UNCERTAINTIES IN MEASUREMENTS PROPERLY PROCESSING DATA
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UNCERTAINTIES IN MEASUREMENTS
PROPERLY PROCESSING DATA
Every measurement has a doubtful digit.
MEASUREMENTS WITH UNCERTAINTIES
This ruler measures to the mm (0.1 cm)
Every measurement made with a ruler like this has an uncertainty of ± 0.05cm. This is half the smallest division.
This measurement is 1.02 ± 0.05 cm.
It lies somewhere between 0.97 cm and 1.07 cm.
Notice that the measurement and the uncertainty have the same level of precision.
Every measurement has a doubtful digit.
MEASUREMENTS WITH UNCERTAINTIES
This stopwatch measures to 0.01 seconds.
This measurement could be 3min and 53.170 seconds or3 min and 53.179 seconds.
This means the uncertainty is ±0.01 seconds and the measurement is 237.17 ± 0.01s.
For IB labs, each measurement needs to be repeated at least 3 times.
Measurement, measurement
For number of repeated values, we find the average. The uncertainty in the average is plus or minus one-half of the range between the maximum value and the minimum value.
Raw data should be presented in an easy to understand data table. The headings should state the name of the quantity, its symbol, the units it is measured in and its uncertainty.
Recording the data
Lengthl/cmΔl = ±0.1 cm
29.4
38.5
48.7
60.5
Of course, you will be measuring at least two variables. Here’s an example of using repeated values.
Recording the data
Lengthl/cm
Δl =±0.1 cm
Time for 10 Periods
10T/sΔ10T = ±0.01s
60.516.77
15.42
15.79
Of course, you will be measuring at least two variables. Here’s an example of using repeated values.
Recording the data
Lengthl/cm
Δl =±0.1 cm
Time for 10 Periods
10T/sΔ10T = ±0.01s
60.516.77
15.42
15.79
Average Time for 10 Periods
16.0
Of course, you will be measuring at least two variables. Here’s an example of using repeated values.
Recording the data
Lengthl/cm
Δl =±0.1 cm
Time for 10 Periods
10T/sΔ10T = ±0.01s
60.516.77
15.42
15.79
Average Time for 10 Periods
16.0
Average Period
1.60
When dividing a measurement by a pure number, divide its uncertainty as well.
Basic Rules
This is called the absolute or rawuncertainty
Basic Rules
cmcmcmwrL
cmcmcmL
rwL
3.02.01.0
5.61.66.12
Basic Rules
%100%100)()()(B
B
A
ABABBAA
tionMultiplica
This is called the fractional or relative uncertainty A
A
The relative uncertainty multiplied by 100% is called the percentage uncertainty.
Basic Rules
Basic Rules
When adding or subtracting uncertainties, you ADD the ABSOLUTE uncertainties.
Summary of the Basics
When multiplying or dividing uncertainties, you ADD the PERCENTAGE uncertainties.
When stating uncertainties the uncertainty must have 1 sig fig and must have the same level of precision of the measurement itself.
To straighten some curves, we use the reciprocal values. The uncertainties MUST be processed properly.
Reciprocal calculation
Some graphs need a variable raised to an exponent to be linearized. Again, the uncertainties MUST be properly processed.
Power Function
Calculate the volume of a sphere whose radius is measured to be
You Try
Gradient Uncertainties
Error bars must be shown on the graph for the variable with the most significant uncertainty.
Data point
Min value the data point could have
Max value the data point could have
Sample Graph
-5
0
5
10
15
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25
30
35
40
0 0.5 1 1.5
Series1
This data has the responding variable with a fixed absolute value of uncertainty of ±2 cm
Sample Graph
0
5
10
15
20
25
30
35
0 0.5 1 1.5
Series1
Here the uncertainty in the responding variable is a fixed percentage of 5%
Sample Graph
y = 23.226x - 0.1833
-5
0
5
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0 0.5 1 1.5
Series1
Linear (Series1)
Here is the best straight line. The variables are proportional since the best straight line passes through the origin (accounting for error) and through each point.
But the slope measurement must have uncertainties associated with it as well.
Sample Graph
To get the uncertainty in the slope value, we will look at the maximum slope and the minimum slope then calculate half the range.
y = 23.226x - 0.1833
y = 25.357x - 2
y = 19.643x + 2
-5
0
5
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0 0.5 1 1.5
Series1
max best fit
min best fit
Linear (Series1)
Linear (max best fit)
Linear (min best fit)
Sample Graph
The max slope is 25.357. The min slope is 19.643. The best slope is 23.226.
y = 23.226x - 0.1833
y = 25.357x - 2
y = 19.643x + 2
-5
0
5
10
15
20
25
30
35
40
0 0.5 1 1.5
Series1
max best fit
min best fit
Linear (Series1)
Linear (max best fit)
Linear (min best fit)We would record the slope as 23.226 ±2.857.
With sig figs, it is 23 ±3.
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