ANALYSIS OF EXPERIMENTAL DATA
DETERMINLNG SIGNIFICANT FIGURES
1. All non-zero digits are always significant.
2. Zeros within non-zero digits are always significant. For example,
Measurement
10.02 g
3.06 mL
Number of Significant Figures
4
3
3. Zeros to the left of the first non-zero digit are not significant. For example,
Measurement
0.02 g
0.0032 g
0.000 415 g
Number of Significant Figures
1
2
3
4. Zeros to the right of the last non-zero digit are usually significant:
Measurement
2.40 g
10.00 g
Number of Significant Figures
3
4
However, it is unclear how many significant figures there are in 3400 g. This confusion can be
avoided by recording the measurement in scientific notation. For example,
Measurement
3 400 g
3.4 x 103 g
Number of Significant Figures
?
2
5. Exact numbers differ from measurements in that they have no uncertainty. These numbers result
from a count of objects. For example, there are 100 cm in a metre or there are twelve eggs in a
dozen. Since exact numbers contain no uncertainty, they are said to have an infinite number of
significant figures. Each digit, including the last one, is known exactly.
RULES FOR ROUNDLNG OFF NUMBERS
1. If the first nonsignificant digit is less than 5, it is dropped, and the last significant digit remains
unaltered.
2. If the first nonsignificant digit is 5 or greater, it is dropped, and the last significant digit is
increased by one.
SIGNIFICANT FIGURES IN CALCULATIONS
When you carry out calculations involving measured values, remember that the precision of the
answer is determined by the precision of the measurements. The following rules outline how to
determine the number of significant figures in the answer of a calculation involving measurements.
(a) Addition and Subtraction
The answer should contain the same number of decimal places as the measurement with the least
number of decimal places. For example,
93.04 g + 8.6 g = 101.6 g
(b) Multiplication and Division
The number of significant figures in the answer should be the same as the measurement with the
fewest significant figures. For example,
20.14 g
Density = -----------25 mL
=
0.81 g•mL-1
UNCERTAINTY IN EXPERLMENTAL DATA
Any data we collect from an experiment has some uncertainty associated with it. The degree of
uncertainty in a measurement is the result of the limitations of the measuring device and the
experimenter's ability to use the device.
Two factors that contribute to the uncertainty of a measurement are precision and accuracy. An
experimental procedure has a high degree of precision if the experiment can be repeated a number of
times to give consistently similar results. Accuracy, on the other hand, refers to how close an
experimental result is to a known or accepted result.
If a true or accepted value is known, the accuracy of a measurement can be obtained using the
formula:
Accepted Value - Experimental Value
Percent Difference =
x 100%
Accepted Value
(The results of this calculation should always be expressed as a positive number)
A high degree of precision does not always imply high accuracy. For example, with practice you
should be able to use a burette to deliver liquid volumes which are within 1% of each other.
However, if the burette has not been properly calibrated, the volumes delivered could differ
significantly from their true value.
UNCERTALNTY IN MEASUREMENT
The following list gives the uncertainties of typical measuring devices found in the lab. The
values are based on the assumption that the experimenter is well versed in using the equipment.
Instrument
Triple beam balance
100 mL graduated cylinder
10 mL graduated cylinder
50 mL burette
10°C to 110°C thermometer
Typical Uncertainty
±0.01 g
±1 mL
±0.1 mL
±0.02 mL
±0.2°C
In recording data it is important to estimate and record the uncertainty. For example, if you weigh a
sample of sodium hydroxide on a triple beam balance, you might record 2.89 ± 0.01 g. The 0.01
suggests that if you repeat the measurement a number of times, the result should lie within the range
2.88 g to 2.90 g the majority of the time. Thus, the uncertainty in a measurement is an estimate of the
reliability or reproducibility of the measuring device. Another way of reporting the uncertainty in a
measurement is its percent uncertainty. For example,
Mass NaOH = 2.89 ± 0.01 g
Uncertainty
Percent Uncertainty =
x 100%
Measurement
0.01
=
x 100%
2.89
= 0.35%
Thus the mass of the sample could be recorded as 2.83 g ± 0.35%
The final method of indicating uncertainty is by the number of significant figures reported. For
example,
Device
100 mL graduate cylinder
10 mL graduate cylinder
50 mL burette
Volume
8 mL
8.2 mL
8.24 mL
Number of Significant Figures
1
2
3
When counting significant figures, record all figures of which you are certain plus the first figure
of which you are uncertain. The limitation of this method is that the amount of uncertainty in the last
figure is not given.
PROPAGATION OF UNCERTAINTY IN CALCULATED RESULTS
Since each measurement has uncertainty, the combination of several measurements in a calculation
also has an uncertainty. The following rules outline how to calculate the uncertainty in calculations
involving experimental data:
1.
Addition and Subtraction
When experimental values are added or subtracted, add the absolute uncertainties in the
measurements.
2.
Multiplication and Division
When experimental values are multiplied or divided, add the percent uncertainties. The
following example illustrates these rules as applied to the calculation of the amount of heat
absorbed when 2.00 g of sodium hydroxide are dissolved in 100 mL of water.
= 21.5 ± 0.2°C
Initial temperature of water (T2)
Maximum temperature reached (T1) = 24.5 ± 0.2°C
∆T
= 3.0 ± 0.4°C
Mass of water
= Density x Volume
= (1.00 g•mL-1)(100 ±1 mL)
= (1.00 g•mL-1)(100 mL ± 1%)
= 100 g ± 1%
= (4.18 J• g-1•°C-1)(Mass of water)( ∆ T)
= (4.18 J• g-1•°C-1)(100 ±1 g)(3.0 ± 0.4°C)
= (4•18 J• g-1•°C-1)(100 g ± 1%)(3.0°C ± 13%)
= 1254J ± 14%
= 1254 ± 180J
The "± 180" suggests that the uncertainty begins in second digit of the number. The remaining
digits in this number are insignificant and should be rounded off. Consequently, the final
answer should be 1300 ± 200 J
Heat absorbed