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Measurement Uncertainty
Using a Gaussian Probability Density Function

- Terms
- Uncertainty
- PDFs
  - Gaussian
  - Uniform
  - Triangular
- Procedure
- Combined
- Example
- References

It is a good practice to make multiple measurements whenever possible to reduced the risk of making a mistake. If only one measurement is made, a mistake could go completely unnoticed. If two measurements are made and they are significantly different, this indicates a problem with one of the measurements and additional measurements are required to determine which is the better value.

However, there are other good reasons for repeating measurements. Sometimes repeat readings give different results due to unavoidable variations in the measurand or in the measurement equipment being used. When repeated measurements give different results, the arithmetic mean of the readings provides the best estimate of the true value of the measurand. Repeated measurements also allow the measurement uncertainty to be evaluated statistically. Knowing the measurement uncertainty allows a judgment to be made about the spread of the values and the quality of the measurements.

Number of Readings

In general, the more repeat measurements, the better the statistical analysis. More measurements will usually result in a better estimate of the true value, and a smaller measurement uncertainty. As a rule of thumb between 4 and 10 readings is usually sufficient. Ten is a popular choice because it makes the math a bit easier. Using 20 would give a slightly better estimate than 10 and using 50 would be only slightly better than 20.

Determining the Mean Value

The first step in reporting the results of data based on repeated measurements is to determine the mean value of the data set.  The mean value is simply the arithmetic average of the set of values.  The equation for calculating the mean value can be expressed as:

and in common statistics notation as: 

This equation indicates that arithmetic mean will be arrived at by summing all the values of Xi from the first one (i =1) to the last one (i = n) and this sum by the number of values (n). The mean value will give a result with less error than any of the individual measurements since some of the random variations will cancel out in the summation.  As n increases and approaches an infinitely large value, the random error will continue to reduce and the value of X will move closer to the true value of the measurand

Determining the Measurement Uncertainty
Once the mean is determined, the next step is to determine the measurement uncertainty using a suitable probability density function (pdf) to model the available knowledge about the measurand. The equation used to make this calculation is shown at the bottom of the page. However, to make the explanation of the process of determining the standard uncertainty more understandable, arriving at this equation will be presented as a series of steps, which include:

  1. Determine the type of pdf to use based on the distribution of readings.
  2. Calculate the variance.
  3. Take the square root of the variance to get the standard deviation.
  4. Divide the standard deviation by the square root of the number of readings to get the standard uncertainty of the measurement.

The first step is to select the pdf that best models the data sample. When making a physical measurement, if the sample set of readings shows a dispersion of readings randomly distributed above and below the mean, it is appropriate to use a Gaussian (normal) pdf. If the data is skewed to one side of the mean or is multi-modal (i.e., with more than one peak), a different pdf should be used. The method described here assumes that you have an unbiased sample that is subject to random deviations, i.e. Gaussian. 

The next step is to characterize the scatter in data. One way to do this is to calculate the average width of the pdf by taking each reading, subtracting it from the mean, adding all the deviations up, and then dividing by the number of readings. However, if there is an equal distribution of readings above and below the mean, then the negative deviations would cancel out the positive deviations and the outcome of this calculation would be zero. Therefore, the square of the deviations is used to calculate the variance and the formula for this calculation is shown below. Notice that the summation is divided by n-1 rather than just n. This is done to remove bias that results from working with a sample rather than a full population of readings. The difference between a sample and a full population is shown in the image below.

The problem with variance is that it produces a value in units that are squared since the deviations were squared. Clearly it is not desirable to have a measurand in some unit and the uncertainty in square units. This is why the symbol for variance is shown as s2. To eliminate this issue, the square root of the variance is take to produce the standard deviation. The equation takes the form shown below.

Standard deviation refers to the amount that an individual reading is expected to vary from the mean. In other words, any single value has an uncertainty equal to the standard deviation. Therefore, if another reading is taken of the same measurand, there is a 68% chance the measurement will fall within one standard deviation of the mean and a 95% chance that it will fall within two standard deviations of the mean. However, when the arithmetic mean of the readings is being used to estimate of the true value of the measurand, it is the uncertainty of this mean that is of interest.

When the mean of the readings in the sample set is used, then the mean value has a much smaller uncertainty than the standard deviation of the sample set. The standard uncertainty of the measurement is the standard deviation of the set of readings (the sample) divided by the square root of the number of readings (see equation below). Standard uncertainty of the mean indicates how much a value averaged from several readings is expected to vary from the true mean. The same probability ranges that apply to the standard deviation of the sample set, also apply to the standard deviation of the mean. The difference between the standard deviation of the readings and the standard deviation of the mean is shown in the image.

(Equation for calculating the standard uncertainty of a measurement using the mean value of a sample set of readings that follow a Gaussian distribution.)
Reading # Value (mm)
1 15.671
2 15.674
3 15.668
4 15.674
5 15.673

Example Calculation
Suppose that ten repeat measurement were made of the same dimension of a rectangular bar using a micrometer and that the readings in the table were produced. What is the the best approximation of the true value and what is the uncertainty in the measurement?

The first step is to calculate the mean and determine if the data is scattered relatively evenly above and below the mean. If it is, then a Gaussian pdf will probably represent the data. The mean is calculated as follows:

It appears that the data is about evenly scattered above and below the mean so a Gaussian pdf should work well. From the table on the previous page (see portion of table copied below), it is given that the standard uncertainty (u) for a Gaussian probability density function, is represented by the following equation where (s) is the standard deviation of the sample readings and n in the number of readings.

which should be rounded to 0.0011mm

After doing the math, it is determined that the standard uncertainty for this measurement is ± 0.00114mm, which was rounded to ± 0.0011mm. The uncertainty was rounded to two significant digits, which is the recommended number and this give the value four decimal places, which is one more than the original readings, which is acceptable for the uncertainty portion of the measurement. The results of the measurement would then be reported as 15.672 ± 0.0011mm with a confidence level of 68% using a Gaussian pdf. The confidence level of the measurand falling within the uncertainty range can be increased to 95% by multiplying 0.00114 by 2 to arrive at 0.0023. Therefore, it would also be correct to express the measurement as 15.672 ± 0.0024mm with 95% confidence using a Gaussian pdf.

Note on Significant Digits and Precision:
The uncertainty should generally be quoted giving two significant figures. The best approximation of the measurand should then be rounded to the same decimal place as the second digit of the uncertainty. However, if using two significant digits causes the uncertain to exceed the precision of the best approximation value by more than one decimal place, then the uncertainty should be rounded up. It should also be noted that when rounding, the uncertainty is always rounded up (never down) to produce a conservatively larger interval.

Evaluation Type
PDF Type
Usually Used When the Measurement is:
Standard Uncertainty (u)
Confidence that the Measurand Lies Within the Uncertainty Range:

Type B

Single value

Gaussian for bell-shaped pdf
A set of repeated and scattered readings
(Std. deviation of the mean)
1u = 68%
2u = 95%
3u = 99%

Last updated on: November 24, 2010