StudySmarter - The all-in-one study app.

4.8 • +11k Ratings

More than 3 Million Downloads

Free

StudySmarter AI is coming soon!

- :00Days
- :00Hours
- :00Mins
- 00Seconds

A new era for learning is coming soonSign up for free

Suggested languages for you:

Americas

Europe

To estimate the error in a measurement, we need to know the expected or standard value and compare how far our measured values deviate from the expected value. The absolute error, relative error, and percentage error are different ways to estimate the errors in our measurements.Error estimation can also use the mean value of all the measurements if there is no…

Content verified by subject matter experts

Free StudySmarter App with over 20 million students

Explore our app and discover over 50 million learning materials for free.

Estimation of Errors

- Astrophysics
- Absolute Magnitude
- Astronomical Objects
- Astronomical Telescopes
- Black Body Radiation
- Classification by Luminosity
- Classification of Stars
- Cosmology
- Doppler Effect
- Exoplanet Detection
- Hertzsprung-Russell Diagrams
- Hubble's Law
- Large Diameter Telescopes
- Quasars
- Radio Telescopes
- Reflecting Telescopes
- Stellar Spectral Classes
- Telescopes
- Atoms and Radioactivity
- Fission and Fusion
- Medical Tracers
- Nuclear Reactors
- Radiotherapy
- Random Nature of Radioactive Decay
- Thickness Monitoring
- Circular Motion and Gravitation
- Applications of Circular Motion
- Centripetal and Centrifugal Force
- Circular Motion and Free-Body Diagrams
- Fundamental Forces
- Gravitational and Electric Forces
- Gravity on Different Planets
- Inertial and Gravitational Mass
- Vector Fields
- Conservation of Energy and Momentum
- Dynamics
- Application of Newton's Second Law
- Buoyancy
- Drag Force
- Dynamic Systems
- Free Body Diagrams
- Normal Force
- Springs Physics
- Superposition of Forces
- Tension
- Electric Charge Field and Potential
- Charge Distribution
- Charged Particle in Uniform Electric Field
- Conservation of Charge
- Electric Field Between Two Parallel Plates
- Electric Field Lines
- Electric Field of Multiple Point Charges
- Electric Force
- Electric Potential Due to Dipole
- Electric Potential due to a Point Charge
- Electrical Systems
- Equipotential Lines
- Electricity
- Ammeter
- Attraction and Repulsion
- Basics of Electricity
- Batteries
- Capacitors in Series and Parallel
- Circuit Schematic
- Circuit Symbols
- Circuits
- Current Density
- Current-Voltage Characteristics
- DC Circuit
- Electric Current
- Electric Generators
- Electric Motor
- Electrical Power
- Electricity Generation
- Emf and Internal Resistance
- Kirchhoff's Junction Rule
- Kirchhoff's Loop Rule
- National Grid Physics
- Ohm's Law
- Potential Difference
- Power Rating
- RC Circuit
- Resistance
- Resistance and Resistivity
- Resistivity
- Resistors in Series and Parallel
- Series and Parallel Circuits
- Simple Circuit
- Static Electricity
- Superconductivity
- Time Constant of RC Circuit
- Transformer
- Voltage Divider
- Voltmeter
- Electricity and Magnetism
- Benjamin Franklin's Kite Experiment
- Changing Magnetic Field
- Circuit Analysis
- Diamagnetic Levitation
- Electric Dipole
- Electric Field Energy
- Magnets
- Oersted's Experiment
- Voltage
- Electromagnetism
- Electrostatics
- Energy Physics
- Big Energy Issues
- Conservative and Non Conservative Forces
- Efficiency in Physics
- Elastic Potential Energy
- Electrical Energy
- Energy and the Environment
- Forms of Energy
- Geothermal Energy
- Gravitational Potential Energy
- Heat Engines
- Heat Transfer Efficiency
- Kinetic Energy
- Mechanical Power
- Potential Energy
- Potential Energy and Energy Conservation
- Pulling Force
- Renewable Energy Sources
- Wind Energy
- Work Energy Principle
- Engineering Physics
- Angular Momentum
- Angular Work and Power
- Engine Cycles
- First Law of Thermodynamics
- Moment of Inertia
- Non-Flow Processes
- PV Diagrams
- Reversed Heat Engines
- Rotational Kinetic Energy
- Second Law and Engines
- Thermodynamics and Engines
- Torque and Angular Acceleration
- Famous Physicists
- Fields in Physics
- Alternating Currents
- Capacitance
- Capacitor Charge
- Capacitor Discharge
- Coulomb's Law
- Electric Field Strength
- Electric Fields
- Electric Potential
- Electromagnetic Induction
- Energy Stored by a Capacitor
- Equipotential Surface
- Escape Velocity
- Gravitational Field Strength
- Gravitational Fields
- Gravitational Potential
- Magnetic Fields
- Magnetic Flux Density
- Magnetic Flux and Magnetic Flux Linkage
- Moving Charges in a Magnetic Field
- Newton’s Laws
- Operation of a Transformer
- Parallel Plate Capacitor
- Planetary Orbits
- Synchronous Orbits
- Fluids
- Absolute Pressure and Gauge Pressure
- Application of Bernoulli's Equation
- Archimedes' Principle
- Conservation of Energy in Fluids
- Fluid Flow
- Fluid Systems
- Force and Pressure
- Force
- Conservation of Momentum
- Contact Forces
- Elastic Forces
- Force and Motion
- Gravity
- Impact Forces
- Moment Physics
- Moments Levers and Gears
- Moments and Equilibrium
- Pressure
- Resultant Force
- Safety First
- Time Speed and Distance
- Velocity and Acceleration
- Work Done
- Fundamentals of Physics
- Further Mechanics and Thermal Physics
- Bottle Rocket
- Charles law
- Circular Motion
- Diesel Cycle
- Gas Laws
- Heat Transfer
- Heat Transfer Experiments
- Ideal Gas Model
- Ideal Gases
- Kinetic Theory of Gases
- Models of Gas Behaviour
- Newton's Law of Cooling
- Periodic Motion
- Rankine Cycle
- Resonance
- Simple Harmonic Motion
- Simple Harmonic Motion Energy
- Temperature
- Thermal Equilibrium
- Thermal Expansion
- Thermal Physics
- Volume
- Work in Thermodynamics
- Geometrical and Physical Optics
- Kinematics Physics
- Air Resistance
- Angular Kinematic Equations
- Average Velocity and Acceleration
- Displacement, Time and Average Velocity
- Frame of Reference
- Free Falling Object
- Kinematic Equations
- Motion in One Dimension
- Motion in Two Dimensions
- Rotational Motion
- Uniformly Accelerated Motion
- Linear Momentum
- Magnetism
- Ampere force
- Earth's Magnetic Field
- Fleming's Left Hand Rule
- Induced Potential
- Magnetic Forces and Fields
- Motor Effect
- Particles in Magnetic Fields
- Permanent and Induced Magnetism
- Magnetism and Electromagnetic Induction
- Eddy Current
- Faraday's Law
- Induced Currents
- Inductance
- LC Circuit
- Lenz's Law
- Magnetic Field of a Current-Carrying Wire
- Magnetic Flux
- Magnetic Materials
- Monopole vs Dipole
- RL Circuit
- Measurements
- Mechanics and Materials
- Acceleration Due to Gravity
- Bouncing Ball Example
- Bulk Properties of Solids
- Centre of Mass
- Collisions and Momentum Conservation
- Conservation of Energy
- Density
- Elastic Collisions
- Force Energy
- Friction
- Graphs of Motion
- Linear Motion
- Materials
- Materials Energy
- Moments
- Momentum
- Power and Efficiency
- Projectile Motion
- Scalar and Vector
- Terminal Velocity
- Vector Problems
- Work and Energy
- Young's Modulus
- Medical Physics
- Absorption of X-Rays
- CT Scanners
- Defects of Vision
- Defects of Vision and Their Correction
- Diagnostic X-Rays
- Effective Half Life
- Electrocardiography
- Fibre Optics and Endoscopy
- Gamma Camera
- Hearing Defects
- High Energy X-Rays
- Lenses
- Magnetic Resonance Imaging
- Noise Sensitivity
- Non Ionising Imaging
- Physics of Vision
- Physics of the Ear
- Physics of the Eye
- Radioactive Implants
- Radionuclide Imaging Techniques
- Radionuclide Imaging and Therapy
- Structure of the Ear
- Ultrasound Imaging
- X-Ray Image Processing
- X-Ray Imaging
- Modern Physics
- Bohr Model of the Atom
- Disintegration Energy
- Franck Hertz Experiment
- Mass Energy Equivalence
- Nuclear Reaction
- Nucleus Structure
- Quantization of Energy
- Spectral Lines
- The Discovery of the Atom
- Wave Function
- Nuclear Physics
- Alpha Beta and Gamma Radiation
- Binding Energy
- Half Life
- Induced Fission
- Mass and Energy
- Nuclear Instability
- Nuclear Radius
- Radioactive Decay
- Radioactivity
- Rutherford Scattering
- Safety of Nuclear Reactors
- Oscillations
- Energy Time Graph
- Energy in Simple Harmonic Motion
- Hooke's Law
- Kinetic Energy in Simple Harmonic Motion
- Mechanical Energy in Simple Harmonic Motion
- Pendulum
- Period of Pendulum
- Period, Frequency and Amplitude
- Phase Angle
- Physical Pendulum
- Restoring Force
- Simple Pendulum
- Spring-Block Oscillator
- Torsional Pendulum
- Velocity
- Particle Model of Matter
- Physical Quantities and Units
- Converting Units
- Physical Quantities
- SI Prefixes
- Standard Form Physics
- Units Physics
- Use of SI Units
- Physics of Motion
- Acceleration
- Angular Acceleration
- Angular Displacement
- Angular Velocity
- Centrifugal Force
- Centripetal Force
- Displacement
- Equilibrium
- Forces of Nature Physics
- Galileo's Leaning Tower of Pisa Experiment
- Inclined Plane
- Inertia
- Mass in Physics
- Speed Physics
- Static Equilibrium
- Radiation
- Antiparticles
- Antiquark
- Atomic Model
- Classification of Particles
- Collisions of Electrons with Atoms
- Conservation Laws
- Electromagnetic Radiation and Quantum Phenomena
- Isotopes
- Neutron Number
- Particles
- Photons
- Protons
- Quark Physics
- Specific Charge
- The Photoelectric Effect
- Wave-Particle Duality
- Rotational Dynamics
- Angular Impulse
- Angular Kinematics
- Angular Motion and Linear Motion
- Connecting Linear and Rotational Motion
- Orbital Trajectory
- Rotational Equilibrium
- Rotational Inertia
- Satellite Orbits
- Third Law of Kepler
- Scientific Method Physics
- Data Collection
- Data Representation
- Drawing Conclusions
- Equations in Physics
- Uncertainties and Evaluations
- Space Physics
- Thermodynamics
- Heat Radiation
- Thermal Conductivity
- Thermal Efficiency
- Thermodynamic Diagram
- Thermodynamic Force
- Thermodynamic and Kinetic Control
- Torque and Rotational Motion
- Centripetal Acceleration and Centripetal Force
- Conservation of Angular Momentum
- Force and Torque
- Muscle Torque
- Newton's Second Law in Angular Form
- Simple Machines
- Unbalanced Torque
- Translational Dynamics
- Centripetal Force and Velocity
- Critical Speed
- Free Fall and Terminal Velocity
- Gravitational Acceleration
- Kinetic Friction
- Object in Equilibrium
- Orbital Period
- Resistive Force
- Spring Force
- Static Friction
- Turning Points in Physics
- Cathode Rays
- Discovery of the Electron
- Einstein's Theory of Special Relativity
- Electromagnetic Waves
- Electron Microscopes
- Electron Specific Charge
- Length Contraction
- Michelson-Morley Experiment
- Millikan's Experiment
- Newton's and Huygens' Theories of Light
- Photoelectricity
- Relativistic Mass and Energy
- Special Relativity
- Thermionic Electron Emission
- Time Dilation
- Wave Particle Duality of Light
- Waves Physics
- Acoustics
- Applications of Ultrasound
- Applications of Waves
- Diffraction
- Diffraction Gratings
- Doppler Effect in Light
- Earthquake Shock Waves
- Echolocation
- Image Formation by Lenses
- Interference
- Light
- Longitudinal Wave
- Longitudinal and Transverse Waves
- Mirror
- Oscilloscope
- Phase Difference
- Polarisation
- Progressive Waves
- Properties of Waves
- Ray Diagrams
- Ray Tracing Mirrors
- Reflection
- Refraction
- Refraction at a Plane Surface
- Resonance in Sound Waves
- Seismic Waves
- Snell's law
- Spectral Colour
- Standing Waves
- Stationary Waves
- Total Internal Reflection in Optical Fibre
- Transverse Wave
- Ultrasound
- Wave Characteristics
- Wave Speed
- Waves in Communication
- X-rays
- Work Energy and Power
- Conservative Forces and Potential Energy
- Dissipative Force
- Energy Dissipation
- Energy in Pendulum
- Force and Potential Energy
- Force vs. Position Graph
- Orbiting Objects
- Potential Energy Graphs and Motion
- Spring Potential Energy
- Total Mechanical Energy
- Translational Kinetic Energy
- Work Energy Theorem
- Work and Kinetic Energy

Lerne mit deinen Freunden und bleibe auf dem richtigen Kurs mit deinen persönlichen Lernstatistiken

Jetzt kostenlos anmeldenNie wieder prokastinieren mit unseren Lernerinnerungen.

Jetzt kostenlos anmeldenTo estimate the error in a measurement, we need to know the expected or standard value and compare how far our measured values deviate from the expected value. The absolute error, relative error, and percentage error are different ways to estimate the errors in our measurements.

Error estimation can also use the mean value of all the measurements if there is no expected value or standard value.

To calculate the mean, we need to add all measured values of x and divide them by the number of values we took. The formula to calculate the mean is:

\[\text{mean} = \frac{x_1 + x_2 + x_3 + x_4 + ...+x_n}{n}\]

Let’s say we have five measurements, with the values 3.4, 3.3, 3.342, 3.56, and 3.28. If we add all these values and divide by the number of measurements (five), we get 3.3764.

As our measurements only have two decimal places, we can round this up to 3.38.

Here, we are going to distinguish between estimating the absolute error, the relative error, and the percentage error.

To estimate the absolute error, we need to calculate the difference between the measured value x0 and the expected value or standard x_{ref}:

\[\text{Absolute error} = |x_0 - x_{ref}|\]

Imagine you calculate the length of a piece of wood. You know it measures 2.0m with a very high precision of ± 0.00001m. The precision of its length is so high that it is taken as 2.0m. If your instrument reads 2.003m, your absolute error is | 2.003m-2.0m | or 0.003m.

To estimate the relative error, we need to calculate the difference between the measured value x0 and the standard value x_{ref} and divide it by the total magnitude of the standard value x_{ref}:

\[\text{Relative error} = \frac{|x_0 - x_{ref}|}{|x_{ref}|}\]

Using the figures from the previous example, the relative error in the measurements is | 2.003m-2.0m | / | 2.0m | or 0.0015. As you can see, the relative error is very small and has no units.

To estimate the percentage error, we need to calculate the relative error and multiply it by one hundred. The percentage error is expressed as ‘error value’%. This error tells us the deviation percentage caused by the error.

\[\text{Percentage error} = \frac{|x_0 - x_{ref}|}{|x_{ref}|} \cdot 100 \%\]

Using the figures from the previous example, the percentage error is 0.15%.

The line of best fit is used when plotting data where one variable depends on another one. By its nature, a variable changes value, and we can measure the changes by plotting them on a graph against another variable such as time. The relationship between two variables will often be linear. The line of best fit is the line that is closest to all the plotted values.

Some values might be far away from the line of best fit. These are called outliers. However, the line of best fit is not a useful method for all data, so we need to know how and when to use it.

To obtain the line of best fit, we need to plot the points as in the example below:

Fig. 1 - Data plotted from several measurements showing variation on the y-axis

Here, many of our points are dispersed. However, despite this data dispersion, they appear to follow a linear progression. The line that is closest to all those points is the line of best fit.

To be able to use the line of best fit, the data need to follow some patterns:

- The relationship between the measurements and the data must be linear.
- The dispersion of the values can be large, but the trend must be clear.
- The line must pass close to all values.

Sometimes in a plot, there are values outside the normal range. These are called outliers. If the outliers are fewer in number than the data points following the line, the outliers can be ignored. However, outliers are often linked to errors in the measurements. In the image below, the red point is an outlier.

Fig. 2 - Data plotted from several measurements showing variation on the y-axis in green and an outlier in pink

To draw the line of best fit, we need to draw a line passing through the points of our measurements. If the line intersects with the y-axis before the x-axis, the value of y will be our minimum value when we measure.

The inclination or slope of the line is the direct relationship between x and y, and the larger the slope, the more vertical it will be. A large slope means that the data changes very fast as x increases. A gentle slope indicates a very slow change of the data.

Figure 3 - The line of best fit is shown in pink, with the slope being shown in light green

In a plot or a graph with error bars, there can be many lines passing between the bars. We can calculate the uncertainty of the data using the error bars and the lines passing between them. See the following example of three lines passing between values with error bars:

Fig. 4 - Plot showing uncertainty bars and three lines passing between them. The blue and purple lines begin at the extreme values of the uncertainty bars

To calculate the uncertainty in a plot, we need to know the uncertainty values in the plot.

- Calculate two lines of best fit.
- The first line (the green one in the image above) goes from the highest value of the first error bar to the lowest value of the last error bar.
- The second line (red) goes from the lowest value of the first error bar to the highest value of the last error bar.
- Calculate the slope
of the lines using the formula below.*m*

\[m = \frac{y_2 - y_1}{x_2-x_1}\]

- For the first line, y2 is the value of the point minus its uncertainty, while y1 is the value of the point plus its uncertainty. The values x2 and x1 are the values on the x-axis.
- For the second line, y2 is the value of the point plus its uncertainty, while y1 is the value of the point minus its uncertainty. The values x2 and x1 are the values on the x-axis.
- You add both results and divide them by two:
\[\text{Uncertainty} = \frac{m_{red}-m_{green}}{2}\]

Let’s look at an example of this, using temperature vs time data.

Calculate the uncertainty of the data in the plot below.

The plot is used to approximate the uncertainty and calculate it from the plot.

Time (s) | 20 | 40 | 60 | 80 |

Temperature in Celsius | 84.5 ± 1 | 87 ± 0.9 | 90.1 ± 0.7 | 94.9 ± 1 |

To calculate the uncertainty, you need to draw the line with the highest slope (in red) and the line with the lowest slope (in green).

In order to do this, you need to consider the steeper and the less steep slopes of a line that passes between the points, taking into account the error bars. This method will give you just an approximate result depending on the lines you choose.

You calculate the slope of the red line as below, taking the points from t=80 and t=60.

\(\frac{(94.9+1)^\circ C - (90.1 + 0.7)^\circ C}{(80-60)} = 0.255 ^\circ C\)

You now calculate the slope of the green line, taking the points from t=80 and t=20.

\(\frac{(94.9- 1)^\circ C - (84.5 + 1)^\circ C}{(80-20)} = 0.14 ^\circ C\)

Now you subtract the slope of the green one (m2) from the slope of the red one (m1) and divide by 2.

\(\text{Uncertainty} = \frac{0.255^\circ C - 0.14 ^\circ C}{2} = 0.0575 ^\circ C\)

As our temperature measurements take only two significant digits after the decimal point, we round the result to 0.06 Celsius.

- You can estimate the errors of a measured value by comparing it to a standard value or reference value.
- The error can be estimated as an absolute error, a percentage error, or a relative error.
- The absolute error measures the total difference between the value you expect from a measurement (X
_{0}) and the obtained value (X_{ref}), equal to the absolute value difference of both Abs = | X_{0}-X_{ref}|. - The relative and percentage errors measure the fraction of the difference between the expected value and the measured value. In this case, the error is equal to the absolute error divided by the expected value \(rel = \frac{Abs}{X_0}\) for the relative error, and divided by the expected value and expressed as a percentage for the \(\text{percentage error per} = \Big(\frac{Abs}{X_0} \Big) \cdot 100\). You must add the percentage symbol for percentage errors.
- You can approximate the relationship between your measured values using a linear function. This approximation can be made simply by drawing a line, which must be the line that passes closest to all values (the line of best fit).

More about Estimation of Errors

How would you like to learn this content?

Creating flashcards

Studying with content from your peer

Taking a short quiz

94% of StudySmarter users achieve better grades.

Sign up for free!94% of StudySmarter users achieve better grades.

Sign up for free!How would you like to learn this content?

Creating flashcards

Studying with content from your peer

Taking a short quiz

Free physics cheat sheet!

Everything you need to know on . A perfect summary so you can easily remember everything.

Be perfectly prepared on time with an individual plan.

Test your knowledge with gamified quizzes.

Create and find flashcards in record time.

Create beautiful notes faster than ever before.

Have all your study materials in one place.

Upload unlimited documents and save them online.

Identify your study strength and weaknesses.

Set individual study goals and earn points reaching them.

Stop procrastinating with our study reminders.

Earn points, unlock badges and level up while studying.

Create flashcards in notes completely automatically.

Create the most beautiful study materials using our templates.

Sign up to highlight and take notes. It’s 100% free.