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Uncertainty in CO2 Data
Every measurement has some uncertainty associated with it, due to the precision of the instrument. Take a ruler for instance. The smallest division on a standard ruler is 1 mm. Let's say we measure a pencil and to the nearest millimeter we find its length to be 21 mm. Now it is pretty unlikely that the pencil is exactly 21 mm long. It is probably a little longer or shorter. If someone else were to measure the pencil, they may state its length to be 22 mm. Who is right?

Within the limits of the precision of the ruler, 1 mm, we both are correct. We should then state the length of the ruler as 21 1 mm, and the other guy should state the length as 22 1 mm. A scientist looking at our data now knows that the precision of our measurement is 1 mm. Some people call the 1 mm an error or probable error in our measurement. It is not an error, but an uncertainty. We took the best measurement we could with the equipment available. If we had made an error, we would have taken the measurement again!

To minimize the uncertainty in a quantity, scientists measure it several times. This also helps to eliminate true errors in a measurement, because the value in error will most likely be substantially different from the others. The value stated is the average value from all the measurements. For example, let's say we took five length measurements of an object:

Image that shows five length measurements of an object.  Please have someone assist you with this.

The average value would be:

Image that shows the average value equation.  Please have someone assist you with this.

The uncertainty is found by first taking the difference between each measurement and the average value. This is called the deviation and is stated as a positive number. Deviation:

Image of five different equations that find the deviation.  Please have someone assist you with this.

The mean deviation, dm, is the sum of the deviations divided by the number of measurements: Mean deviation:

Image that shows the mean deviation equation.  Please have someone assist you with this.

The probable uncertainty is then found by dividing the mean deviation by the square root of the number of measurements:

Image that shows the probable uncertainty equation.  Please have someone assist you with this.

The measurements of the length would then be reported as

Image that shows the measurements of the length.  Please have someone assist you with this.

Sometimes uncertainties are given in percents. The percent uncertainty is found by dividing the probable uncertainty by the average value and multiplying by 100.

Image that shows the percent uncertainty equation.  Please have someone assist you with this.

What if we add, subtract, multiply, or divide several numbers with uncertainties? Let's say we have three numbers:

Image that shows three sets of numbers.  Please have someone assist you with this.

For addition and subtraction, the numerical uncertainty is equal to the square root of the sum of the squares of the probable uncertainties. If we add our three numbers, we get 24.41 with an uncertainty of

Image that shows the numerical uncertainty equation.  Please have someone assist you with this.

So the result would be written as Image that shows the equation results.  Please have someone assist you with this.

For multiplication and division, the numerical percent uncertainty is equal to the square root of the sum of the squares of the percent uncertainties. If we multiply our three numbers, we get 510 with an percent uncertainty of

First, turn probable uncertainties into percent uncertainties:

Image that shows the numerical percent uncertainty equation.  Please have someone assist you with this.

Then, Image that shows a square root equation.  Please have someone assist you with this.

So the result would be written as Image that shows the equation results.  Please have someone assist you with this.

If many measurements are taken, the uncertainty is sometimes measured by the standard deviation. One standard deviation means that 68% of all measurements will be within this value. Two standard deviations mean 95% of all measurements will be within this value. Calculation of the standard deviation is beyond the scope of this note.

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