Lab 2 Uncertainty!
!
!
Objective!
The purpose of this lab is to introduce the concept of uncertainty in physical measurements.!
Every measurement has an amount of uncertainty associated with it. The uncertainty can tell us
the likely value of a repeated measurement. It can also tell us how close we are to the true
measured value. !
We are going to practice characterizing the uncertainties of three types of measurements by
measuring the value of pi using various circular objects’ diameters and circumferences.!
Part 1 Single measurement!
If you can make a measurement only once, you will have to estimate the uncertainties of the
measurements.!
Take a ruler and measure the diameter of the circular object. Here are the issues that can make
you reading uncertain.!
a.! Reading the position of the ruler could be off by about 0.5 mm. (instrument error)!
b.! The placement of the object may be off by about 0.5 mm. (method error)!
Take a string, wrap it around the circumference of the object, and measure the length of the string
required to wrap around the object. Besides the above errors,!
c.! The string may stretch or shrink by about 2 mm. (method error)!
d.! The marking of the string may be off by 1 mm. (method error)!
This is how you would report the measurements.!
diameter = 20.3 ± 1.0 mm !
circumference = 63.9 ± 3.0 mm !
To calculate the value of pi, you divide the circumference by the diameter.!
π=
circumference
63.9 ± 3.0 mm
=
!
diameter
20.3 ± 1.0 mm
The value and its uncertainty are calculated this way.
formulas. There is a list at the end of this file.!
63.9 63.9
π=
±
20.3 20.3
These are called error propagation
⎛ 1.0 ⎞⎟2 ⎛ 3.0 ⎞⎟2
⎜⎜
⎟ + ⎜⎜
⎟ = 3.1478 ± 0.2142 !
⎜⎝ 20.3 ⎟⎟⎠
⎜⎝ 63.9 ⎟⎟⎠
To the correct number of significant digits, the result is!
!
!
!
π = 3.15 ± 0.21 !
page 1
Part 2 Repeated measurements!
If you can make multiple measurements of the same value, you can use statistics to tell you what
the uncertainty is. !
Measure and record the diameters and circumferences of 10 circular objects and calculate the
value of pi for each of them without regard for the uncertainties of each measurement. You should
end up with 10 values of the same thing, namely pi.!
In aggregate, they tell you the value of pi through their mean. The uncertainty is found through
calculating the standard error. The mean is calculate using the AVERAGE function in Excel. The
standard error is found in two steps. First, you can calculate the standard deviation using the
STDEV function. The standard error is just the standard deviation divided by the square-root of
the number of samples. Your measurement is presented this way.!
(mean ± standard error) unit !
For example, here is a list of experimental pi values derived from the ratio of the circumferences
divided by their respective diameters. By the way, what I am showing are intermediate results so I
am keeping all of the digits of each number.!
pi
3.0000
3.4900
2.7200
3.4900
2.9300
3.2100
3.0000
3.1400
3.8400
2.8600
The mean is 3.1680. The standard deviation is 0.3448. The standard error is 0.1090. To the right
number of significant digits, which is 3, !
!
π = 3.17 ± 0.11 !
page 2
Part 3 Trend Measurement!
A trend comprises of measurements at different input variable (X) values to see if it affects the
output variable (Y) values. !
What we want to know is how the output changes with different values of the input. This means
we want to look at the actual diameters and circumferences. Here, let’s take the diameter as the
input variable and the circumference as the output variable. What is relationship between the
input and the output?!
circumference = f (diameter) !
In this case, we have a model of what the relationship should be. This makes thing much easier.
We expect the circumference to be proportional to the diameter, or, in the language of algebra,!
circumference = constant ⋅ diameter = slope ⋅ diameter + y -intercept !
We know that the slope is pi. Let’s see how the data tell us that. Here is the data.!
diameter (cm)
circumference (cm)
2.36
7.08
3.04
10.61
4.36
11.86
5.04
17.59
5.88
17.23
7.16
22.98
7.88
23.64
8.72
27.38
10.16
39.01
10.80
30.89
With trend data, you should always plot it first. This will reveal any error in the data.!
Circumference vs Diameter
Circumference (cm)
40
30
20
10
0
0
2
4
7
Diameter (cm)
9
11
!
Here, the next-to-last data point looks off. You should check to see if any error was made. If no
obvious error were made, you will have to decide whether to include or exclude it. You may also
decide to repeat the measurement and add addition data points to the graph. There is no rule
against having multiple data points for a certain x or y value. I will keep it for instructional
purposes.!
page 3
The information we are looking for is in the slope of the trend. The trend is basically the equation
of the best line through the data. In Excel, a quick way to do this is to use the TRENDLINE
function. Select the graph, then select LAYOUT-TRENDLINE-MORE TRENDLINE OPTIONS….
Check the box that shows you the equation. Also, check the box the lets you set the y-intercept to
zero. Since we have a model that tells us the y-intercept should be zero, let’s match the equation
to out model.!
I find the slope to be 3.2008.!
Circumference vs Diameter
Circumference (cm)
40
30
y = 3.2008x
20
10
0
0
2
4
7
9
Diameter (cm)
11
!
The slope itself has uncertainty since it does not go through the data perfectly. Notice that this
method of finding the slope does not tell you the uncertainty of the slope. To find the uncertainty
of the slope, we have to use the LINEST function. !
Select a 2 cell by 2 cell area. Type “=linest(“. Select the y data. Type “,”. Select the x data. Type
“,”. Select the y-intercept type. Type “1” for a non-zero y-intercept or type “0” for a zero yintercept. In this case, we want a “0”. Finally, finish with a “,1)”. Do not hit enter, but, instead do
this “CONTROL-SHIFT-ENTER”.!
The top-left number is the slope. The bottom-left number is the standard error of the slope. The
top-right number is the y-intercept and below it is its standard deviation.!
I get a standard error of 0.1231. This mean the value of pi is!
π = 3.20 ± 0.12 !
Finally, between an average and a trend, we always favor the trend.
page 4
Part 4 What Does It Mean?!
When you have a result like this, what can you say about the quality of the data.!
π = 3.20 ± 0.12 !
Fist, we can measure the accuracy of the method. For this we calculate the percent error.!
percent error =
expected value − experimental value
×100% !
expected value
For our result, this is!
percent error =
3.14 − 3.20
×100% = 1.9% !
3.14
For most of the labs we do, 5% is a good result.!
Second, we can get a sense of the precision of the results. This is what the standard error tells
us. One way to quantify this value is to also compare it to the expected value. Let me call this the
percent uncertainty.!
percent uncertainty =
experimental uncertainty
×100% !
expected value
For our result, this is!
percent uncertainty =
0.12
×100% = 3.8% !
3.14
A 100% percent uncertainty is a bad result. It means we would have a result of 3.20±3.14. At
50%, the result would still be bad at 3.02±1.57. As a rule of thumb, a percent uncertainty of 5%
would be a good precision.!
Third, another consideration is whether the expected value is within, usually, 2 standard errors of
the experimental value. If the standard error is small (which is a good thing), but the accuracy is
not there, then there may be no overlap between the experimental results and the expected result.
This is fixed by producing a higher accuracy.!
For out result, the expected value is within 1 standard error of the experimental value so the
measurement is a good one.!
3.14
2.96
3.08
3.20
3.32
3.44
!
In summary, we want small percent error and small percent uncertainty, and we want overlap
between the range of the experimental result and the expected value.
page 5
Uncertainty Formula!
!
!
Addition!
(A ± a)+ (B ± b) = (A + B)± a 2 + b 2 !
Subtraction!
(A ± a) − (B ± b) = (A − B) ± a 2 + b 2 !
Multiplication!
2
2
! a$ ! b$
(A ± a)(B ± b) = (AB) ± (AB) # & + # & !
" A% " B%
Division!
2
2
! a$ ! b$
(A ± a) / (B ± b) = (A / B) ± (A / B) # & + # & !
" A% " B%
Square (same as multiplication)!
2
2
!a$ !a$
a
(A ± a)(A ± a) = (A )± (A ) # & + # & = (A 2 )± (A 2 ) 2 !
" A% " A%
A
2
2
Square-root!
(A ± a) = ( A ) ±
a
!
2A
Note: treat a constant as though it has a zero uncertainty.
page 6