Error Analysis

The deviation between the actual measured value and the true value of the measurand is what is regarded as error. However, this definition has limitation in that the true value of the measurement remains unknown and cannot be exactly determined. The act of quantifying doubt about the measured value is uncertainty. Any measurement has a margin of doubt and this doubt should be quantified.

Basically, the measurements of any quantity say x is stated as,

(Measured value of x)=xbest ±δx (1)

Where xbestis the best estimate of the quantity concerned and δx is the uncertainty or error margin. From (1) the confidence of the measurement lies between the highest probable value measured xbest+δx and the lowest valuexbest−δx.

Absolute error

The magnitude of the physical error in a measured value. That is, the difference between the measured and the accepted value.

Eabsolute=|xmeasured−xaccepted| (2)

Relative error

The relative size of the error is defined as the ratio of absolute error to the accepted value.

Erelative=|xmeasured−xaccepted|

xaccepted(3)

Expressing relative error as a percentage gives the per cent of error.

There are several sources of random and systematic errors and uncertainties. These errors range from the incorrect measuring technique, the measuring instrument as well as the personnel who perform the experiment. Systematic errors are usually the difficult types of errors to identify. This is because they are the built in errors in the measuring instruments either though design or wrongly calibration. Bias from the person performing the experiment would also bring the errors. Repetition of the observation won’t bring out the spread of the error in measurement.

Random errors, however, leads to a spread/distribution of the results and therefore they are easy to identify and quantify. These arise due to fluctuations of the physical parameters with a particular statistical nature as well as from the person who is collecting the values. These uncertainties are reduced by collecting multiple measurements.

This is because the influence affects the results of the repeated measurement.

Standard deviation

The set of input quantitiesx1 , x2,… xn are categorised as quantities whose uncertainty is determined from current measurement and also as quantities whose uncertainties are brought into measurement from other sources such as reference data.

In a nonlinear function of the input values,x1, x2, … xn the arithmetic mean of individual observation x i ,k is computed as,

x=1n∑k=1

n

x i , k (4)

The experimental standard deviation, which is the positive square root of the variance, is calculated from the equation (5).

σ=s=√ 1n−1∑i=1

n

(x i−x )2 (5)

Thus for n observations with x as mean, then the variance is

s2= 1n−1∑(x−x)2 (6)

The average value x is derived from a finite number of samples n and thus this won’t be the exact mean if an infinite sample values were considered. Therefore, this mean value has an associated uncertainty which is known as experimental standard deviation of the mean. The experimental standard deviation of the mean is basically obtained from the standard deviation. As shown by (7)

s¿) =σ√n (7)

Where n is the number of measurements contributing to the mean.

Standard uncertainty is used to express the level of confidence associated with a particular distributed data values from a set of several measured values.

Accuracy is a measure of how close is the measured value is to the ‘true” or accepted value which differ with precision in that precision is a measure of how reproducible a measurement is.

Determining uncertainties and their associated magnitude is not relatively easy. There

are errors or uncertainties which are due to irregularities and natural fluctuation.

Various mathematical methods are used to estimate these uncertainties. However, in

some cases, uncertainty is estimated by assigning it a value which is half the finest value

of scale associated with the measuring instrument. Therefore, the tolerance margin of

error is within the range of plus or minus half the value of the precision of the

measuring instrument.

Propagation of errors

Measured values are used to calculate other values, such as area of a rectangle or the power of induction machine which depends with speed and torque. Each measured value has uncertainty associated with. Thus this should be put into consideration during calculation.

If a calculated variable z is a function of two variables x and y, where x and y are measured, then

z=z(x,y) (8)

With x and y having associated uncertainty of δx∧δy respectively, then the propagated uncertainty, δz in z is,

δz=√( ∂ z∂ x )2

(δx)2+( ∂ z∂ y )2

(δy)2 (9)

When considering uncertainties, deviation from the mean is of importance, in a set of measured valuables, deviation can be defined as

δ x i=x i+x (10)