Experimental Measurement and Error
Part 1. Density Determination and Significant Figures
Density is the ratio of mass to volume.
density 
1.
mass
volume

m
V
Which of the properties above are extensive? Which are intensive?
Mass and volume are extensive properties (depend on size of sample), whereas density is an
intensive property (depends only on identity and not size of sample).
2. Which property can usually be measured to the highest degree of precision? Mass
3. What is the level of precision of most balances in the COM chemistry lab, and what is the
greatest level of precision achievable by balances at COM?
Our cheapest balances measure to the nearest 0.01g (centigram), most of our balances
measure to the nearest 0.001g (milligram), and our expensive analytical balances measure to
the nearest 0.0001g.
4. Rank the following volume-measuring devices from least precise (1) to most precise (5):



Beakers can be used for very crude estimates, but in addition to poor precision, the
markings are often inaccurate.
Moderate precision can be achieved by using graduated cylinders for larger volumes
and graduated pipets for smaller volumes.
The greatest precision is achieved by using volumetric flasks for large volumes and
volumetric pipets for small volumes.
5. What is the purpose of significant figures?
Significant figures are used to communicate the uncertainty in a measurement or to estimate
the uncertainty in a result calculated from measurements. Although significant figures
provide very limited information about measurement uncertainty, their use is simple and
straightforward as compared to other approaches.
Chem 131
Experimental Measurement and Error
Page 1
Example: Density determination of a solid aluminum cylinder
An aluminum bar was placed on a standard milligram balance, which had previously been tared
(zeroed). Some water was then poured into a 25mL graduated cylinder. Finally, the aluminum
bar was placed into the graduated cylinder and submerged in the water. Each of these steps is
pictured below. Determine the density of aluminum according to these measurements.
14
18
13
17
11.357 g
Mass of Al = 11.357g
VWater + Al
– VWater
VAl
17.74 mL
13.57 mL
4.17 mL
Volume of Water = 13.57mL
Volume of Water + Al = 17.74mL
Al = mAl / VAl = 11.357g / 4.17mL = 2.72 g/mL
If the reported literature value for the density of Al at 25C is 2.702 g/mL, comment on the
accuracy and precision of the result above.
The result is accurate when compared to the literature value, given the precision of the
measurements. Significant figures imply that we should expect agreement in the ones and tenths
place (i.e., up to 2.7) but uncertainty in the hundredths place, which is exactly what we find.
The experimental result above has poorer precision than whatever technique was used to
establish the literature value, which has its uncertainty in the thousandths place.
If you had multiple samples of aluminum, what might you do to improve the precision of your
density determination using the same equipment as above?
The only way to improve the precision (i.e., significant figures) in the resulting density is to
improve the precision of the volume measurement. Using the same graduated cylinder, this
could be accomplished if the volume of the aluminum sample were greater than 10mL (giving 4
sig figs for VAl rather than 3). Assuming each aluminum bar is roughly the same size, we could
use 3 bars instead of just one, giving a volume of ~12mL, and a density of 2.7xx g/mL.
Part 2. Accuracy and Precision in Repeated Measurements
Although the use of significant figures is the simplest and therefore perhaps most common way
to characterize the uncertainty of a result, the application of statistics to repeated measurements
provides a more complete picture of experimental error and uncertainty.
Chem 131
Experimental Measurement and Error
Page 2
Example: Calibration of Volume-Measuring Devices
As pointed out earlier, precision is usually much greater with measurements of mass than of
volume. For certain substances such as water, the density is also known and reported in the
literature to a high degree of precision. This provides an alternative high-precision approach for
determining the volume dispensed by a device, and hence can be used to calibrate the device.
Suppose that various types of lab glassware were used to measure 25 mL of water as accurately
as possible. The water was dispensed into a tared container, then weighed on a milligram
balance (resulting in 5 sig figs). Volumes could then be calculated from the masses using the
density of water at 25C, which is 0.99707 g/mL.
If the mass measured was 25.121g, what would the corresponding volume be?
1
0.99707
25.121
25.195
Complete data is shown below for 5 measurements made in a 50-mL beaker:
“ xi ”
Trial
mass (g)
volume (mL)
1
2
3
4
5
Sum
27.375
29.435
27.028
28.441
26.758
27.455
29.521
27.107
28.525
26.837
139.446
Mean  x  
 xi
St. Dev.   
n

139.446 mL

5
  x  x i 2
n 1

x  xi (mL)
x  xi 2 (mL2)
0.434
-1.632
0.782
-0.635
1.052
0.188
2.665
0.611
0.404
1.108
4.975
27.889 mL
4.975 mL2

4
Report Measured Volume as: 28 ± 1 mL
1.115 mL
(27.9 ± 1.1 mL would also be acceptable here)
Why do you think a 50mL beaker does not give precise volume measurements?
Since a beaker is short and squat, slightly different volumes produce imperceptible differences in
liquid heights.
Why do you think a 50mL beaker does not give accurate volume measurements?
A likely explanation is that the markings on the container were not accurately calibrated. (Note
that the price of a beaker is 1/10th or less that of a graduated cylinder or volumetric flask.)
Chem 131
Experimental Measurement and Error
Page 3
Data is provided below for the volumes of water (values have already been converted from
masses) measured using a 50-mL graduated cylinder. Complete the calculation of results.
50mL Graduated Cylinder
Trial
Volume (mL)
x  xi (mL)
x  xi 2 (mL2)
1
23.916
-0.072
0.005
2
23.979
-0.135
0.018
3
23.567
0.277
0.077
4
23.627
0.217
0.047
5
24.131
-0.287
0.082
Sum
119.220
Mean  x  
 xi
St. Dev.   
n

0.230
119.220 mL

5
  x  x i 2
n 1

23.844 mL
0.230 mL2

4
0.240 mL
Report Measured Volume: 23.8 ± 0.2 mL
Data is provided below for volumes measured using a 25-mL volumetric pipet. The mean and
standard deviation have been calculated, and are shown.
25mL Pipet
Trial
Volume (mL)
1
24.979
2
25.033
3
25.015
4
25.028
5
24.995
Reported Volume: 25.01 ± 0.02 mL
Which is the most precise device for measuring 25mL of solution? The 25mL volumetric pipet
What systematic error do you think may have occurred that caused the inaccuracy of the volumes
dispensed by the graduated cylinder?
One likely error is that a small amount of water would always be left behind on the walls of the
graduated cylinder when the water is poured into the container for weighing.
How could you change your technique to improve the accuracy of the graduated cylinder?
Use the difference between two volumes (initial and final) to determine the volume transferred.
Chem 131
Experimental Measurement and Error
Page 4
Part 3. The Use of Linear Regression (Graphing a Straight Line) to Determine Values
In many cases, multiple measurements are made in which two different properties vary, but one
or more others remain constant. In such cases, a graphical approach can often be used to
determine the value of one or more of the constants.
For example, suppose that for our density determination of aluminum, we had measured the
masses and volumes of several different aluminum bars. Although the samples would have
different values of mass and volume, they should in theory all have the same density. Why?
Density is an intensive property; it is the same for all pure samples of aluminum, regardless of
the size of the sample.
Any straight line can be described by an equation of the form y = mx + b, where y is the
dependent variable, x is the independent variable, and m (the “slope”) and b (the “y-intercept)
must both be constants. When plotting a graph of data described by this equation, y values are
usually plotted on the vertical axis, while x values are plotted on the horizontal axis.
Rearrange the equation relating density, mass, and volume into an equation that has the form
y = mx + b. (Note that one of these may be equal to zero.) Indicate which variable you would
plot on the vertical, and which on the horizontal axis.
m=V + 0
so if V is plotted on the horizontal and m on the vertical, the line should
have slope  and zero intercept.
Use a graphical approach to determine density from the following data that was collected for 4
different Al samples.
mass Al (g)
Volume Al (mL)
17.209
6.34
8.780
3.26
13.984
5.14
11.870
4.39
Density Determination for Aluminum
18.0 –
17.0 –
16.0 –
Density = slope
2– 1
2– 1
17.5 – 8.5
6.42 – 3.12
Mass of Sample (g)
15.0 –
14.0 –
13.0 –
12.0 –
11.0 –
9.0
3.30
10.0 –
= 2.7 g/mL
9.0 –
8.0 –
3.0
4.0
5.0
6.0
7.0
Volume of Sample (mL)
Chem 131
Experimental Measurement and Error
Page 5