Невизначенність

Every time that we do an experiment we take measurements. When we measure something we do our best to make an exact observation. But, can we be certain that our measurements are exact? No. All measurements are inexact. We show that our measurements are inexact by giving our measurement uncertainty. The measurement uncertainty is the doubt we have about our measurement. To show our doubt we write our uncertainty as a range of values. We are confident that our exact measurement is within our uncertainty range.

###What Causes Uncertainty? All instruments have uncertainty in their measurements. It does not matter if you are using the ten-dollar scale in a middle school chemistry lab or the million-dollar instrument in a professional lab. Often, the instrument's owner's manual will tell you its uncertainty. However, if it does not tell you, a general rule of thumb is to use one unit of the smallest significant digit. For example, if you measure something to be 52.5 grams your uncertainty would be 0.1 grams.

The person taking the measurement is also a source of uncertainty. We can never be certain that we have measured something properly. For example, let's imagine that you are measuring the length a piece of string with a yardstick. A few possible sources of uncertainty would be...

You did not stretch the string all the way.
You overstretched the string
The string length was between the markings on the yardstick and you had to estimate the measurement.
You were so bored in class that you were not really paying attention. Do not mention this in your lab report.

Any of these scenarios would produce doubt about our measurement.

Writing Uncertainties

When we write our uncertainty we use the format: measurement $\pm$ uncertainty. For example, let's say we measure the mass of a substance to be 5.05 grams and our uncertainty is 0.04 grams. We would write our value as 5.05g $\pm0.04$ g. Our uncertainty tells us that we are confident that our measurement lies between 5.01g and 5.09g.

Uncertainties and Significant Figures

Uncertainties follow the rules of significant figures. It is always the uncertain digit that sets the number of significant figures. For example, you could not have a mass of 5.05g $\pm0.1$ g. In this case, we would be saying that the tenths place is uncertain. However, we were able to measure to the hundredths place. That does not make sense!

Calculating Uncertainties

Eventually, you will use statistics to calculate your uncertainty. You calculate uncertainty by taking multiple measurements and then using the standard deviation. The number of measurements you should take will depend on your level. However, professional scientists can sometimes take hundreds of measurements. The easiest way to calculate the standard deviation is to use a Microsoft Excel sheet.

Uncertainties in Calculations:

In some experiments, you will measure multiple variables to use in a calculation. However, each measurement will have its own uncertainty. So how do we handle uncertainties when we have to add, subtract, multiply or divide?

Uncertainties in Addition and Subtraction:

When you have to add or subtract measurements, the units stay the same. When you have the same units, you add the absolute uncertainties. Absolute uncertainties have the same units as the measurement. Let's look at a couple of examples.

Addition: 5.05g $\pm0.04$ g + 1.01g $\pm0.03$ g = 6.01g $\pm0.06$ g

Subtraction: 5.05g $\pm0.04$ g - 1.01g $\pm0.03$ g = 4.04g $\pm0.06$ g

In the examples, we are adding or subtracting masses. Our units stay the same throughout. Therefore, we first add or subtract our measurements and then add our uncertainties together.

Uncertainties in Multiplication and Division:

When we do multiplication or division, the units in our final answer are different than our measurements. When we multiply or divide we have to convert our absolute uncertainties to relative uncertainties. Relative uncertainties are the uncertainties as a percentage. To calculate relative uncertainties we use the formula: relative uncertainty= $\frac{uncertainty}{measurement}$ $\times$ 100%. Once we have our relative uncertainties we add them together.

Let's look at an example.

Let's say we wanted to calculate the volume of a box with Length=2.00m $\pm0.01m$ , Width= 2.01m $\pm0.04$ m and Height= 2.31m $\pm0.05$ m.

First, we calculate the volume: V=LxWxH= 2.00m x 2.01m x 2.31m= 9.29 $m^{3}$

Second, we calculate the relative uncertainties:

Relative Uncertainty of L= $\frac{0.01}{2.00}$ $\times$ 100% = 0.50%

Relative Uncertainty of W= $\frac{0.04}{2.01}$ $\times$ 100% = 1.99%

Relative Uncertainty of H = $\frac{0.05}{2.31}$ $\times$ 100% = 2.16%

Third, we add the relative uncertainties: Total Relative Uncertainty: 0.50% + 1.99% + 2.16% = 4.65%

Finally, we convert our relative uncertainty back to an absolute uncertainty:

Final Answer: 9.29 $m^{3}$ $\pm4.65$ % $\rightarrow$ 9.29 $m^{3}$ $\pm0.44$ $m^{3}$