Types of Measurement & Reasons of Measurement Error

Types of Measurement

Direct Measurement such as measuring liquid density using the hydrometer in which we take a direct reading without calculation or using any law , Measuring volume using the graduated cylinder .

There is one measuring tool that is used , There is one measurement process , No mathematical  relation is applied and there is one measurement error .

Indirect Measurement such as Determining the liquid density via measuring mass by a balance and volume by a graduated cylinder , Then , dividing mass by volume , Measuring volume by multiplying length , width and height .

More than one measuring tool are used , More than one measurement process , There is a mathematical relation is applied to find the quantity and there is more than one measurement error ( cumulative error ) .

Error in Measurements

Man has been interested throughout history to improve measurement techniques and develop its instruments because of its obvious impact on the scientific and technological process , But , no measurement is accurate 100 % , There must be an error even if it is small .

Reasons of measurement error

There are several probable reasons for measurement error , from which :

Choosing improper tool : For example , using the beam balance instead of the sensitive balance in measuring the mass of a golden ring .

A defect in the measuring tool : Examples of defects may be the magnet inside is partially demagnetized because it is outdated , The pointer has a zero error when there is no electric current .

Wrong procedure due to unexperienced persons : Ignorance of using graduated devices like the multimeters , Looking at the device pointer or the scale at an oblique line instead of being perpendicular to the scale .

Environmental conditions such as : Temperature , Humidity , Air currents , Example : When using the sensitive balance , the air currents may produce an error , Because of this the sensitive balance is kept inside a glass box .

Estimating error in direct measurement

To find the error in direct measurement , There are two types of error which are Absolute error ( Δ X ) and Relative error ( r ) , Absolute error ( Δ X ) is the difference between the real ( actual ) value ( X₀ ) and the measured value ( X ) .

Δ X = | X₀ – X |

The sign |  | indicates that the result is always positive even if the actual value is less than the measured value , It has a measuring unit which is the same measuring unit of the physical quantity , Relative error ( r ) is the ratio between the absolute error ( Δ X ) to the real value ( X₀ )  .

r = Δ X / X₀

The relative error is a better indication for the measurement accuracy than the absolute error , The measurement accuracy is considered higher as the relative error decreases , It has no measuring unit because it is the ratio between two quantities having the same measuring units .

Although the absolute error in measuring the classroom length is greater than that in measuring the pencil length , the relative error in measuring the classroom length is less than in measuring the pencil length .

This meant that the relative error is a better indication for measurement than the absolute error , As the relative error decreases , the measurement accuracy increases .

Estimating error in case of indirect measurement

The procedure of calculating error in case of indirect measurement depends on the mathematical operation applied :

Example : Measuring the volume of two amounts of a liquid , V = V₁ + V₂ , Finding the volume of a coin by subtracting the volume of water before dropping the coin into the measuring cylinder from that after dropping it , V_coin = V₂ – V₁  .

The absolute error = The absolute error in first measurement + The absolute error in second measurement 

Δ X = Δ X₁ + Δ X₂ = | X₀₁ – X₁ | + | X₀₂ – X₂ |

The relative error ( r ) :  r =Δ X / X₀

Example : Finding the area of a rectangle by measuring its length and its width then multiplying them , Finding the density of a liquid by measuring its mass and its volume then dividing them .

The relative error = The relative error in first measurement + The relative error in second measurement

r = r₁+ r₂ = Δ X₁ / X₀₁ + Δ X₂ / X₀₂

The absolute error ( Δ X ) :  Δ X = r + X₀