Module 0 day 2 Exploration worksheet
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Physics
Date
Dec 6, 2023
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MODULE 0
EXPLORATION: MARGIN OF ERROR AND STATISTICAL ANALYSIS
Names of Group members: ____________________________
Date: _________
____________________________
____________________________
➢
Attach any calculations and work done on scratch paper to this worksheet!
PART A:
Uncertainty in a Single measurement
In this part you will be measuring the length of a piece of wood and determining the uncertainty
in that measurement.
Exercise A1: Uncertainty in measurement using centimeter ruler
(30 cm ruler that is green and tan)
1.
Using the piece of wood that is on your table and the ruler, measure the length of the
wood and estimate the uncertainty in that measurement, enter values below.
Have each
member of your group measure and determine the values below as a group.
Length of wood (in centimeters):
________________
Estimated Uncertainty in measurement of length: ___________
Width of wood (in centimeters):
________________
Estimated Uncertainty in measurement of width: ___________
Exercise A2: Uncertainty in measurement using clear ruler
2.
Using the piece of wood that is on your table and the clear ruler, measure the length of
the wood and estimate the uncertainty in that measurement, enter values below. Have
each member of your group measure and determine the values below as a group.
Length of wood (in centimeters):
________________
Estimated Uncertainty in measurement of length: ___________
Width of wood (in centimeters):
________________
Estimated Uncertainty in measurement of width: ___________
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3.
Write the length of the wood in the proper form
of “Value” ± “Uncertainty” for both the
Centimeter ruler and clear ruler.
Centimeter Ruler:
________________________________
Clear Ruler:
______________________________
4.
Is the uncertainty in the measurement of the length the same as the uncertainty in
measuring the width, explain?
5.
Which ruler has the larger uncertainty and why?
6.
Which measurement has more significant figures and why?
Part B:
Uncertainty in a Calculated Measurement
In this exploration you will measure the diameter of a tennis ball using two different methods,
calculate the volume of the tennis ball and
create a “margin of error” for
the calculated value in
each method.
The volume of a sphere is given by:
𝑉 =
4
3
𝜋𝑟
3
but we will be measuring the diameter of the ball not the radius (r), therefore using the
diameter r = d/2 the equation we will be using to calculate volume is
𝐕 =
𝟏
𝟔
𝛑𝐝
𝟑
𝐄𝐪?𝐚?𝐢?? 𝟏
where π = 3.14
and d is the diameter of the ball
The Uncertainty in the calculated value of the volume using the “Max/Min Method” is given by
Uncertainty =
(Maximum value of volume − Minimum value of volume)
2
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Exercise B1: Uncertainty in volume using the clear ruler
1.
Have each member of your lab group measure the diameter of the ball using the clear
ruler.
Write down each value in a table and enter the average value below. (It helps to
close one eye to make the measurement)
Do NOT use any other devices or methods to
reduce the uncertainty only use your eye!
Average diameter of tennis ball:
________________
The smallest marking on your ruler
is millimeters, which means that your data should be “precise
to +/- 0.5
mm.”
or less. However, just because your measurement device is capable of
measuring to within 1mm, doesn’t mean YOU were capable of using it to that precision. To
report a more descriptive “margin of error” on your measurement we need to think of how
accurately you were able to make the measurement and determine an Estimated uncertainty.
2.
Explain some specific reasons why it was difficult for you to measure the diameter of the
tennis ball using the ruler.
3.
As a group, decide on an upper limit for the diameter of the ball and a lower limit for the
diameter of the ball. By doing this the average diameter and the uncertainty in your
measurement can be determined.
Upper limit of diameter of tennis ball:
______________
Lower limit of diameter of tennis ball:
______________
4.
The average diameter of the ball is obtained by adding the upper and lower limits of the
diameter of the ball then divided by 2.
Average diameter =
Upper limit+lower limit
2
= _____________________
5.
The uncertainty in the measurement of the diameter of the ball is obtained by subtracting
the upper and lower limits and dividing by two.
Calculate and record the value below.
Uncertianty in diameter =
Upper limit−lower limit
2
= ___________________
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Page
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6.
Record your measurement of the diameter of the ball using the ruler as both an average
value +/- a margin of error AND as a range.
Diameter of tennis ball with uncertainty:
______________
7.
Using the average diameter of the ball as the diameter calculate the volume of the ball in
cm
3
using
Equation 1
above
Volume of tennis ball:
______________
Calculate the maximum and minimum values of the volume that the tennis ball could
have using your values for maximum diameter and minimum diameter.
Max Volume of tennis ball:
______________
Min. Volume of tennis ball:
______________
Calculate the uncertainty in the volume of the tennis ball
Uncertainty in volume of tennis ball using ruler:
_____________
Write the volume of the tennis ball with uncertainty in proper form in the space below
Volume of tennis ball using ruler:
________________________________
Exercise B2: Uncertainty in volume using the Calipers
How to use the Calipers:
The Caliper will turn on automatically when it is moved or press the on
button.
Close the Caliper and zero the reading by pressing the zero button.
Make sure the
reading is in millimeters (mm).
Expand the Caliper until the ball will fit inside the jaws, gently
move the jaws until they are touching both sides of the ball, read the digital scale. Have each lab
member practice using the calipers measuring the diameter of the tennis ball.
For a digital device the uncertainty in its reading is given by the manufacturer, for the Calipers
you are using the uncertainty is ± 0.01 mm.
8.
Do you think this is the uncertainty in your measurement of the diameter of the ball?
Why would it not be the uncertainty of the diameter? explain why or why not
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MARGIN OF ERROR for the Calipers
9.
Place the Caliper around the tennis ball, move the Calipers in until you feel that it is
measuring the smallest diameter that is reasonably possible. DO NOT clamp down hard
on the ball with the Calipers!
Write this minimum value below.
Lower limit of diameter of tennis ball:
______________
10.
Place the Caliper around the tennis ball, move the Calipers out until you feel that it is
measuring the largest diameter that is reasonably possible.
Write this maximum value
below.
Upper limit of diameter of tennis ball:
______________
11.
Subtracted the maximum value from the minimum value and divide this value by two.
This gives the margin of error for the measurement of the ball for this method.
Record
your uncertainty below (show work!)
Average diameter of tennis ball with uncertianty:
______________
12.
Using the average diameter of the ball from the Caliper calculate the volume of the ball in
cm
3
using equation 1 above for method 2
(
be careful to convert values to centimeters
before calculating!!!)
Volume of tennis ball:
______________
Calculate the maximum and minimum values that the tennis ball could have using your
values of margin of error
Max Volume of tennis ball:
______________
Min. Volume of tennis ball:
_____________
Calculate the uncertainty in the volume of the tennis ball
Uncertainty in volume of tennis ball using Calipers:
_____________
Write the volume of the tennis ball with uncertainty in proper form in the space below
Volume of tennis ball with uncertainty using Calipers:
_____________________________
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Comparing Exercise B1 and B2
Now we want to report our findings and see if the volume of the tennis ball is consistent
between the two measuring methods.
As a scientist you will need to decide the best way
to communicate your findings to others.
Reporting your values as a range in table format
:
13.
Fill in the table with your findings:
Average value
Uncertainty
Volume of tennis ball from ruler in cm
3
Volume of tennis ball from calipers in cm
3
14.
Calculate the percent difference between the two volumes of the tennis ball
15.
Reporting your values in a bar graph with error bars:
Excel can be a great tool to create simple yet visually appealing graphs to convey your findings.
It can also be frustrating if you have little experience with the program.
You will be using it
often, so follow these steps to generate a bar graph with custom error bars.
Creating a graph that
looks like the one below is your end goal.
If you know how to do this, go for it.
If you don’t,
use the “
Making a Bar chart in Excel with error bars
”
page 19 of
the “Introduction”
guide from
the first week of lab.
Figure 1: This is an example of how your bar graph might look.
The numbers will be different
depending on your data.
16.
Attach a copy of the bar graph to this worksheet.
Caluclated Volume of Tennis Ball
Usiing Centimeter ruler
Volume (cm^3)
110
Uncertianty (cm^3)
70
Usiing Micrometers
Volume (cm^3)
117.73
Uncertianty (cm^3)
0.06
110
117.73
0
20
40
60
80
100
120
140
160
180
200
1
Volume (cm^3)
Volume of Tennis Ball
Ruler and Micrometer
Ruler
Micrometer
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17.
Form a statement about whether the two volumes measurements agree with each other or
not.
If the uncertainty ranges (error bars) of the two measurements overlap, then they
agree.
If the ranges of the two values do not overlap, then they do not agree.
It is
important to include a statement like this in addition to a table or bar graph in your
reports.
18.
List at least 2 sources of uncertainty in using the ruler for determining the volume of the
tennis ball and circle what you feel is the Dominate error in the method.
19.
List at least 2 sources of uncertainty in using the calipers for determining the volume of
the tennis ball and circle what you feel is the Dominate error in the method.
PART C:
Statistical Uncertainty
Exercise C1: Statistical Uncertainty in rolling dice
In this exploration you are going to toss a set of dice (two) and
record the sum of the
numbers that come up on the two dice
.
1.
What is the highest sum that could occur?
2.
What is the lowest sum that could occur?
3.
Do you think that it is equally likely to get any sum?
Explain your reasoning.
4.
Now have each of your lab members toss the dice and record the sum that results from
each toss.
Record each individual sum in Excel (use Column A) since you will need
them later.
Have a total of 100 tosses
with their sums.
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5.
Use Excel to count how many times each sum appeared in your dataset. DO NOT
ATTEMPT TO COUNT THE OCCURENCES WITHOUT EXCEL.
One way to do this is to use the
COUNTIF(range, criteria)
function.
This function scans a
certain range of cells and reports how many times the specified criteria occurs. Follow the
remaining instructions if you don’t know how to use this function to count your data.
a.
In your Excel workbook, starting in cell 2 of column C, type the numbers 2 thru
12 into the rows of that column.
Then, in column D row 2, type
=COUNTIF(A1:A100, 2).
Repeat this step working your way down the column,
but changing the function to =COUNTIF(A1:A100, 3)… until you reach 12.
b.
Go back to cell 1 of columns C and D and add labels to identify what these
columns represent.
c.
Your workbook might look like this:
6.
Go to the Google spreadsheet and enter your group’s raw data
into column A
of the sheet
(don’t overwrite anyone’s data; just go to the next available cell in column A).
7.
Back in your own spreadsheet: Create a vertical bar graph to display how many times each
sum appeared in your data.
The x-axis should show the possible sums 2 thru 12.
The y-axis
should show the number of occurrences.
Make sure to include axis labels.
If you need help
creating the bar graph, use the same guide included in this manual that you used earlier (this
graph doesn’t need error bars).
Copy and paste
a screenshot of your bar graph on the next
page.
8.
Based on your graph, what was the sum that occurred the most often?
9.
After all the groups have entered their data into the Google spreadsheet, copy and paste the
full dataset into your own Excel workbook and then repeat the steps you did earlier to count
the number of times each sum 2 thru 12 appeared in the entire class’ dataset.
Make a vertical
bar graph, just like you did before, using all of the groups’ data.
Copy and paste that graph
below.
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10.
According to this new graph, what was the sum that occurred the most often?
11.
Do you believe that each sum is equally likely to occur?
If not, which sum is the most likely
to occur?
We are going to call this the theoretical or “true” value.
Data
can be “distributed” or spread out in many different ways.
This lab has been an example of
data that occurs due to chance and forms a “
Normal Distribution
”.
The sum of the dice tends to
be around a central value, which you determined above.
A Normal Distribution is also known as
a “Bell Curve”, as seen to the right.
A Normal Distribution has some very important features
that we can use to analyze our data.
•
Mean
–
the average; sum of all data points divided by the number of data points
•
Mode
–
most commonly occurring data point
•
Median
–
the central data point; half the data falls below the median and half the data
falls above the median
•
Standard deviation
, σ –
a measure of how spread out the data is from the average; if the
data follows a standard distribution, then 68% of the data are with 1 standard deviation of
the mean, 95% of the data is within 2 standard deviations of the mean, and 99.7% of the
data are within 3 standard deviations of the mean.
In Excel, you can calculate the average of a
dataset by using the function “AVERAGE (range)”.
In the parentheses, you should input the range of cells where your data is located.
You can also
calculate the standard deviation of a set of data by using the “STDEV (range)” function.
Again,
in the parentheses, you should input the range of cells where your data is located.
12.
Calculate the average and standard deviation using Excel functions for the original 100
dice tosses your group performed.
Report those values below.
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13.
How does the average
value from your group’s original 100 tosses compare with the
theoretical or “true” value of 7?
Is your average within 1 standard deviation? If not, how
many standard deviations is it within?
14.
Calculate the average and standard deviation using the Excel functions for
all of the data
points
using the entire class’ dice tosses.
Report those values below.
15.
How did the additional data from the class change the average and standard deviation
fro
m your group’s original dataset, if at all?
(i.e. did the average get closer to the
theoretically value with the addition of more data points? Did the standard deviation get
smaller or larger?)
Another statistical quantity that is useful to describe your data in relation to other experiments is
called the
“standard error of the mean”
.
Standard error describes how well your subset of data
describes the larger dataset, possibly taken by additional experimenters.
It is a good way to
compare your experimental values to other groups who have tested the same quantities.
To
interpret standard error,
you might say “The average value we measured has a 68% chance of
falling within 1 standard error of the average value another group who carried out the same
experiment found.
Additionally, the average value we measured has a 95% chance of falling within 2 standard
errors of the average value another group who carried out the same experiment found. And
finally, the average value we measured has a 99.5% chance of falling within 3 standard errors of
the average value another group who carried out the same experiment found.”
To calculate the standard error of the mean, you must first calculate the standard deviation σ of
your data.
Then use the following equation:
𝑆𝐸 =
𝜎
√𝑁
, where N is the number of data points in
your dataset.
16.
Calculate the standard error for your 100 original data points.
Page
11
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17.
Calculate the standard error for the entire class
’s data
.
18.
What do you notice about the standard error as the number of data points increases?
The effect of
statistical
errors (also known as “random” errors) can be reduced through
additional data collection.
If an error is truly statistical, then it has an equally likely chance of
being above the “true” value as being below the “true” value.
With enough measurements, t
hese
statistical errors should cancel out and the average value will be unaffected.
Another type of
error is called
systematic
error.
This type of error affects a measurement in the same way every
time.
No matter how many measurements you take, the error will not cancel out and will skew
the data above or below the “true” value every time.
Exercise C2: Go to your instructor and ask for a new set of dice.
19.
Repeat your experiment with the new set of dice.
Record the sum that occurs for 100
new rolls of your dice in Excel and create a vertical bar graph to show the number of
occurrences for the sums 2 thru 12 for your data only.
20.
Calculate the average and standard deviation for your new dataset.
21.
Based on what you have learned, do you believe that the new set of dice follows a normal
distribution? Use qualitative and quantitative arguments based on your average and
standard deviation.
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22.
Give an example of a possible source of statistical error in your experiment today.
(“Human error” is
never
an acceptable answer.)
23.
Give an example of a possible source of systematic error in your experiment today.
Attach your Excel spread sheet for both sets of dice to the worksheet.
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