NOT discard or round any figures
until the final number is reported.
SIGNIFICANT FIGURES AND CALCULATIONS THAT REQUIRE
MULTIPLE STEPS
2 Significant
Figures
∞ Significant Figures
1 Significant Figure
1 Significant Figure
2 Significant Figures
1 Significant Figure
0 Significant Figures
• What is average and standard deviation for the following 3
measurements of the same sample?
SOURCES
• American Public Health Association, American Water Works Association,
Water Environment Federation. 1995. Standard Methods for the Examination
of Water and Wastewater. 19th ed. Washington, DC: American Public Health
Association.
• Barnes, D.S., J.A. Chandler. 1982. Chemistry 111-112 Workbook and
Laboratory Manual. Amherst, MA: University of Massachusetts.
• Christian, G.D. 1986. Analytical Chemistry, 3rd ed. New York, NY: John Wiley
& Sons, Inc.
• Frisbie, S.H., E.J. Mitchell, A.Z. Yusuf, M.Y. Siddiq, R.E. Sanchez, R. Ortega,
D.M. Maynard, B. Sarkar. 2005. The development and use of an innovative
laboratory method for measuring arsenic in drinking water from western
Bangladesh. Environmental Health Perspectives. 113(9):1196-1204.
• Morrison Laboratories. 2006. Meniscus Madness. Available:
http://www.morrisonlabs.com/meniscus.htm [accessed 25 August 2006].