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phy131studiof15:lectures:chapter1 [2015/07/27 16:02] mdawber [Error and Uncertainty] |
phy131studiof15:lectures:chapter1 [2015/08/24 09:19] (current) mdawber [Scientific Notation] |
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Examples: | Examples: | ||

- | $0.000056$ m = $5.6 \times 10^{-5}$ m or $5.6 \times 10^{-2}$ mm. | + | $0.000056$m = $5.6 \times 10^{-5}$m or $5.6 \times 10^{-2}$mm. |

- | $795,000$ g = $7.95 \times 10^{5}$ g or $7.95 \times 10^{2}$ kg or $795$ kg. | + | $795,000$g = $7.95 \times 10^{5}$g or $7.95 \times 10^{2}$kg or $795$kg. |

In general it is not correct to give more significant figures for a number than the precision to which you know it. However you should not round off numbers too early in a calculation, as this can affect the accuracy of the final answer. | In general it is not correct to give more significant figures for a number than the precision to which you know it. However you should not round off numbers too early in a calculation, as this can affect the accuracy of the final answer. | ||

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While the words accuracy and precision sound like they refer to similar things their meanings in physics are actually slightly different. | While the words accuracy and precision sound like they refer to similar things their meanings in physics are actually slightly different. | ||

- | Precision refers to how well a quantity can be determined. This determination of this quantity be the result of multiple independent measurements, which presumably would improve the precision, but when we talk of precision we are **not** considering how the value of the quantity compares to a "known" or "established" value. | + | Precision refers to how well a quantity can be determined. This determination of this quantity may be the result of multiple independent measurements, which presumably would improve the precision, but when we talk of precision we are **not** considering how the value of the quantity compares to a "known" or "established" value. |

Accuracy, on the other hand, does make this comparison. The accuracy of a measurement refers to how well a measured value agrees with a a "known" or "established" value. | Accuracy, on the other hand, does make this comparison. The accuracy of a measurement refers to how well a measured value agrees with a a "known" or "established" value. | ||

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But here is where it gets confusing... When you estimate the uncertainty of your measurement, as you will do frequently in the lab component of this course, you should consider the possible sources of error that contribute to the uncertainty. This way if there are large sources of error in your experiment, you will have a large uncertainty which will not exclude the accurate value of the quantity you are trying to measure. | But here is where it gets confusing... When you estimate the uncertainty of your measurement, as you will do frequently in the lab component of this course, you should consider the possible sources of error that contribute to the uncertainty. This way if there are large sources of error in your experiment, you will have a large uncertainty which will not exclude the accurate value of the quantity you are trying to measure. | ||

- | ===== Stating errors ===== | + | ===== Stating uncertainty ===== |

Because we cannot know any experimentally measured value to absolute certainty we state such values in the format | Because we cannot know any experimentally measured value to absolute certainty we state such values in the format | ||

- | $\mathrm{value}\pm\mathrm{error in value}$ | + | $\mathrm{value}\pm\mathrm{uncertainty\,in\,value}$ |

- | This is equivalent to saying that the actual value of the quantity could be anywhere between the value minus the error and the value plus the error. | + | This is equivalent to saying that the actual value of the quantity could be anywhere between the value minus the uncertainty and the value plus the uncertainty. |

- | | + | |

+ | Much of our activity in the lab will be about the best way to realistically determine and estimate for the uncertainty in any value we obtain. | ||

+ | |||

+ | ===== 1.P.030 ===== | ||

+ | | ||

===== Example of systematic error ===== | ===== Example of systematic error ===== | ||

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{{http://ic.sunysb.edu/Class/phy133s/vidspics/bullsyefourframesd.png}} | {{http://ic.sunysb.edu/Class/phy133s/vidspics/bullsyefourframesd.png}} | ||

- | Think of the round object as an archery target. The archer shoots some number of arrows at it, and each dot shows where one landed. Now think of the "bull's eye" -- the larger black dot in the center -- as the "true" value of some quantity that's being measured, and think of each arrow-dot as a measurement of that quantity. The problem is that the one doing the measurements does not know the "true" value of the quantity; s/he's trying to determine it experimentally, and this means there must be uncertainty associated with the experimentally determined value. Note that each archery target -- we'll call them 1,2,3,4 from left to right -- shows a different distribution of arrow-hit/measurements. | + | Think of the round object as an archery target. The archer shoots some number of arrows at it, and each dot shows where one landed. Now think of the "bull's eye" -- the larger black dot in the center -- as the "true" value of some quantity that's being measured, and think of each arrow-dot as a measurement of that quantity. The problem is that the one doing the measurements does not know the "true" value of the quantity; they are trying to determine it experimentally, and this means there must be uncertainty associated with the experimentally determined value. Each archery target shows a different distribution of arrow-hit/measurements. |

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- | we can derive some useful rules for common operations you will carry out in the labs. You can find more details about these equations in the [[phy131studio:labs:error|error manual]] | + | we can derive some useful rules for common operations you will carry out in the labs. You can find more details about these equations in the [[phy131studiof15:lectures:error|error manual]] |

if $S=aX$ then $\Delta S=a\Delta X$ | if $S=aX$ then $\Delta S=a\Delta X$ | ||

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if $S=A^n$ then $\large \frac{\Delta_S}{S}=|n|\times \frac{\Delta_A}{A}$ | if $S=A^n$ then $\large \frac{\Delta_S}{S}=|n|\times \frac{\Delta_A}{A}$ | ||

+ | ===== Measurement Exercises ===== | ||

+ | |||

+ | Using the meter stick | ||

+ | |||

+ | 1. Determine the length of the perimeter of one of the carpet tiles on the floor. Estimate the uncertainty in your value by considering the uncertainty in your measurement of each side. | ||

+ | |||

+ | 2. Determine the area of one of the carpet tiles on the floor. Estimate the uncertainty in your value by considering the uncertainty in your measurement of each side. | ||

+ | |||

+ | 3. Determine the area of one of the round tables you are sitting at. Estimate the uncertainty by considering the uncertainty in your measurement of the diameter of the table. | ||

+ | Finally, let's collect the values from the different groups and average them and then estimate the uncertainty in our average. | ||

===== Why is error and uncertainty so important? ===== | ===== Why is error and uncertainty so important? ===== | ||