# Significant Figures Lab

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```					Significant Figures Lab

Purpose         To apply the rules of significant figures when measuring and predicting experimental values.
To determine absolute and relative uncertainty in measurements.
To determine the absolute and relative errors in results.
To compare experimental uncertainties and errors.

Theory          With every measurement, x, there is some uncertainty, Δx, due to the reading limits of the measuring device
or the fluctuations in the value itself (noise). The relative uncertainty, x , is the ratio of the absolute
x
uncertainty of the measurement, Δx, to the measured value itself, x.

When the same set of conditions are repeated for several trials, the results should be constant. The precision
of these results can be expressed as the range of values or by the sample standard deviation of the values.
The range is the difference between the largest value, xmax, and the smallest value, xmin, for a set of repeated
trials.
range = xmax - xmin
The sample standard deviation, s, indicates how tightly clustered all the values are and minimizes the effect of
outliers. About 70% of all values for the repeated trials should lie within one standard deviation of their
average value, x .
 ( x  x i )2
                             where n is the number of repeated trials.
n 1

Uncertainty in calculations

Consider the following rectangle whose
length, l, is 40.0  0.5 mm and whose
width, w, is 10.0  0.5 mm.
When adding or subtracting values, the absolute uncertainty ∆x of the result is the sum of the absolute
uncertainties of the values.
The perimeter, P, of a rectangle is the sum of the sides:
P = 2 (l + w) = 2 (40.0 mm + 10.0 mm) = 100.0 mm
The absolute uncertainty of the perimeter, ∆P, could be calculated as follows
∆P = 2 (∆l + ∆w) = 2 (0.5mm + 0.5mm) = 2.0 mm = 2 mm (uncertainty only has one digit)
The perimeter would be expressed as P = 100.  2 mm = 10.0  0.2 cm = 0.100  0.002 m

When multiplying or dividing values, the relative uncertainty, x , of the result is the sum of the relative
x
uncertainties of the values used.
The area, A, of a rectangle is the product of the length, l, and width, w, of the sides:
A = l  w = (40.0 mm) (10.0 mm) = 400.0 mm2  

The relative uncertainty of the area, A , could be calculated as follows
A
A     l w
0.5 0.5
             0.010.05  0.06 = 1% + 5% = 6% relative uncertainty
A  l  w 50.0 10.0

The absolute uncertainty of the area, ∆A, could be determine as follows:
A 
        A    A  0.06(500)  30 mm2 Therefore, area, A = 500  30 mm2 = 5.0  0.3 cm2
 A 


Materials

Chemicals                  copper (II) sulfate, CuSO4, solution, dilute (unknown concentration) – referred to as “blue water”

Measuring Devices          30 cm ruler (± 0.5 mm)        graduated cylinder, 10 cm3 ( ± 0.05 cm3)    metrestick (± 0.001 m)
digital scale (200 ± 0.01g)           stopwatch or Pasco Xplorer GLX datalogger

Other                      pennies (8)       human volunteer (1)

Safety           Lab aprons and goggles must be worn at all times in the Chemistry Lab.
The “blue water” contains dissolved chemicals including copper (II) ions; this liquid is not highly toxic but
students should avoid contact with their skin, eyes, mouth, and nose. If any “blue water” is spilled, wipe up
the spill immediately and rinse thoroughly with water.

As with all experiments, students are to behave courteously and carefully. Students should wash their hands
well immediately after the completing the experiment. No food or drink should be brought into the lab area.

Procedure
Part 1 : Time interval for heartbeats
   To ensure similar conditions, all timings were conducted without delay in the same location using the same subject, timers,
and stopwatches.
   Exactly ten (10) heartbeats were counted by the subject while the time interval for those ten heartbeats was measured by
three timers using stopwatches.
   This procedure was repeated, providing a total of six data values for 10 heartbeats.
   This procedure was performed once for 60 beats (3 data values for 60 heartbeats)
   The data was recorded using ink in Table 1a.

Part 2 : Dimensions of a penny
   Two pennies were given to each group member
   The diameter and height of each penny was measured using a 30-cm ruler
   The height and length of all pennies together was measured using a 30-cm ruler.
   The data was recorded using ink in Table 2a.

Part 3 : Density of “blue water”
   All objects were removed from the 200g digital scale and the scale was reset to zero.
   A small graduated cylinder (10 mL) was placed on the scale and its mass was recorded.
   Liquid that was labelled “blue water” was placed in the graduated cylinder until the
meniscus (lowest point on the top of the surface) was at the 10 mL mark.
   The mass of the graduated cylinder and the 10-mL of “blue water” was measured using the 200 g digital scale.
   The same procedure was repeated using a 50-mL, then a 100-mL graduated cylinder and the maximum volume of “blue
water”

   The data was recorded using ink in Table 3a.
Raw Data

Table 1a.          Time intervals, t, for exactly10 heartbeats and 60 heartbeats. The uncertainty will be determined in Table 2a.

time interval, t
TRIAL
10 beats                                             60 beats

1

2

3

Table 2a.          Height and diameter of individual pennies and the combined height and length for all pennies.
Single penny                                                       all 8
A                           B                           C                         D                 pennies

height, h
 0.5 mm

diameter, d
 0.5 mm

Table 3a.          Masses of empty and full graduated cylinders. The uncertainty for volume is half of the smallest increment on the
cylinder used (analog device). The uncertainty of the mass of the empty graduated cylinder is the smallest place value
of the digital scale used (0.1g or 0.01g).

volume of “blue water”,             mass of empty graduated cylinder,             mass of graduated cylinder and “blue water”,
Vbw /     mL                             mgc /    g                                       mgc+bw /  g
Data Processing

Table 1b.        Analysis of time intervals for 10 beats and 60 beats

10 heartbeats
60 heartbeats (predicted)                 60 heartbeats (measured)                            error
(measured)
mean time       sample standard            relative          mean time                                     mean time        sample standard
uncertainty                                              absolute error           relative error
interval,         deviation             uncertainty          interval,                                     interval,          deviation
s                                                          s                        %
x/s               s/s                      %                 x/s                                           x/s                s/s

Table 2b.        Analysis of height and diameter of pennies

single penny (measured)                         all 8 pennies (predicted)               all 8 pennies (measured)                       error
standard                                                                                          standard
mean,                                relative              mean                                    mean                         absolute error /        relative error /
deviation  /                                                  uncertainty                          deviation
x / cm              cm
uncertainty %            x / cm                                  x / cm           / cm
cm                       %

height, h
diameter, d

Table 3b.        Density of “blue water”

volume of “blue water”                mass of “blue water”,                 density of “blue water”              absolute uncertainty             relative uncertainty
V / ± mL                            mbw / ± g                                ρ / kg dm-3                          kg dm-3                             %
Sample Calculations
 For each TYPE of calculation, ONE sample calculation must be shown
o Generally, the first set of data that uses the calculation is used to demonstrate the calculation
o Communicate the variables and formula clearly for the reader
o Substitute using a designated sample set and solve
o Show proper number of significant digits and appropriate units in the final answer
 Sample calculations should be shown near the table where the data is presented

Calculations to be shown include, but are not limited to:

Mean Time Interval, x , for 10 heartbeats (from data in Table 1a)
x = (sum of all time intervals,  xi ) ÷ (the number of samples taken, n)
xi
x        
n

Sample Variance, s2, for 10 heartbeats (from data in Table 1a)
( x  xi )2
s 
2

n 1

Sample Standard Deviation, s, for 10 heartbeats (from data in Table 1a)
( x  xi ) 2
s s     2

n 1

Relative Uncertainty for 10 heartbeats (from data in Table 1b)
relative uncertainty        = (absolute uncertainty of the measurement) ÷ (measurement value) = x
x
s
or        = (sample standard deviation) ÷ (sample mean) =    100% 
x       

Absolute Error for 60 heartbeats
absolute error = difference between the experimental value and expected value
= (time for 60 heartbeats, t60) – (6 times the time for 10 heartbeats, 6 t10)
= | t60 – 6t10 |
=

Relative Error for 60 heartbeats
relative error = the ratio of the absolute error to the accepted value, expressed as a percent
= (absolute error for 60 heartbeats) ÷ (actual time interval for 60 heartbeats)
| t60  6t10 |
=                    100% 
t60
Conclusion and Evaluation

   comment on the purpose / goals of the experiment
   state the significant findings of the experiment
   state the general relationships demonstrated in the experiment
   comment on the possible implications of these conclusions
   compare the measured and expected values
   evaluate the random and systematic errors in the experiment
   identify specific strategies to improve the investigation

Heartbeats
 Compare the actual (measured) time for 60 heartbeats and the expected time for 60 heartbeats based on the average time
for 10 heartbeats.
 Evaluate the difference between these two values.
 Identify specific reasons why the expected value was higher/lower than the actual value
 Identify specific methods or materials that could improve the precision (reduce random error), the accuracy (remove/reduce
a source of systematic error), the efficiency, or the meaning of this or a similar investigation.

Pennies
 Compare the actual (measured) height and diameter all of your pennies and the expected height and diameter of an
average penny.
 Evaluate the differences between these values.
 Identify specific reasons why the expected value was higher/lower than the actual value
 Identify specific methods or materials that could improve the precision, accuracy, efficiency, or meaning of this or a similar
investigation.

Density of Blue Water
 Compare the density of the “blue water” using each size of graduated cylinder
 Evaluate which graduated cylinder provides the best data for determining these values.
 State your concluded value and uncertainty (including relative uncertainty) for “blue water”
 Compare this concluded value to the density of pure water (1.00 g / mL).
 Identify specific methods or materials that could improve the precision, accuracy, efficiency, or meaning of this or a similar
investigation.

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