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Instructions:Exercises 1 to 8 are 2 points each.Each question asks mean value or standard deviation/uncertainty one at a time. Don’t group them together yet. This will be done in Exercise 9. Also, when the standard deviation/uncertainty is written by itself, you don’t put plus or minus (±) in front of it.Exercise 9 is 4 points.Report each [mean] ± UncertaintyGraph is 10 points.Make sure to complete everything as explained in the videos such as having chart title (no units), titles for axes (units required), plot points with the uncertainty bars, three linear lines (average, max, and min). You will also need the find the slope and its uncertainty.Make sure to use correct units and significant figures as they will also affect your gradeYou may either write directly on the worksheet if you have a digital pen or print it, write normally, and then upload the scanned copy of your pages.Show work on a separate paper or document. Only put answers on The pages cannot be too big, too small, blurry, rotated, or upside down. It shouldn’t be too difficult to read your handwriting. Anything that makes my grading of your paper will result in point deductions.
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Activity
Intro
Exercises
For questions (1) – (8) use the following information. You have measured the length of a
table to be 205.0 cm, 205.8 cm, 205.4 cm, 204.6 cm, and 204.9 cm five independent times.
You measured the width of the same table to be 60.1 cm, 60.4 cm, 60.2 cm, 60.0 cm, and
60.5 cm five independent times.
1) Calculate the mean length L of the table.
________________________________________________________________________
2)
L of the table.
________________________________________________________________________
3) Calculate the mean width W of the table.
________________________________________________________________________
4)
W of the table.
________________________________________________________________________
5) Calculate the area A = L x W of the table.
________________________________________________________________________
6) Using the correct equation for propagation of error, calculate the uncertainty of the area
A of the table.
________________________________________________________________________
7) Calculate the perimeter P = 2L + 2W of the table.
________________________________________________________________________
8) Using the correct equation for propagation of error, calculate the uncertainty of the
P of the table.
________________________________________________________________________
9) Report the mean length, mean width, area, and perimeter including their uncertainties in
the correct format discussed above.
________________________________________________________________________
Activity
Intro
Activity
Intro
Introduction to Measurements
When making quantitative measurements that involve continuous variables, the level of
uncertainty must be reported. Better instruments and laboratory procedures will yield results
closer to the actual result. It is important to note that obtaining the exact answer is not as
important as learning how to report a experimental value along with the level of uncertainty. In
other words, you must be honest when reporting values.
OBJECTIVES
In this activity, you will interpret and analyze data.
PART 1 – RULES AND DEFINITIONS
1. Measurement error – The difference between the experimental value of a quantity and the
accepted value of the quantity.
Error = Experimental value – Accepted value
Example 1 –
Solution – Error = 3.16 – 3.14 = 0.02
Relative error – The error of a quantity divided by the accepted value of the quantity. If xK
is the actual/accepted value and xE is the experimental value, then
xK xE
Relative error
xK
3. Percent error
xK
xE
100% This is used when comparing a experimental value to an
xK
accepted value. The percent error provides an idea of the accuracy of the measurement.
Example 2 – What is the percent error in the measurement of
= 3.16?
Solution – We found the error to be 0.02 in the previous example. The relative error is then
0.02
0.00637 to a few decimal places. The percent error is then:
3.14
0.00637 100% 0.64%
Activity
Intro
x2 x1
100%. When comparing two experimental values, the
( x1 x2 )
2
percent difference provides a measure of the precision of the experiment. Notice the
denominator is the average of the values.
5. Personal errors are mistakes made by the experimenter when taking data or in calculating.
4. Percent difference
6. Systematic errors result from incorrectly calibrated equipment, poor laboratory habits,
and/or incorrect zero point positioning. Repeating the measurement will not reduce the
error. Systematic errors cannot be analyzed using statistics.
7. Random errors that are produces by unpredictable and unknown variations. All personal
and systematic errors can be eliminated, but some random errors will remain. Random errors
can be analyzed using statistics.
8. Accuracy – The ability of a measurement to match the actual value of the quantity being
measured or how close the measurement is to the true value.
Example 3 – If the actual value of gravity is accepted to be 9.8 m/s2, then which measured value
is more accurate, 9.7 m/s2 or 9.5 m/s2?
Solution – 9.7 m/s2 is the correct answer since it is closer to the accepted value.
9. Precision – The ability of a measurement to be consistently reproduced. The number of
significant figures (discussed below) in the reported value indicates the level of precision of
the measuring instrument. Small random errors lead to higher precision.
Example 4 – Which group of measured values has a greater precision, (25 m, 26 m, and
24 m) or (22 m, 28 m, and 32 m)?
Solution – (25 m, 26 m, and 24 m) is a more precise grouping since the repeated measurement is
closer in each case.
Accuracy vs. Precision – Consider the three images below. Ten shots are fired at a target three
separate times. Each shot is considered a single measurement. The goal is to hit the target’s
center.
Activity
Intro
Case 1 – This data set is not precise (the repeatability of the
measurements is low). None of the measurements are accurate,
though the average of the data set may seem accurate (it may land
near the center). Arbitrarily chosen measurements should have
high percent difference. Without precision, the data set is not
reliable. This is an example of using a tool beyond its limit or
beyond the abilities of the user, such as firing too far from the
target, or trying to measure the thickness of a mosquito’s wing
with a meter stick.
Case 2 – This data set is precise, but not accurate. The repeatability
of the measurements is high (they are grouped closer together).
The average of the data set is far from the center, though. Arbitrarily
chosen data points will have low percent difference, but the average
will have a high percent error. This is an example of a systematic
error, such as incorrect sighting of the device, or not zeroing the tool
properly.
Case 3 – This data set is precise and accurate. The measurements
are repeatable and the average is near the center. Arbitrarily chosen
data points will have low percent difference, and the average will
have a low percent error.
10. Significant figures – All the digits in a measurement that are certain plus one that is
estimated.
Rules for counting significant figures:
a. The most significant digit is the leftmost nonzero digit. In other words, zeros at the
left are never significant.
b. If there is no decimal point explicitly given, the rightmost nonzero digit is the
least significant digit.
c. If a decimal point is explicitly given, the rightmost digit is the least significant
Activity
Intro
digit, regardless of whether it is zero or nonzero.
d. The number of significant digits is found by counting the places from the
most significant to the least significant digit.
Example 9 – How many significant figures are in each value?
Value
Number of Significant
Figures
232
3
23200
3
0.230
3
4.0012
5
2004
4
203.20
5
0.000030
2
Note that zeros can cause some confusion when counting significant figures. To clear this
confusion, write potentially ambiguous values in scientific notation.
Example 6 – How many significant figures does 8000 have?
Solution – By the above method 8000 should have one significant figure.
Example 7 – How can you report the value 8000 to have two significant figures?
Solution – Rewrite 8000 as 8.0 x 103.
When measurements are added or subtracted, the answer can contain no more decimal places
than the measurement with the left-most decimal place. When measurements are multiplied or
divided, the answer can contain no more significant figures than the measurement with the
fewest significant figures.
Example 8 – 9.001 cm + 2.1 cm = 11.101 cm, but is reported as 11.1 cm, since 2.1 ends at the
tenths place.
Example 9 – 9.001 cm x 2.1 cm = 18.9021 cm2, but is reported as 19 cm2, since 2.1 only has
two significant figures.
Activity
Intro
11. Precision of the measuring tool – The smallest subdivision that can be read directly. If a
single value is measured to be 25.0 cm with a tool of precision 1 mm = 0.1 cm, then the
value should be reported as (25.0 ± 0.1) cm.
Reporting Values and Dealing With Random Errors
12. Mean and Standard Deviation – Random errors have an equal likelihood to be low or high
compared to the true value. So, taking the mean x of many measurements
x1, x2, …, xn is a natural way to reduce the effect of random errors. The mean is defined as
1 n
x
xi
ni1
and is the best value obtained from all the measurements. (Note: If several values are averaged,
a general rule is to assign one more significant figure to the mean value.)
Statistical analysis will show that the sample standard deviation
1
n 1
n
n 1i1
( xi
x )2
is a good measure of the precision of the measurements.
13. Standard error
n 1
n
measures the precision of the mean.
14. Reporting the uncertainty – The standard deviation (or standard error if many
measurements are made) will substitute as the uncertainty for the mean of many
measurements. It is necessary to report it correctly to the reader. Use the following format
x
n 1
or x
It is important to note that it is necessary to keep no more than one significant figure in the
standard deviation and the standard error. (Some texts will say that the standard deviation
and standard error should be no more than two significant figures.) Be sure to keep the same
decimal place in the mean as in the standard deviation and standard error, even if this
means rounding the mean to a lower decimal place (you can remove certainty to ensure the
decimal places match). Never add digits to the mean in order to match the decimal place of the
standard deviation and standard error (you cannot add certainty). If the standard deviation or
standard error is too small, then use the precision of the measuring tool.
Example 10 – Given the following measurements find the mean and standard deviation and
report it in the correct format. 2.45 m, 2.47 m, 2.43 m, 2.51 m, 2.44 m.
Activity
Intro
Solution – The mean is:
2.45 m 2.47 m 2.43 m 2.51 m 2.44 m
5
x
2.46 m
The standard deviation works out to be 0.0316 m. The correct form for reporting is:
2.46 ± 0.03 m
since the uncertainty is rounded to one significant digit (0.03) and the mean is rounded to match
the decimal place. In this case, 2.46 ends at the hundredths place, which matches 0.03. If the
uncertainty was calculated to be 0.3, for instance, then the correct reporting would be:
2.5 ± 0.3 m.
Sometimes the standard deviation will be calculated to be too small and will seem to be zero. In
this case, we must use the precision of the measuring tool and the measurer’s technique to
estimate the uncertainty. In other words, the uncertainty would be the smallest value that the
measurer can read directly.
Propagation of Error
It is not entirely trivial how to include the uncertainties in calculations involving more than one
quantity with an uncertainty. It is important to use the method of propagation of error. There are
two forms of equations that will be discussed here.
A) P
kx a y b z c
B) S
ax by cz
For equation (A) P
kx a y b z c the propagation will be found using the following equation:
(A1)
For equation (B) S
2
P
2
a2
P
x
2
x
2
b2
y
2
y
2
c2
z
2
z
2
ax by cz the propagation will be found using the following equation:
(B1)
2
S
a2
2
x
2
2
y
Example 11 – The equation for volume of a cylinder is V
c2
2
z
d 2L
, where d = diameter
4
and L = length are the only two measured values. Since the V = volume is to be calculated from
measured values, to find the uncertainty of the volume, you must propagate the error.
Activity
Intro
Solution – The volume equation matches the form of equation A, since the measured values are
being multiplied. The first step is to put the volume equation in the form of equation A:
P = V (calculated value)
x = d (first measured value)
b = 2 (exponent of the first measured value)
y = L (second measured value)
c = 1 (exponent of the second measured value)
z = 1 (no third measured value)
d = 0 (no third measured value)
Substituting the above into the matching error equation (A1), we find:
V
2
V
2
2
2
d
2
d
2
2
1
2
L
2
L
where the values of d and L would be the mean values. Solving for
V
yields the uncertainty of
the volume.
Graphs
Several experiments will require you to construct a graph or a curve. Unless otherwise specified,
these are to be done by hand into your notebook. The following items should be considered:
1. The Axes – The horizontal axis is known as the axis of the abscissa, and the vertical axis is
known as the axis of the ordinate. In most cases, the instructions should illustrate which
quantities are to be plotted horizontally and which are to be plotted vertically. Generally, the
independent variable, typically horizontal, is taken as the abscissa and the dependent
variable, typically vertical, is taken as the ordinate. Often you are asked to plot the ordinate
versus the abscissa. For example, if you are asked to plot F vs. x, then you construct a graph
with F on the vertical axis and x on the horizontal axis. Always label the axes with the
variables and their units.
2. The Table – A table of quantities to be plotted should be made for convenience in plotting
and to aid in selecting a scale. The units of each quantity should be identified at the top of
the column.
3. The Scale – The scale of the graph is the number of units that correspond to one space or
block on the graph paper. The scale should be chosen for both ordinate and abscissa that the
curve, when drawn, will extend over most of the paper. Remember that the larger the space in
which the data fits, the more precisely the points can be plotted. At the same time, a
convenient scale should be chosen which is not awkward. Consult your table to determine a
suitable range. Before deciding on a scale, try it out to see if points can be plotted easily. In
almost all cases, values should increase from left to right and from below to upward. Indicate
the scale plainly by numbering the divisions. You do not have to number every division.
Activity
Intro
4. The Plotted Points – The plotted points should be small but identifiable. If several curves
are to be drawn on the same set of axes, use different identification around the points of
each curve, circles for the first, triangles for the second, squares for the third, etc… This
should be done before attempting to draw the curve itself.
5. The Curve – After the points have been plotted, a curve corresponding to the theoretical
expectations should be drawn. If, for example, a straight line is expected, it should be drawn
in such a way that about one half of the points miss the line on the same side. This is the line
of best fit. A linear least squared fit of the data can be performed. Use a ruler or straight edge
to draw a straight line.
6. The Title – Write a title above the graph and caption below it if necessary.
Activity
Intro
Graphing Exercise
Suppose an experiment were conducted on the stretching of a spring as a function of the force
applied to the spring yielding the data in the following table
(Note: The values above are held to a few digits without consideration to the rules of reporting
values discussed on pages 4 and 5. This is common for data and calculations tables to avoid
roundoff errors in future calculations. Following the rules for reporting values, the correct
reporting of the value x in the fourth row above would be = 2 ± 1 cm and the correct reporting
for row 5 above would be = 3.9 ± 0.3 cm, and so on.)
First, you will need to decide where to draw the axes and what the scale should be. There are no
absolute right or wrong choices, but some choices are better than others. Label the axes and mark
convenient intervals
Next, plot a graph of F vs. x. This means that F is on the vertical axis (ordinate) and x is on the
horizontal axis (abscissa).
Affix error bars to each point. Draw a straight line of length 2 2
horizontally centered on each point.
1
Next, draw the theoretical curve. The theory states the relationship between F and x is a straight
line through the origin. Draw the best straight line you can through the origin and the plotted
points. Try to leave about as many points above and below the line. This is the line of best fit.
(Ask your instructor for the best fit ruler).
Finally, draw the MAX and MIN lines. Draw two further lines through the origin with the largest
and smallest slopes respectively that are reasonably near the plotted points. These lines should
pass through the edges of some of the error bars. The MAX line will have a larger slope and the
MIN line will have a smaller slope.
Find the slopes of the three straight lines on your graph. You may use any two points on the line,
but points well separated will provide better precision. To find the slope, find the “rise over the
run” – the change in vertical over the change in horizontal.
Activity
Intro
Calculate the uncertainty of your line of best fit. Statistically, it is accepted to subtract the slope
of the MAX line and the slope of the MIN line and divide by 2. This will give the uncertainty of
the slope of the line of best fit.
A plot of the data with error bars is shown below. The slope of the best fit is calculated to be 93
N/m. The slopes of the MAX line and MIN line, respectively, are 110 N/m and 82 N/m. The
uncertainty of the line of best fit is then found by taking the difference of these two slopes and
112 82
dividing by 2. This gives an uncertainty of
14 N/m . Considering the rules of
2
reporting significant digits, we write slope 90 10 N/m .
How do your hand-generated graph results compare?
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