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defect in the measuring tool : Examples of defects may be the magnet inside is partially demagnetized because it is outdated , The pointer has a zero error when there is no electric current .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

**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 )**

** It is the difference between the real ( actual ) value ( X _{0 }) and the measured value ( X ) .
**

**Δ X** = | **X**_{0 } – **X** |

_{0 }

**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 )**

** It is the ratio between the absolute error ( Δ X ) to the real value ( X _{0 }) .**

**r = Δ X / X**_{0}

_{0}

**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 .**

**Note : 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 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 = V _{1 }+ V_{2} , 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} = V_{2 }– V_{1} .**

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

**Δ X = Δ X**_{1 }+ Δ X_{2} = | X_{01 }– X_{1} | + | X_{02 }– X_{2} |

_{1 }+ Δ X

_{2}= | X

_{01 }– X

_{1}| + | X

_{02 }– X

_{2}|

**The relative error ( r ) : **

**r =Δ X / X**_{0}

_{0}

**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 = r**_{1}+ r_{2 }= Δ X_{1 }/ X_{01 }+ Δ X_{2 }/ X_{02}

_{1}+ r

_{2 }= Δ X

_{1 }/ X

_{01 }+ Δ X

_{2 }/ X

_{02}

**The absolute error ( Δ X )**** : **

**Δ X = r + X**_{0}

_{0}

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