⚖️     Measurements

Absolute uncertainty vs Error 

Percentage error and Percentage uncertainty 

Accuracy as binary concept - Need for measurement , uncertainty of measurement, and accepted value 

Resolution of a  measuring device is the smallest step or increment or the distance between two divisions.

A physical quantity is specified  by the measured value and  absolute uncertainty

Absolute uncertainty of a measurement:

• Single measurement

  • • For analog meter the absolute uncertainty = half the resolution

    • For a digital meter the absolute uncertainty = resolution

    • Improvement of poor resolution by measuring  N  items at once   - ->  this will improve the absolute uncertainty

• Many measurements

  • •  absolute uncertainty = Half the spread    or  Maximum distance from the  mean

The quality of the measurement is specified by percentage uncertainty:

• It can be improved  if physical  quantity is way bigger than the resolution of the meter.  At least 10 times. 

• Choose the smallest possible range consistent with the measured value.

Precision is relevant only for many measurements:

• It is about the  spread

• Half the spread is the absolute uncertainty

• Increasing the absolute  uncertainty corresponds to decreasing    the precision

Accuracy:

• Accuracy is the deviation of the mean from the true value

• Single measurement    [ Measured  - True] OVER  [ True]

• Many measurements   [ Mean - True] OVER  [ True]

• Accuracy can be view as the number of correct significant figures

Sources of random error:

• Random error will change the mean

• Noise

• Real things are never identical.         Truly identical are only the elementary particles    

• Manufacturing process  variation     

• Fluctuation of temperature      

Reducing random error:    

• If there is no systematic error then averaging will  improve  the mean - consequently the accuracy        Averaging improves the accuracy not the precision

• Improvement by discarding   anomalous readings  

• Reducing random error by Larger or Longer or Bigger quantities

• If measurement is accurate but not precise then there is random error

Types of systematic errors:

• Parallax error     

• Zero error      

• Miscalibration

Properties of systematic errors:

• Systematic errors are directional

• Systematic error will shift the mean 

• If measurement is precise but not accurate then there is systematic error

Dealing with systematic error: 

•  Resetting           

• Use gradient

Time:        Stopwatch         Light-gate        Data logger for fast measurements

    Systematic error:   Reaction time      

Distance:  

• Ruler - 0.5mm        Vernier calliper - 0.1 mm          Micrometre -0.01 mm

• Systematic error:  Parallax error

Volume of liquid :  

• Graduated beaker

Mass:           

• • Digital scale         Balance      

  • Random error:  Variation of earth g - field 

  • Systematic error:  Buoyancy force    

Temperature: 

• Thermometer

• Systematic error: Time lag of thermometer      Leakage of heat          Gradient of the temperature

• Temperature can affect other physical quantities

Radioactivity:

• GM counter

• Systematic error: Dead time

• Random error:    Background radiation                

• Different types of radiation have different attenuation coefficients

If the thing measured is not a constant, spatial or time variation then performing many measurements leads to better estimation of the range and consequently to the absolute uncertainty.

Range of  a meter is about the minimum  and maximum value.

Sensitivity or the minimum detectable signal

Error propagation

Parallel and perpendicular lines: set triangle

Vertical line: plumb line / weight on vertical string