Measurement uncertainties

Editor's Notes

• #3 To test whether the recognized patterns are consistent, Physicists perform experiments, leading to new ways of understanding observable phenomena in nature.
• #4 Ask the class to express opinions on what the effect of the measurement tool might have on the true value of a measured physical quantity. What about the skill of the one measuring?
• #7  Is the measurement repeatable?
• #8 For instance, with the use of a tape measure, a length measurement may vary due to the fact that the tape measure is not stretched straight in the same manner in all trials.
• #9 So what is the height of a table? A volunteer uses a tape measure to estimate the height of the teacher’s table. Should this be reported in millimeters? Centimeters? Meters? Kilometers?
• #10 The choice of units can be settled by agreement. However, there are times when the unit chosen is considered most applicable when the choice allows easy access to a mental estimate. Thus, a pencil is measured in centimeters and roads are measured in kilometers. How high is mount Apo? How many Filipinos are there in the world? How many children are born every hour in the world?
• #11 This is how we report the accuracy of a measurement. The maximum and minimum provides upper and lower bounds to the true value.
• #12 The measurement can also be presented or expressed in terms of the maximum likely fractional or percent error. Thus, 52 s ± 10% means that the maximum time is not more than 52 s plus 10% of 52 s (which is 57 s, when we round off 5.2 s to 5 s). Here, the fractional error is (5 s)/52 s.
• #13 Discuss that the uncertainty can then be expressed by the number of meaningful digits included in the reported measurement. For instance, in measuring the area of a rectangle, one may proceed by measuring the length of its two sides and the area is calculated by the product of these measurements.
• #14 Note that since the meterstick gives you a precision down to a single millimeter, there is uncertainty in the measurement within a millimeter. The side that is a little above 5.2 cm or a little below 5.3 cm is then reported as 5.25 ± 0.05 cm. However, for this example only we will use 5.25 cm. Since the precision of the meterstick is only down to a millimeter, the uncertainty is assumed to be half a millimeter. The area cannot be reported with a precision lower than half a millimeter and is then rounded off to the nearest 100th.
• #15  Note that since the original number has 3 figures, the conversion to cubic inches should retain this number of figures
• #17 A measurement is limited by the tools used to derive the number to be reported in the correct units as illustrated in the example above (on determining the area of a rectangle). Now, consider a table with the following sides: What about the resulting measurement error in determining the area?
• #18 Note: The associated error in a measurement is not to be attributed to human error. Here, we use the term to refer to the associated uncertainty in obtaining a representative value for the measurement due to undetermined factors. A bias in a measurement can be associated to systematic errors that could be due to several factors consistently contributing a predictable direction for the overall error. We will deal with random uncertainties that do not contribute towards a predictable bias in a measurement. The above indicates that the best estimate of the true value for x is found between x – Δx and x + Δx (the same goes for y).
• #19 How does one report the resulting number when arithmetic operations are performed between measurements?
• #20 How does one report the resulting number when arithmetic operations are performed between measurements? The estimate for the compounded error is conservatively calculated. Hence, the resultant error is taken as the sum of the corresponding errors or fractional errors. Thus, repeated operation results in a corresponding increase in error.
• #21 The arithmetic average of the repeated measurements of a physical quantity is the best representative value of this quantity provided the errors involved is random. Systematic errors cannot be treated statistically. For measurements with associated random uncertainties, the reported value is: mean plus-or-minus standard deviation. Provided many measurements will exhibit a normal distribution, 50% of these measurements would fall within plus-or-minus 0.6745(sd) of the mean. Alternatively, 32% of the measurements would lie outside the mean plus-or-minus twice the standard deviation.
• #22 The standard error can be taken as the standard deviation of the means. Upon repeated measurement of the mean for different sets of random samples taken from a population, the standard error is estimated as: