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Measurement and Uncertainty

Accuracy, Precision, and Uncertainty

The degree of accuracy and precision of a measuring system are related to the uncertainty in the measurements. Uncertainty is a quantitative measure of how much your measured values deviate from a standard or expected value. If your measurements are not very accurate or precise, then the uncertainty of your values will be very high. In more general terms, uncertainty can be thought of as a disclaimer for your measured values.

For example, someone might ask you to provide the mileage (a number of miles travelled or covered) on your car. In this case, you might say that it is 45,000 miles, plus or minus 500 miles. The plus or minus amount is the uncertainty in your value. That is, you are indicating that the actual mileage of your car might be as low as 44,500 miles or as high as 45,500 miles, or anywhere in between. All measurements contain some amount of uncertainty.

In our example of measuring the length of the paper, we might say that the length of the paper is 11 in., plus or minus 0.2 in. The uncertainty in a measurement, \(A,\) is often denoted as \(\delta A\) (“delta \(A\)”). So the measurement result would be recorded as \(A ± \delta A.\) In our paper example, the length of the paper could be expressed as \(11\;\mathrm{in}. ± 0.2.\)

Factors Contributing to Uncertainty

The factors contributing to uncertainty in a measurement include:

  1. Limitations of the measuring device,
  2. The skill of the person making the measurement,
  3. Irregularities in the object being measured,
  4. Any other factors that affect the outcome (highly dependent on the situation).

In our example, such factors contributing to the uncertainty could be the following:

  • the smallest division on the ruler is 0.1 in.,
  • the person using the ruler has bad eyesight, or
  • one side of the paper is slightly longer than the other.

At any rate, the uncertainty in a measurement must be based on a careful consideration of all the factors that might contribute and their possible effects.

Making Real-World Connections: Fevers or Chills?

Uncertainty is a critical piece of information, both in physics and in many other real-world applications. Imagine you are caring for a sick child. You suspect the child has a fever, so you check his or her temperature with a thermometer. What if the uncertainty of the thermometer were \(3.0°\mathrm{C}\)? If the child’s temperature reading was \(37.0°\mathrm{C}\) (which is normal body temperature), the “true” temperature could be anywhere from a hypothermic \(34.0°\mathrm{C}\) to a dangerously high \(40.0°\mathrm{C}\). A thermometer with an uncertainty of \(3.0°\mathrm{C}\) would be useless.

Percent Uncertainty

One method of expressing uncertainty is as a percent of the measured value. If a measurement \(A\) is expressed with uncertainty, \(\delta A\), the percent uncertainty (%unc) is defined to be

\(\%\;\mathrm{unc} = \cfrac{\delta A}{A} × 100\%.\)

Example on Calculating Percent Uncertainty: A Bag of Apples

A grocery store sells \(5\text{-lb}\) bags of apples. You purchase four bags over the course of a month and weigh the apples each time. You obtain the following measurements:

  • First Week: \(4.8\text{-lb}\)
  • Second Week: \(5.3\text{-lb}\)
  • Third Week: \(4.9\text{-lb}\)
  • Fourth Week: \(5.4\text{-lb}\)

You determine that the weight of the \(5\text{-lb}\) bag has an uncertainty of \(±0.4\;\text{lb}\). What is the percent uncertainty of the bag’s weight?

Strategy

First, observe that the expected value of the bag’s weight, \(A\), is 5 lb. The uncertainty in this value, \(\delta A,\) is 0.4 lb.  We can use the following equation to determine the percent uncertainty of the weight:

\(\%\;\mathrm{unc} = \cfrac{\delta A}{A} × 100\%.\)

Solution

Plug the known values into the equation:

\(\%\;\mathrm{unc} = \cfrac{0.4\;\mathrm{lb}}{5\;\mathrm{lb}} × 100\%.\)\( = 8\%.\)

Discussion

We can conclude that the weight of the apple bag is \(5\;\mathrm{lb} ± 8\%.\) Consider how this percent uncertainty would change if the bag of apples were half as heavy, but the uncertainty in the weight remained the same. Hint for future calculations: when calculating percent uncertainty, always remember that you must multiply the fraction by 100%. If you do not do this, you will have a decimal quantity, not a percent value.

Uncertainties in Calculations

There is an uncertainty in anything calculated from measured quantities. For example, the area of a floor calculated from measurements of its length and width has an uncertainty because the length and width have uncertainties. How big is the uncertainty in something you calculate by multiplication or division? If the measurements going into the calculation have small uncertainties (a few percent or less), then the method of adding percents can be used for multiplication or division.

This method says that the percent uncertainty in a quantity calculated by multiplication or division is the sum of the percent uncertainties in the items used to make the calculation. For example, if a floor has a length of 4.00 m and a width of 3.00 m, with uncertainties of 2% and 1%, respectively, then the area of the floor is 12.0 m2 and has an uncertainty of 3%. (Expressed as an area this is 0.36 m2, which we round to 0.4 m2 since the area of the floor is given to a tenth of a square meter.)

Example

A high school track coach has just purchased a new stopwatch. The stopwatch manual states that the stopwatch has an uncertainty of \(±0.05\;\text{s}.\) Runners on the track coach’s team regularly clock 100-m sprints of \(11.49\;\text{s}\) to \(15.01\;\text{s}.\) At the school’s last track meet, the first-place sprinter came in at \(12.04\;\text{s}.\) and the second-place sprinter came in at \(12.07\;\text{s}.\) Will the coach’s new stopwatch be helpful in timing the sprint team? Why or why not?

Solution

No, the uncertainty in the stopwatch is too great to effectively differentiate between the sprint times.

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