Direct and indirect measurement

For direct Measurement

The Mean Value x ̅ : It is obtained by dividing the total sum of measurements (x_i) by the total number of measurements (n). Its expression is given as follows:

x = i = 1 n x i / n x csup{ %Ux2013 } = sum from{i=1} to{n} x_{i} / n

the Mean Absolute Uncertainty (∆x ̅): This refers to the average of the variances between the measured values and the mean value. Its expression is given as follows:

Δ x = i = 1 n | x i x | n %DELTA x csup{ %Ux2013 } = { sum from{i=1} to{n} lline {x}_{i} - x csup{ %Ux2013 } rline } over {n}

The Mean Squared Uncertainty (σ): It denotes the average of the squared variances between the measured values and the mean value. Its expression is given as follows

σ = i = 1 n ( x i x ) 2 n 1 %sigma = sqrt{ { sum from{i=1} to{n} ( x_{i} - x csup{ %Ux2013 } ) ^{2} } over { n - 1 } }

The Absolute Uncertainty (∆x): This represents the absolute magnitude of the largest difference between the measured values and the mean value. It s expression is given as follows:

Δ x = max | x i x | %DELTA x=max lline x_{i} - x csup{ %Ux2013 } rline

Or the maximum value that an error can take

Δ x = max | δ e | %DELTA x=max lline %delta e rline

For Indirect Measurement

In such situations, we determine a quantity's value through mathematical relationships with other variables. Indirect measurement is chosen when it's not feasible to directly measure the quantity or when more detailed analysis is needed to explore how specific variables affect it. We can evaluate the absolute or relative uncertainty of a physical quantity x=f (a ,b ,c,..) , which is expressed as a function of other variables, using the differential method for uncertainty estimation as follows:

Δ x = | δ f δ a | Δ a + | δ f δ b | Δ b + | δ f δ c | Δ c %DELTA x= lline { %delta f} over { %delta a } rline %DELTA a+ lline { %delta f } over { %delta b } rline %DELTA b+ lline { %delta f} over { %delta c} rline %DELTA c

The Relative Uncertainty: It signifies the proportion of the absolute Uncertainty in relation to the mean value. Its expression is given as follows :

Δ x x { %DELTA x} over x csup{ %Ux2013 }

Note

The measured value is written in the following form:

x = x Δ x x= x csup{ %Ux2013 } %Ux2213 %DELTA x