Measurement of quantities such as length, volume, mass, temperature, light absorbance, and time is the heart of experimental science. Ultimately, the most elegant theory rests on numbers obtained in the laboratory by a scientist using a measuring device or instrument. No device or instrument is capable of producing an exact result--a result with no uncertainty. When a scientist reports the result of a single measurement, such as a length or volume, s/he also reports the uncertainty in the result, a number indicating the reliability of the measuring device used. For example, the mass of a sample obtained using a balance which weighs to the nearest 0.1 g may be 32.0 g. The uncertainty in this number is 1 unit in the tenths decimal place, and indicates that the balance reading is uncertain by 1 unit in this place. The mass is reported as 32.0 ± 0.1 g, and the result of the weighing is known to 1 part in 320. Scientists sometimes communicate uncertainty implicitly rather than explicitly. For example, the mass above might be reported as 32.0 g. Reporting the tenths decimal place implies an uncertainty of 1 digit in this place. The mass is known to 3 significant figures, which is the number of digits written (except in numbers like 0.0032, where leading zeros are place holders, not significant figures). Weighing this sample on a more sensitive balance might give the result 32.0492 g. Now the uncertainty is 1 in the ten-thousandths place, and the mass is known to 6 significant figures. The result of this weighing is known to 1 part in about 320000, much better than with the less sensitive balance.

Most quantities can be expressed in more than one system of units. For example, length can be expressed in English units or metric units; temperature can be expressed in Celsius degrees, Fahrenheit degrees, or Kelvins. Generally, uncertainties are related by the same conversion factors as relate the quantities in two scales. If a length measured in inches has an uncertainty of 1 inch, then the same length expressed in centimeters (cm) has an uncertainty of 2.54 cm, because 1 in = 2.54 cm. If a temperature on the Celsius scale has an uncertainty of 0.1 ^oC, the same temperature on the Fahrenheit scale will have an uncertainty of 0.18 ^oF because there are 1.8 Fahrenheit degrees per Celsius degree. Some examples will make this clear.

Example. A man is 74 inches tall. What is his height in meters?

Height = 74 inches, with an implicit uncertainty of ± 1 inch. There are 2.54 cm per inch. Height in cm is

74 x 2.54 = 187.96 cm

The uncertainty in this number is ±2.54 cm. Since the uncertainty occurs first in the 1's place, we report the height as 188 ± 3 cm. Height in meters is 1.88 ± 0.03. The usual significant figure rule would force us to report this as 1.9 m, because the height in inches is given to 2 significant figures. But this would imply that we know the man's height to only 1 part in 19 in meter units, whereas we know it to 1 part in 74 in inches. Reporting the height as 1.9 m would mean a loss of information!

Example. What is 9 ^oC on the Fahrenheit scale?

The relationship between the two scales is

This formula says to 1) calculate the deviation of each individual measurement from the mean and square it; 2) add up all the squared deviations; 3) divide the sum by 1 less than the number of measurements; and 4) take the square root of the result. The standard deviation in our Spags to Uno distance is

s = {[(-0.1)² + (-1.3)² + (1.3)²]/(2)}^1/2 = 1.3

Note that in this example, this is the same as half the range. Although 1/2 the range and standard deviation will generally be similar, they will not in general be equal. The standard deviation represents a statistically more significant measure of precision than does 1/2 the range.

The error in a measured result is the difference between the measured value and the true value of the quantity being measured. Error indicates the accuracy of the measured value. Error is to accuracy what precision is to reproducibility. An extension of the example above shows the meanings of these terms clearly.

Example. An accurate road survey gives the distance between Spags and Uno as 23.9 miles. What is the error in our estimate of this distance?

Error = measured value - true value = 22.3 - 23.9 = -1.6 miles. The negative sign indicates that the measured value is smaller than the true value.

In summary:

Precision--the reproducibility of a measurement, indicated by writing the measured value ± range/2 or ± S.D.;

Accuracy--the difference between the measured value of a quantity and the true value. Indicated by writing the error (measured value - true value).

The uncertainty in a measured number carries into calculations made using the number. A calculated result is expressed consistently with the uncertainties in the numbers used in the calculation. How this is done depends on the arithmetic operation performed.

Addition and Subtraction. The uncertainty in the result of adding or subtracting two numbers is the sum of their uncertainties.

Suppose that a buret is filled with solution to a volume reading of 2.64 mL. Solution is delivered from the buret to a flask. The final buret reading is 27.48 mL. What volume of solution was delivered? The uncertainty in a single buret reading is ± 0.01 mL. The volume delivered is the difference between final and initial readings, and has an uncertainty which is the sum of the uncertainties in the two readings:

Volume delivered to flask = 24.84 ± 0.02 mL.

Multiplication and Division. The relative uncertainty in a product or quotient is the same as that of the least-well-known factor. The relative uncertainty is expressed using the "x parts in y" approach introduced above. If a number known to 1 part in 100 is multiplied by a number known to 1 part in 1000, the product can be known to at best 1 part in 100.

For example, suppose a 10.00 mL sample of water (delivered from a 10-mL volumetric pipet) has a mass of 9.9643 g (measured with an analytical balance). What is its density? Since density is mass per unit volume, we calculate

r = 9.9643 g/10.00 mL = 0.99643 g/mL

This must be reported to a number of significant figures consistent with the uncertainties in mass and volume:

Uncertainty in mass is 0.0002 g, or 2 parts in 99643.

Uncertainty in volume is 0.01 mL, or 1 part in 1000.

The volume is less certain than the mass. The density cannot be known more precisely than the least-precisely-known of mass and volume. At best, density is known to about 1 part in 1000. We should report density as 0.996 ± 0.001 g/mL. (The usual significant figure rule would allow us to report density to 4 significant figures, as 0.9964 g/mL, since volume is known to 4 sig figs. However this implies that we know density to 1 part in 9964! How can this be if we only know volume to 1 part in 1000??) A common sense approach to significant figures using the "x parts in y" method avoids the pitfalls of the textbook rules.

Precisions carry through calculations according to the same rules.

Logarithms. Frequently we need to take the logarithm of a measured number to make a plot. How do we express the uncertainty in the logarithm of a measured value?

Suppose we need the logarithm of (1.25 ± 0.01) x 10^-4. We first find the log of 1.25 x 10^-4 to be

-3.9031. To find the uncertainty in this, we compute logs of the maximum and minimum values our measured value could have, based on its uncertainty:

log (1.26 x 10^-4) = -3.900

log (1.24 x 10^-4) = -3.907

Comparing with 3.9031, the variation is in the 3rd decimal place. Uncertainty in the log is 1/2 of 0.007, or 0.004. The result is 3.903 ± 0.004. The number of digits following the decimal point in the log is the same as the number of significant figures in the original measured value. This is always true.

Exponentiation and Trigonometric Functions. The same approach used for logarithms can be used when raising a number to a power or finding the values of trigonometric functions of a number:

For example, raise 10 to the power -2.4 ± 0.2.

1. 10^-2.4 = 3.981 x 10^-3

2. Extremes of measured value are -2.2 and -2.6.

3. 10^-2.2 = 6.310 x 10^-3; 10^-2.6 = 2.512 x 10^-3

4. Range of function = 6.3-2.5 x 10^-3 = 3.8 x 10^-3

5. Uncertainty in function = 1.9 x 10^-3.

The value of the function is reported (4 ± 2) x 10^-3. Note how an uncertainty of 1 part in 12 in the original value translates to uncertainty of 2 parts in 4 in the exponential. Exponential functions magnify uncertainty!

Many laboratory measurement devices are graduated--marked with equally spaced lines corresponding to incremental values of the quantity measured. For example, a meter stick is 1 meter long, with large lines for centimeter subdivisions, and smaller lines for millimeter subdivisions. It is graduated in millimeters. A 100-mL graduated cylinder is marked with large lines every 10 mL and smaller lines every mL. It is graduated in mL. Similarly, burets are graduated in 0.1 mL, 10-mL measuring pipets to 0.1 mL, and standard laboratory thermometers to 1 ^oC.

There is a standard procedure for reading a value from a graduated device, summarized as follows: estimate the value to one decimal place more than the level of graduation. When reading a temperature with a thermometer graduated in degrees, estimate to tenths of a degree. Record a buret reading to the nearest hundredth of a mL (if the liquid level falls exactly on a major graduation, say at 32.00 mL, write 32.00, not 32). Record a volume measured with a 100-mL graduate to the nearest 0.1 mL. Please adhere strictly to this procedure. If you do not, you will introduce into your measurement more uncertainty than actually exists.

1."Error, Precision, and Uncertainty" by Charles J. Guare. Journal of Chemical Education, 1991, 68, 649-652.

2. "Is 8 ^oC Equal to 50 ^oF?" by H. Bradford Thompson. Journal of Chemical Education, 1991, 68, 400-402.

3. "Basic Principles of Scale Reading" by Gavin D. Peckham. Journal of Chemical Education, 1994, 71, 423-424.