Uncertainty and Measurement
Introduction
Suppose you have a summer job monitoring the pollution in a local lake. You are instructed to collect three
100-mL water samples at certain locations at set times each day. To each sample, you add 5 mL of a coloring agent
that reacts and changes color intensity in proportion to the amount of pollutant in the water. You then check each
sample with an instrument that detects color intensity and gives a quantitative, or numerical, measure of the amount
of pollutant in the water. Unfortunately, your measurements of similar samples vary by 10 to 20 percent. How could
you increase the accuracy and precision if your measurements?
Every measurement has an uncertainty, or a built-in error. This error is due to limitations in the
measurement scale, the manufacturing process, and the ability of the human eye to detect small differences. For
example, when measuring volume with a graduated cylinder, the width of the scale lines, variations in glass
thickness, and slight changes in the angle of sight when reading the scale are some of the factors that cause
uncertainty. Because of this uncertainty, no measurement should be thought of as an exact value, but rather as a
value within a range that varies with the uncertainty. For example, the uncertainty of the volume measurement made
with 100-mL graduated cylinder may be ±0.5 mL. Thus if you measure 100 mL of water, the actual volume would
be 100.0 ±0.5 mL, or within a range of 99.5 mL to 100.5 mL. Although ±0.5 mL represents only a ±0.5% error for a
100.0 mL measurement, it becomes a much larger error of ±10% when you measure a smaller quantity, such as
5.0mL.
There are two important lessons you should learn about making measurements. First, you should
familiarize yourself with a scale of each piece of lab equipment and learn to read each scale as accurately as
possible. Second, you should know the uncertainty of your measurements, because your results cannot be more
accurate than the built-an error allow.
In this laboratory investigation, you will become familiar with the measurement scale of balances,
graduated cylinders, and a thermometer. Then you will determine the uncertainty of measurements made with this
equipment. If you really do the job monitoring water pollution, you will know how to increase the accurate and
precision of your measurements so that they are scientifically useful.
Pre-Lab Discussion
Read the entire investigation and the relevant pages of your textbook. Then answer the questions that follow.
1. Why is it important to wear eye protection at all times in the chemistry laboratory, even whne you are not using
an open flame or dangerous chemicals?
2. In the diagram of the graduated cylinder shown in Figure-1, what fraction of a mL does each division, or
increment, between the 1- mL marking represent?
3. What is the volume of the liquid shown in Figure-1?
4. Which do you think will have a more predictable impact on the measurements you make in the laboratory, human
error or uncertainty in the measurement scales of the lab equipment? Explain.
Problem
How large are the uncertainties in measurements made with common lab equipment?
Materials
safety goggles
distilled water (at 20°C)
laboratory apron
Dropper
laboratory balance
Celsius thermometer
standard masses
heat-proof gloves
2 objects of unknown masses
beaker containing boiling water
graduated cylinder, 10-mL
plastic tub
graduated cyrlinder, 100-mL
Ice
Safety
Wear your goggles and lab apron at all times during the investigation. Your eyes are fragile, and you should always
protect them from potential laboratory hazards such as shattering glass or splashing boiling water.
Procedure
Part A: Estimating the Uncertainty of a Balance
1. Put on your goggles and lab apron. Obtain a laboratory balance and use the zeroing adjustment so the scale reads
zero with no mass on the pan. Gently disturb the pan by touching it, and check to make sure that the balance returns
to zero with no visible deviation.
2. Study the balance scale that has the smallest counterweight. Determine the mass increment size (in grams)
represented by any one of the smallest scale divisions (between two adjacent marks), and record this value. Once
you know this value, determine the mass represented by one half and one fifth of this scale increment. Record these
vales. (Note: If you are using an electronic balance, skip this step.)
3. Obtain a standard and place it on the balance pan. Adjust the counterweights to find its mass. The mass should be
equal or very close to the standard’s given value. Once the exact balance point is found, record the mass as
accurately as the smallest scale increment allows (e.g., 10.00g)
4. Shift the smallest counterweight just slightly until you observe the slightest deviation from the zero point. The
shift may be less than one scale division. Do the same for a shift in the opposite direction. These slightly lower and
harder readings represent the apparent uncertainty range. Record the masses for the upper and lower ends, or limits,
of the range. (Note: If you are using an electric balance, the apparent uncertainty is represented by the range between
the higher and lower masses that may flicker on the display.)
5. As exactly as possible, record the mass of each of the two objects of unknown mass. Contribute your
measurements to the class data bank for later use. While waiting for your turn with the two unknowns, you can
proceed with part B of the investigation.
Part B: Estimating the Uncertainty of Graduated Cylinders
6. Using the laboratory balance, measure and record the mass of a dry 10-mL graduated cylinder and a dry 100-mL
graduated cylinder. CAUTION: If glass cylinders are being used, take care not to knock them over and break them.
If a cylinder does shatter, do not pick up the broken pieces with your bare hands.
7. Record the volume represented by the smallest volume increment on each of the cylinders. Also determine and
record the volume represented by one half and one fifth of the smallest volume increment.
8. Use a dropper to add 10.0 mL of distilled water to each cylinder. Add the last few drops to each cylinder carefully
so that the bottom curve of the meniscus is on the 10.0-mL mark.
9. With the laboratory balance, measure and record the mass of each cylinder containing 10.0 mL of water.
Part C: Estimating the Uncertainty of a Thermometer
10. Obtain a Celsius thermometer. Determine and record the temperature represented by the smallest scale
increment. Also determine and record the temperature represented by one half and one fifth of the smallest scale
increment. CAUTION: thermometers are fragile. Handle with care.
11. Put on a pair of heat-proof gloves and place the thermometer in the beaker of boiling water provided by your
teacher. For 1 to 2 minutes, hold the thermometer in the boiling water so that the tip is not touching the beaker
bottom. Remove the thermometer from the boiling water and quickly read it. Record the temperature, estimating to
tenths of a degree. Contribute your measurement to the class data bank. CAUTION: Do not touch the beaker or hot
plate with your bare hands.
12. Turn off the hot plate. Allow the thermometer to cool to room temperature. Then place it in a tub of ice water
provided by your teacher. Leave the thermometer in the ice bath for 1 to 2 minutes. Record the temperature and
contribute your measurement of the freezing point of water to the class date bank.
13. Return all equipment to the supply area. Clean up your work area and wash your hands before leaving the
laboratory.
Observations
Part A: Laboratory Balance
Smallest mass scale increment ________
One half of smallest mass scale increment ________
One fifth of smallest mass scale increment_______
Mass of standard weight ________
Mass of standard weight ________
Highest limit________
Lowest limit ________
Mass of unknown #1 _________
Mass of unknown #2_______
Part B: Graduated Cylinders
Smallest volume scale increment
One half of smallest volume scale increment
One fifth of smallest volume scale increment
Mass of empty cylinder
Mass of cylinder with 10.0 mL of water
Mass of 10.0 mL of water
Part C: Thermometer
Smallest temperature scale increment
One half of smallest temperature scale increment
One fifth of smallest temperature scale increment
Temperature of boiling water
Temperature of freezing water
10-mL
_____
_____
_____
_____
_____
_____
100-mL
_______
_______
_______
_______
_______
_______
_________
_________
_________
_________
_________
Calculations
Part A: Uncertainty for Laboratory Balance
1. Use your data for the mass of the standard weight to find the apparent uncertainty of your balance. Subtract the
lower limit mass if the standard weight from the higher limit and divide this difference by 2. Round to one
significant digit to get the apparent uncertainty (e.g., 0.0125 g rounds off to 0.01 g). Write the mass of the standard
weight followed by the uncertainty (e.g., 10.00 g ± 0.01 g).
2. With classmates, evaluate the class measurement of the masses if the unknown. Discard any values that are much
greater or much smaller than the majority of values. List and average the remaining masses for each unknown to
find an average mass for each.
3. Determine the practical uncertainty for the lab balance by doing the following steps:
a. To find the deviation in mass measurements for each unknown compute the difference (absolute value) between
the average mass value (see answer to Question 2) and each of the mass measurements in the data list for Question
2.
b. Average the list of mass deviations for each unknown and round to one significant figure (e.g., 0.012 g rounds off
to 0.01 g). This is the practical uncertainty.
Avg deviation (uncertainty)
Unknown #1 ______
Unknown #2 ______
c. Report the average mass of each unknown followed by its uncertainty (e.g., 5.25 ± 0.01 g).
Avg mass with uncertainty
Unknown #1 ______
Unknown #2 _____
Part B: Uncertainty for Graduated Cylinders
1. With classmates, evaluate the class measurements of the masses of 10 mL of water. Discard and values that are
much greater or much smaller than the majority of measurements. List the remaining measurements and compute an
average mass for the water in each cylinder.
2. Determine the practical uncertainty of the mass measurements made in the 10-mL and the 100-mL graduated
cylinders by doing the following steps:
a. Take the difference (absolute value) between the avg mass of water and each individual mass of water in the data
list for the 10-mL graduated cylinder. This gives a list of data deviations. Do the same for the 100-mL cylinder.
b. Average the deviations for each cylinder and round to one significant digit (e.g., 0.133 g rounds off to 0.1 g). This
practical uncertainty of the mass measurement is equivalent to the practical uncertainty of the cylinder volume
because 1.00 g of water at 20°C has a volume of 1.00 mL
Average deviation (uncertainty) 10 mL cylinder100mL cylinder
c. Report the volume of water (10.0 mL) in each cylinder, followed by the calculated for each cylinder from Part B.
Average volume with uncertainty 10-mL cylinder:
100-mL cylinder:
Part C: Uncertainty of a Celsius Thermometer
1. With classmates, evaluate the class measurements of the boiling points and freezing points of water. Discard any
values that are much higher or lower than the majority of temperature in each list. List the remaining temperatures
and compute an average boiling point temperature and an average freezing point temperature.
Class Data Bank: Water Temperature (°C)
Boiling point
Freezing point
Avg boiling point ________ Avg freezing point______
2. Determine the practical uncertainty of the thermometers by doing the following steps:
a. Take the difference (absolute value) between the average boiling point and each boiling point in the data list. This
gives a list of data deviations. Do the same with the average freezing point data.
Deviations from Averages temperature (°C)
Boiling Point
Freezing Point
b. Average the deviations for the boiling points and the freezing points and round to one significant figure. These
are the practical uncertainties for the thermometers.
Average deviation (uncertainty)
Boiling Point ______________________
Freezing Point ____________________
c. Report the average boiling point and freezing point of water followed by the calculated uncertainty of each.
Average boiling point with uncertainty
Average freezing point with uncertainty
____________________________
____________________________
Critical Thinking: Analysis and Conclusions
1. Why do students measuring the mass of the same object on similar balances report slightly different
masses?
2. Which did you find to have a smaller uncertainty, the 10 mL or 100 mL graduated cylinder? Give a reason
why one has a smaller uncertainty.
3. Based on your uncertainty determinations, tell whether balances or graduated cylinders appear to be more
accurate measuring devices.
4. Assuming that the equipment was functioning properly, explain the probably source of error in data values
that were discarded because of their large deviations.
5. Do the uncertainties you calculated for each type of lab equipment more closely match the size of the
smallest scale division, one-half division, or one-fifth division?
6. Based on your answer to Question 5, what uncertainty would you assign to each type of equipment used?
Balance ______________
10 mL graduated cylinder ________
Thermometer __________
100 mL graduated cylinder _______
Critical Thinking: Applications
1. If the uncertainty of a balance is ±0.005 g, how many significant figures would you use to report a scale
reading when the counterweights lie exactly on the 8 gram mark? Explain your answer.
2. Suppose a students asks your advice about how to measure 9 mL of a liquid as accurately as possible using
a graduated cylinder. Would you recommend a 10 mL or a 100 mL graduated cylinder? Support your
answer using the results of the investigation.
3. What procedural change would you recommend to increase the accuracy and precision of the
measurements discussed in the Introduction to this lab?
Going Further
1. Determine the uncertainty of a carpenter’s tape measure, a tape measure used for sewing, or a measuring
cup used for cooking. To make this determination, use methods similar to those used in this investigation.
Present your findings to your class.