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Measurement Errors

All measurments will have errors. Either random errors or systematic errors. These errors have to be represented well in writing down the value of the quan- tity. We must also be aware of how errors propagate through the system.

The effect of taking more measurements

Most errors have a normal distribution, meaning it follows the probability density curve of Gauss

\begin{equation} f_{n}\left ( x \right )=\frac{1}{\sigma \sqrt{2\pi}}e^{-\frac{1}{2}\left ( \frac{x-\mu }{\sigma } \right )^{2}} \label{eq:NormalDistribution} \end{equation}

with the standard deviation $\sigma$ and the average $\mu$. The Gauss curve was already visible in figure ## of the Basic Measurement Theory chapter. By taking sufficient measurements, for example $N$, we can determine the shape in the Gauss curve. The location of this peak corresponds to the average $\mu$ and the width of the curve to the standard deviation $\sigma$. For a reasonable number of measurements ($N>15$), $95\%$ of the measurements lies between $\mu-2\sigma$ and $\mu+2\sigma$. The standard deviation $\sigma$ decreases with the square root of $N$, and so the precision increases with the square root of $N$. We can now see that with random errors, the precision can be increased by taking more measurements. For systematic errors, this averaging does not help, we still have the same offset $\mu$.

For a systematic error of zero ($\mu=0$), we can say that the random error is equal to $\pm 2\sigma$. When endlessly repeated measuring, the real value $x_{0}$ is equal to the average $\mu$. When measured $N$ times, the formula for the real value $x_{0}$ with a probability of $95\%$ is \begin{equation} x_{0} = \mu \pm \frac{2 \sigma}{\sqrt{N}}. \label{eq:NinetyFiveConfidence} \end{equation} The systematic error can be approximated by $\left | x_{0}-\mu \right |$ for sufficient high $N$. However, because we do not know the real value $x_{0}$, we have to use the independent reference (calibration) measurement that has a ten times higher accuracy.

Significant digits

The accuracy of a measured value is represented in the number of significant digits (‘meaningfull digits’). So from the number of digits we can recognise the accuracy of the number. The number of significant digits is the total number of digits without noticing the comma, where a zero on the left side does not count.

For example $6.34$ has three significant digits. This means that the real value lies between $6.335$ and $6.345$ and $0.2$ has one significant digit. Note that $0.02$ also has only one significant digit because the leading zeroes are not significant!

  • The value of $3000 m$ lies between $2999.5$ and $3000.5 m$.
  • The value of $3 km$ lies between $2.5$ and $3.5 km$.
  • When a value is measured with a certain instrument, the accuracy can be denoted explicitly, e.g. a force can be measured as $23.4N \pm 0.3 N$.

Once the standard deviation $\sigma$ of a measurement is known, we can use that for the representation of the number

  • Take the highest power of ten smaller than $\sigma/2$:
    • For example, when $\sigma=0.03 \rightarrow \sigma/2=0.015 \rightarrow accuracy= 0.01$
    • For example, when $\sigma=0.01 \rightarrow \sigma/2=0.005 \rightarrow accuracy= 0.001$
  • Round to a multiple of this:
    • For example, when when $\sigma=0.03$ and $y_{m}=8.314$, then $accuracy = 0.01$ and $y_{m}$ must be written as $y_{m}=8.31$
  • Last digit 5 round to even number (avoid bias)
    • For example: $\sigma = 0.03$ and $y_{m} = 8.315$, then $accuracy = 0.01$ and $y_{m} = 8.32$
    • For example: $\sigma = 0.03$ and $y_{m} = 8.345$, then $accuracy = 0.01$ and $y_{m} = 8.34$
  • When more than 1 decimal goes away than round in one step.
    • For example: $\sigma = 0.3$ and $y_{m} = 8.345$, then $accuracy = 0.1$ and $y_{m} = 8.3$

In case of a calculation, do not round the intermediate results. Otherwise, you are summing up errors.

Error can be represented as relative errors (as a percentage). Take care of the exact meaning:

  • Absolute error: $d = 5.19 \pm 0.06 mm$ is equivalent to the
  • Relative error with respect to the measured value: $d = 5.19 mm \pm 1.2 \%$ but also
  • Relative with respect to a full scale (for example of $200 mm$): $d = 5.19 \pm 0.03 \%$

Error propagation

In the measurement chain (or in our model), the reading may be the result of a mathematical operation on two input variables. For example, the length of a bar may be the sum of the first part plus a second part. Or, as another example, the output of a sensor is the product of the quantity to be measured times the sensitivity of the sensor. The question is what happens to the error of the output if both values (length 1 and length 2, or sensitivity and quantity) have noice and uncertainty. There are some basic rules to determine the error propagation under mathematical operations for the 'worst-case' estimation:

  • If two quantities are added or subtracted, the individual absolute uncertainty is added in the result
  • If two quantities are multiplied or divided, the percentages of uncertainty are added to get the percentage of uncertainty in the result
  • When finding the square root of a quantity, we divide the percentage of uncertainty by two. For squaring, the percentage uncertainty is multiplied by two.

Note that when dealing with error propagation one has to handle random errors and systematic errors strictly separate. In case of a systematic error one has to take the sign into account with a difference or quotient of quantities. And, also with systematic errors, one has to subtract the errors (absolute respectively relative) with a sum or product of quantities.

In case of a calculation, for example on a calculator, we normally take a simpeler approach:

  • With a product or quotiënt the number of significant digits of the result is equal to the smallest number of significant digits of the original numbers.
    • For example: $R = U/I = 21.3/0.2061= 103.3478893740902 \Omega \rightarrow R = 103 \Omega$
  • With a addition or subtraction the number of digits after the comma is equal to the smallest number of digits after the comma of the original numbers.
    • For example: $I = 2.5 + 0.357 = 2.9 A$


Add or subtract Multiply or divide
Error Absolute errors add up Relative errors add up
Number of digits Lowest number of decimals Lowest number of digits
Tab. 1: Propagation of errors in a worst-case approach

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theory/sensor_technology/st2_measurement_errors.txt · Last modified: 2017/10/10 18:37 by glangereis