• Accuracy is the closeness of a measured value to the true Accuracy is the closeness of a measured value to the true value.value.

• For example, the measured density of water has become more For example, the measured density of water has become more accurate with improved experimental design, technique, and accurate with improved experimental design, technique, and equipment.equipment.

CH 103: LABORATORY MEASUREMENTSCH 103: LABORATORY MEASUREMENTS

ACCURACYACCURACY

Density of HDensity of H22O at 20° CO at 20° C

(g/cm(g/cm33))

11

1.01.0

1.001.00

0.9980.998

0.99820.9982

0.998200.99820

0.9982030.998203

• Percent error is used to estimate the accuracy of a Percent error is used to estimate the accuracy of a measurement.measurement.

• Percent error will always be a positive.Percent error will always be a positive.

• What is the percent error if the measured density of titanium (Ti) What is the percent error if the measured density of titanium (Ti) is 4.45 g/cmis 4.45 g/cm33 and the accepted density of Ti is 4.50 g/cm and the accepted density of Ti is 4.50 g/cm33??

ACCURACYACCURACY

Average = 15.3 μg/LAverage = 15.3 μg/LStandard Deviation = 2.1 μg/LStandard Deviation = 2.1 μg/L

What is the true concentration of As in this experiment?What is the true concentration of As in this experiment?

Estimate the accuracy of this method.Estimate the accuracy of this method.

How precise is this method?How precise is this method?

• Precision is the agreement between repeated measurements of the Precision is the agreement between repeated measurements of the same sample. Precision is usually expressed as a standard deviation.same sample. Precision is usually expressed as a standard deviation.

• For example, the precision of a method for measuring arsenic (As) was 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 determined by measuring 7 different solutions each containing 14.3 μg/L of As.As.

PRECISIONPRECISION

Measured Measured ConcentrationConcentration

(μg/L)(μg/L)

18.418.4

13.613.6

13.613.6

14.214.2

16.016.0

13.613.6

17.817.8

14.3 μg/L14.3 μg/L

2.1 μg/L2.1 μg/L

• Describe the accuracy and precision of these 4 targets.Describe the accuracy and precision of these 4 targets.

ACCURACY AND PRECISIONACCURACY AND PRECISION

Accurate, and Accurate, and preciseprecise

Precise, but Precise, but not accuratenot accurate

Accurate, but Accurate, but not precisenot precise

Not accurate, Not accurate, and not and not preciseprecise

• Systematic (or determinate) errors are reproducible and Systematic (or determinate) errors are reproducible and cause a bias in the same direction for each measurement.cause a bias in the same direction for each measurement.

• For example, a poorly trained operator that consistently makes For example, a poorly trained operator that consistently makes the same mistake will cause systematic error. Systematic error the same mistake will cause systematic error. Systematic error can be corrected.can be corrected.

• Random (or indeterminate) errors are caused by the natural Random (or indeterminate) errors are caused by the natural uncertainty that occurs with any measurement.uncertainty that occurs with any measurement.

• Random errors obey the laws of probability. That is, random Random errors obey the laws of probability. That is, random error might cause a value to be over predicted during its first error might cause a value to be over predicted during its first measurement and under predicted during its second measurement and under predicted during its second measurement. Random error cannot be corrected.measurement. Random error cannot be corrected.

ERRORSERRORS

• By convention, a measurement is recorded by writing all exactly By convention, a measurement is recorded by writing all exactly known numbers and 1 number which is uncertain, together with known numbers and 1 number which is uncertain, together with a unit label.a unit label.

• All numbers written in this way, including the uncertain digit, are All numbers written in this way, including the uncertain digit, are called significant figures.called significant figures.

• For example, the blue line is 2.73 cm long. This measurement 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 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 known. The third digit (0.03 cm) is uncertain because it was interpolated or estimated 1 digit beyond the smallest interpolated or estimated 1 digit beyond the smallest graduation.graduation.

INTERPOLATION AND SIGNIFICANT FIGURESINTERPOLATION AND SIGNIFICANT FIGURES

• What is the volume of water in this graduated cylinder? Always What is the volume of water in this graduated cylinder? Always measure the volume of a liquid at the bottom of the meniscus. measure the volume of a liquid at the bottom of the meniscus. The units are mL.The units are mL.

• The volume of water is 52.8 mL. The 52 mL are exactly known, 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 and the 0.8 mL is uncertain because it was interpolated or estimated 1 digit beyond the smallest graduation.estimated 1 digit beyond the smallest graduation.

INTERPOLATION AND SIGNIFICANT FIGURESINTERPOLATION AND SIGNIFICANT FIGURES

• Zeros between nonzero digits are significant. That is, 508 cm Zeros between nonzero digits are significant. That is, 508 cm has 3 significant figures.has 3 significant figures.

• Leading zeroes merely locate the decimal point and are never Leading zeroes merely locate the decimal point and are never significant. That is, 0.0497 cm equals 4.97 x 10significant. That is, 0.0497 cm equals 4.97 x 10-2-2 cm and has 3 cm and has 3 significant figures.significant figures.

• Trailing zeros are significant as follows: 50.0 mL has 3 Trailing zeros are significant as follows: 50.0 mL has 3 significant figures, 50. mL has 2 significant figures, and 50 mL significant figures, 50. mL has 2 significant figures, and 50 mL has 1 significant figure.has 1 significant figure.

SIGNIFICANT FIGURES AND ZEROSSIGNIFICANT FIGURES AND ZEROS

Datum Datum (grams)(grams)

Number of Number of Significant Significant

FiguresFigures

Datum Datum (milliliters)(milliliters)

Number of Number of Significant Significant

FiguresFigures

10,03410,0341.9081.9080.320.320.000460.000461501500.00001600.0000160

150.150.0.7050.7050.0540.0545.86 x 105.86 x 10-7-7

304030400.00007300.0000730

554422222233

333322333333

• When adding or subtracting do NOT extend the result beyond When adding or subtracting do NOT extend the result beyond the first column with a doubtful figure. For example, …the first column with a doubtful figure. For example, …

SIGNIFICANT FIGURES, ADDITION, AND SUBTRACTIONSIGNIFICANT FIGURES, ADDITION, AND SUBTRACTION

• What is 16.874 + 2.6?What is 16.874 + 2.6?

• What is 16.874 - 2.6?What is 16.874 - 2.6?

SIGNIFICANT FIGURES, ADDITION, AND SUBTRACTIONSIGNIFICANT FIGURES, ADDITION, AND SUBTRACTION

• When multiplying or dividing the answer will have the same When multiplying or dividing the answer will have the same number of significant digits as the least accurate number used number of significant digits as the least accurate number used to get the answer. For example, …to get the answer. For example, …

2.005 g / 4.95 mL = 0.405 g/mL2.005 g / 4.95 mL = 0.405 g/mL

• What is 16.874 x 2.6?What is 16.874 x 2.6?

• What is 16.874 / 2.6?What is 16.874 / 2.6?

SIGNIFICANT FIGURES, MULTIPLICATION, AND DIVISIONSIGNIFICANT FIGURES, MULTIPLICATION, AND DIVISION

• An An averageaverage is the best estimate of the true value of a is the best estimate of the true value of a parameter.parameter.

• A A standard deviationstandard deviation is a measure of precision. is a measure of precision.

• Averages and standard deviations require several steps to Averages and standard deviations require several steps to calculate. You must keep track of the number of significant calculate. You must keep track of the number of significant figures during each step. Do figures during each step. Do NOTNOT discard or round any figures discard or round any figures until the final number is reported.until the final number is reported.

SIGNIFICANT FIGURES AND CALCULATIONS THAT SIGNIFICANT FIGURES AND CALCULATIONS THAT REQUIRE MULTIPLE STEPSREQUIRE MULTIPLE STEPS

SIGNIFICANT FIGURES AND CALCULATIONS THAT SIGNIFICANT FIGURES AND CALCULATIONS THAT REQUIRE MULTIPLE STEPSREQUIRE MULTIPLE STEPS

2 Significant Figures

1 Significant Figure

2 Significant Figures

0 Significant Figures

1 Significant Figure

1 Significant Figure

Significant Figures ∞

• What is average and standard deviation for the following 3 What is average and standard deviation for the following 3 measurements of the same sample?measurements of the same sample?

• American Public Health Association, American Water Works Association, American Public Health Association, American Water Works Association, Water Environment Federation. 1995. Standard Methods for the Water Environment Federation. 1995. Standard Methods for the Examination of Water and Wastewater. 19th ed. Washington, DC: American Examination of Water and Wastewater. 19th ed. Washington, DC: American Public Health Association.Public Health Association.

• Barnes, D.S., J.A. Chandler. 1982. Chemistry 111-112 Workbook and Barnes, D.S., J.A. Chandler. 1982. Chemistry 111-112 Workbook and Laboratory Manual. Amherst, MA: University of Massachusetts.Laboratory Manual. Amherst, MA: University of Massachusetts.

• Christian, G.D. 1986. Analytical Chemistry, 3rd ed. New York, NY: John Christian, G.D. 1986. Analytical Chemistry, 3rd ed. New York, NY: John Wiley & Sons, Inc.Wiley & Sons, Inc.

• Frisbie, S.H., E.J. Mitchell, A.Z. Yusuf, M.Y. Siddiq, R.E. Sanchez, R. 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 Ortega, D.M. Maynard, B. Sarkar. 2005. The development and use of an innovative laboratory method for measuring arsenic in drinking water from innovative laboratory method for measuring arsenic in drinking water from western Bangladesh. Environmental Health Perspectives. 113(9):1196-western Bangladesh. Environmental Health Perspectives. 113(9):1196-1204.1204.

• Morrison Laboratories. 2006. Meniscus Madness. Available: Morrison Laboratories. 2006. Meniscus Madness. Available: http://www.morrisonlabs.com/meniscus.htm [accessed 25 August 2006]. [accessed 25 August 2006].

SOURCESSOURCES