How Things Work: Standard Deviation as a Tool for Measuring Precision and Accuracy
Professional Surveyor Magazine -
April 2007Geomatics Industry Association of America (GIAA)
Whether surveying with total stations, levels, or GPS, doing a plane survey or a geodetic one, doing a control survey or making a contour map, surveyors of all types use the statistical concept of standard deviation to evaluate the quality of their surveying measurements. This column however is not intended to be a primer on statistics in general or standard deviation in particular. It is actually a reminder to surveyors that standard deviation is a standard tool used by manufacturers of hardware and software as well. As we mentioned in the trigonometric leveling column, surveying has become "black box," and while the benefits to surveyors are many, we need to remember that we are responsible for the results these tools generate. To do so requires a general understanding of how some of the intermediate and final results computed (and sometimes displayed) by your measuring technology are determined.
When measurements are repeated, most surveyors know that this is done to improve the reliability of measurements. It is worth saying that independent repeated measurements do this best. A statistical tool like standard deviation is a simple method for evaluating the quality of the measurements. Initially, this is only a measure of precision. If the biases (systematic errors) in the measurements are understood or compensated for, then the standard deviation could be a reflection of the accuracy also.
EDM Example
With phase resolving EDM, it has been common for them to make several hundred or even several thousand measurements and display the average as the result every time the "Measure" button is pressed. While different manufacturers use different systems for evaluating the data, one method is to break up the sequence of hundreds or thousands of discrete measurements into blocks. The number of measurements in a block may be 100. The standard deviation of the block will be computed and stored in memory. Then the next block will be measured and its standard deviation computed and stored, and so on. As the blocks are measured the standard deviation (and mean) of the blocks are compared, one to another. Any significant differences are noted, and the software will either direct the instrument to make more blocks of measurements to improve the quality based on this evaluation or shut down the measuring process and indicate to the user that the measurement is not feasible.
Standard deviation is computed with the following equation:
σ = √(Σν²) ÷ (n-1)
where
- σ is the uncertainty in a single measurement with a confidence level of 68%,
- ν is the residual, the difference between each measurement in the sample and the mean of the sample,
- Σν² is the sum of the squared residuals, and
- n is the number of measurements in the sample.
If the standard deviation of each block is measured, the expected differences between the means of each block at the 68% level can be computed by:
σm = σ ÷ √n
where
- σm is the standard deviation of the mean,
- n is the number of measurements in the block, and
- σ is the standard deviation for each measurement in the block.
If the EDM calculates that the standard deviation ( ) is 4.2 mm, and there are 100 measurements in the block, then the expected difference in the standard deviation (i.e. 68% confidence) of the next block would be 0.42 mm m ). Instrument designers can use this information to decide whether any particular block needs to be re-measured.
Regardless of the direction events take the instrument on during a single measurement, the evaluation, decisions, and results displayed to the user are a result of various techniques used to manage the quality of the measurement. Standard deviation computations are an important part of these. When the decision is made to take more blocks of measurements, this is apparent to the user by the longer- than-normal time it takes to complete the full measurement. Such a situation may occur when the infrared light path from the instrument to the prism is broken due to passing traffic or people, or perhaps foliage is brought into the path intermittently be the wind. Another situation is when scintillation or "heat shimmer" occurs, preventing the infrared light from following the same path to the prism, causing more-than-normal variations in the distance measured. In all of these cases, if the interruptions or changes in light path appear excessive as detected through examining the standard deviations of the individual blocks, then the internal parameters set by the instrument designer (in some instruments, this may actually be controllable by the user) will cause the instrument to shut down the measuring process and display a message that it is not possible to complete the measurement. In actuality many hundreds of measurements may have been made, but the uncertainty in them exceeds the preset parameters, and as a matter of quality control the result is not displayed.
Time-of-flight EDMs can perform a similar analysis. As they are often used in reflectorless mode, the changes in the shape of the return pulse due to atmospheric disturbances add to the "drama" of variability in the measurements. When measurements are made to objects that move, whether that is a non-prism target on a handheld pole, or swaying trees, moving branches, swinging electric wires, or even the rustling grass on the ground, not only does the "range" from instrument to target constantly change significantly, the configuration of the surfaces reflecting the light can dramatically change the shape and intensity of the return pulses, making analysis of the measurement much more complex.
Coming In A Future Column:
Use of Standard Deviation in GPS Technology
About the Author
Geomatics Industry Association of America (GIA) is an organization of manufacturers, suppliers, and distribution partners, encompassing the present and emerging technologies which address customer needs in surveying, GPS, engineering, construction, GIS/LIS, and related fields by providing leadership in training and education to enhance the efficiency and effectiveness of their related business. www.giaamerica.org
» Back to our April 2007 Issue