Collecting and Processing Data
Processing uncertainties
Collecting Data
An important part of practical work in Physics is recording numerical data and effectively analysing it.
In finding the average speed of this cart, the distance travelled and time taken between A and B must be measured.
Both measurements will have errors.
Measuring Variables
These are some common variables you will measure in this course.
Precision in Measuring
The following data is recorded when measuring the distance travelled by the cart, using a ruler:
d1 = 0.82 m
d2 = 0.85 m
d3 = 0.82 m
This could be due to an imperfection in the ruler.
The “true” distance is 0.75 m. The results are precise, but not accurate.
Precision in Measuring
A result is precise when similar measurements are obtained using the same method.
The results may not be “true” or accurate, but they are repeatable.
Precise
Not accurate
Precision in Measuring
The following data is recorded when measuring the time taken by the cart, using a stopwatch:
t1 = 1.23 s
t2 = 1.15 s
t3 = 1.37 s
Average t = 1.25 s
This usually arises from random fluctuations.
The “true” time taken is 1.25s. The results are accurate, but not precise.
Accuracy
Accuracy relates to the quality of a result and the closeness to it’s true value.
The marksman has approached the true value, although without great precision
Accurate
Not precise
Systematic Errors
Systematic error can be caused by an imperfection in the equipment being used or from mistakes the individual makes while taking the measurement.
A voltmeter might show a reading of 1V even when it is disconnected from any circuit.
This type of error can be offset by simply subtracting the value of the zero error.
Example
Random Errors
Random errors arise from random fluctuations in measurements
You may encounter random errors due to variation of temperature in the liquid.
This type of error can be reduced by repeating readings (and stirring!)
Example
Absolute Uncertainty
Every measurement has an associated uncertainty, which shows the range of possible results for that measurement.
Example
Here, 1 cm is the absolute uncertainty in the length of the stick.
length of the stick, d =
Analogue Instruments
The absolute uncertainty is half the smallest division on the measuring instrument (reading error).
Quantifying Uncertainties (Length)
Measuring length means measuring the left AND right position of the object. Thus both uncertainties in length are added.
In this example, the uncertainty in position is 0.5 cm, thus:
Digital Instruments
The absolute uncertainty is the smallest division that the instrument can read.
Repeated Readings
Measurements are often repeated to reduce random error and increase reliability.
Now, the uncertainty in the
velocity is half the range of
the readings.
Significant Figures
Uncertainties are limited to 1 or 2 significant figures. The uncertainty cannot be more precise (more significant figures) than the best estimate of the measured value.
Average Value:
1.50 ms-1 (3sf)
Calculated Uncertainty:
0.345 ms-1 (3sf)
Quoted Uncertainty:
0.35 ms-1 (2sf)
! When taking repeated readings, if the uncertainty is 0, it is good practice to add the reading error in the measuring equipment, In this case, it is 0.01 ms-1.
Fractional + Percentage Uncertainty
It is easier to compare the scale of uncertainty in a measurement by considering the fractional or percentage uncertainty:
fractional uncertainty
percentage uncertainty
The fractional / percentage uncertainty gives an idea of the random error (and reliability) of the result.
Propagating Uncertainties
When calculating the average speed of the cart, the following data is used:
d = 0.83 ± 0.02m
t = 1.25 ± 0.07s
The uncertainties must be combined to find a value for average speed (d/t), including its uncertainty.
Multiplication and Division
Add the fractional uncertainties of all quantities:
d = 0.83 ± 0.02 m
t = 1.25 ± 0.07 s
Addition and Subtraction
Add the absolute uncertainties of all quantities:
Example
Circumference:
Exponents
Multiply the fractional uncertainty with the exponent
Example
Area:
25 ± 0.1 cm
25 ± 0.1 cm
Uncertainty (Error) Bars
The error bars produce, in effect, an error box around the point.
The value could exist anywhere within the error box.
Uncertainty (Error) bars give a visual representation of the uncertainty on the point of a graph.
Lines of Best Fit
The maximum line of best fit is the line that passes through all error boxes at the steepest possible angle.
The minimum line of best fit is the line that passes through all error boxes at the least steep possible angle.
The line of best fit through points on a graph, should pass through all the error “boxes”.
Uncertainty in Gradient
The uncertainty in the gradient of a graph can be found by considering half the difference between the gradients of the maximum and minimum lines of best fit.
Best Gradient:
Uncertainty in Gradient:
Gradient:
Uncertainty in Intercept
The uncertainty in the y-intercept can be found by considering half the difference between the intercepts of the maximum and minimum lines of best fit
Best Intercept:
Uncertainty in Intercept:
Intercept:
Percentage Difference
The percentage difference gives an idea of the accuracy of a result. It tells us how close it is to the “true value”.
Measured distance: 0.82m
True distance = 0.75m
This result is not accurate
TASK 1. Data Analysis Lab - Determining the acceleration of free-fall