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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, if someone asked you to provide the mileage on your car, 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 $\mathrm{\delta A}$ (“delta $A$ ”), so the measurement result would be recorded as $A\pm \mathrm{\delta A}$ . In our paper example, the length of the paper could be expressed as $11\text{in.}\pm \text{0.}2\text{.}$
The factors contributing to uncertainty in a measurement include:
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.
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 $3\text{.}\mathrm{0\xba}\text{C}$ ? If the child’s temperature reading was $\text{37}\text{.}\mathrm{0\xba}\text{C}$ (which is normal body temperature), the “true” temperature could be anywhere from a hypothermic $\text{34}\text{.}\mathrm{0\xba}\text{C}$ to a dangerously high $\text{40}\text{.}\mathrm{0\xba}\text{C}$ . A thermometer with an uncertainty of $3\text{.}\mathrm{0\xba}\text{C}$ would be useless.
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