Video Transcript
A value measured by a digital timer is 1.23 seconds. What is the uncertainty of the time value? Plus or minus 0.23 seconds, plus or minus 0.115 seconds, plus or minus 0.01 seconds, plus or minus 0.005 seconds.
We are given a measured value, 1.23 seconds, and we are asked to determine the associated uncertainty.
Now, the uncertainty in a measurement, generally speaking, does not depend on the value that was measured. That said, we can still use our measured value to figure out the uncertainty. The key observation is that the measured value has two places to the right of the decimal point. However, most intervals of time cannot be exactly expressed with only two places to the right of the decimal point. For example, 1.227 seconds and 1.231 seconds both require three places to the right of the decimal point to express exactly.
Because the timer can only display values to two decimal places, that is, to the nearest hundredth of a second, whenever the exact interval of time the timer is measuring requires more than three decimal places to express exactly, the value displayed on the timer will be the exact value rounded to the nearest hundredth.
We can see this visually by drawing a number line. On this number line, we have marked several values that can be expressed exactly with only two decimal places. These are values that the timer can display. We have also marked several values that cannot be expressed exactly with only two decimal places. These are values that the timer cannot display because the timer only displays values to two decimal places.
If the exact length of the interval of time that we are measuring is represented by this value here on the number line, then the value that the timer will display is 1.23 seconds, because this value is closer to 1.23 than it is to 1.22. Similarly, this value is closer to 1.23 than it is to 1.24, the next value that the timer could display. So this value will also be displayed by the timer as 1.23 seconds.
Any interval of time whose exact length is between 1.225 seconds and 1.235 seconds will have its length displayed on the timer as 1.23 seconds. This is because any interval of time whose length is between 1.225 seconds and 1.235 seconds is closer to 1.23 seconds than any other value that the timer can display.
But what this means is that when the timer does display the length of the interval of time as 1.23 seconds, the exact length of the interval could be as short as 1.225 seconds or as long as 1.235 seconds. In other words, the true length of the interval of time could be up to 0.005 seconds shorter or 0.005 seconds longer than the measured value.
This is the uncertainty that we are looking for. When we measure 1.23 seconds, the true value is 1.23 seconds plus or minus 0.005 seconds. So the uncertainty of our measurement is plus or minus 0.005 seconds. And again this is the maximum amount by which the true value could differ from, but still be consistent with, the measured value.
Finally, it’s worth mentioning that 0.005 is exactly half of 0.01, the difference between adjacent possibilities for the measured value. We call this distance the resolution. And it is generally true that the uncertainty in the value of a measurement is half of the measurement’s resolution.
