# How To Calculate Uncertainty in Physics

## What is uncertainty in physics?

Even though science is known to able to make pretty accurate predictions, they are never exact. We use mathematical models to describe the world around us, but once we go ahead and make measurements of those predictions, we find that we cannot measure everything with a 100% accuracy. This is what we call uncertainty.

If we measure the length of an object with a ruler, the length that we read could be around 203 mm. But given that rulers are not perfect, and also that we might not be reading the measurement accurately enough, the actual length might actually be just a bit larger or smaller. Since the smallest distance we can measure with a ruler is 1 mm, our uncertainty is +/- 1 mm, and we can write our measurement as 203 mm +/- 1 mm. This means that the length is 203 mm, but it could vary up to 1 mm.

In the following sections we will see the process of calculating uncertainty.

## Uncertainty formula

Definition: The uncertainty of several measurements is given by the standard deviation of the values, which is a measure of how spread out the measurements are.

## How To Calculate Uncertainty

Step 1: Calculate the mean of all the measurements.

Step 2: Calculate the square of each sample minus the mean.

Step 3: Sum all those squares for all measurements.

Step 4: Divide the sum by N and take the square root.

Step 5: State the final measurement.

### Example: Calculate the standard deviation with the equation above

The following measurements are given:

5.4 6.7 3.8 5.1 6.3

Step 1: Calculate the mean of all the measurements.

The number of measurements is N=5, so the mean is:

(5.4  +  6.7  +  3.8  +  5.1  +  6.3) / 5  =  5.46

Step 2: Calculate the square of each sample minus the mean.

(5.4 – 5.46)^2  =  0.0036

(6.7 – 5.46)^2  =  1.5376

(3.8 – 5.46)^2  =  2.7556

(5.1 – 5.46)^2  =  0.1296

(6.3 – 5.46)^2  =  0.7056

Step 3: Sum all those squares for all measurements.

0.0036  +  1.5376  +  2.7556  +  0.1296  +  0.7056  =  5.132

Step 4: Divide the sum by N and take the square root.

Sqrt(5.132/5) = 1.0131

We want to have the same number of decimal points as for the mean, so we round it to 2 decimal places: 1.01

Step 5: State the final measurement.

The final measurement is the mean +/- the standard deviation, so our answer is 5.46 +/- 1.01.

Here you can use an online tool to calculate the standard deviation.

### Example in excel

The same example done in excel:

Step 1: Calculate the mean of all the measurements.

First you write all the measurements in the first column, and then you use the function SUM to add all those together, and then divide it by 5, which is the number of measurements there are.

Step 2: Calculate the square of each sample minus the mean.

Here it is important to write \$B\$3 instead of B3, because we want that cell to be fixed when we pull down that formula for every row. A3 stays that way because that cell will change according to each row.

After we write the equation, we pull it down so that it applied to each row.

Step 3: Sum all those squares for all measurements.

Then we simply use the function SUM again to add all the values from the last step.

Step 4: Divide the sum by N and take the square root.

After doing all those steps we arrive at our answer, which is the same we obtained when doing the calculations by hand:

## Arithmetic Operations with Uncertainty

Often you will need to perform basic arithmetic operations of measurements with different uncertainties. We will see the rules that need to be applied when adding, subtracting, multiplying or dividing measurements.

### Absolute and relative uncertainty

It is first important to understand the distinction between the two. Absolute uncertainty is what we have seen so far, where the uncertainty is given in units of the original measurement, for example 5.6 mm +/- 0.3 mm.

But is 0.3 mm a lot of uncertainty? It surely depends what we are measuring. If we are measuring the distance to the sun, then 0.3 mm is a very small number. On the other hand, if we are measuring the width of a hair, then 0.3 mm becomes relevant. This is where relative uncertainty comes into play. It gives the uncertainty as a percentage of the original measurement.

For our example 5.6 mm +/- 0.3 mm, the relative uncertainty would be (0.3/5.6)*100 = 5.4%

So, we can write 5.6 mm +/- 5.4%

(5.1 mm +/- 0.1 mm) + (4.3 mm +/- 0.2 mm) =

= (5.1 mm + 4.3 mm)  +/-  (0.1 mm + 0.2mm) =

= 9.4 mm +/- 0.3 mm

### Subtraction: The absolute uncertainties are added (not subtracted!)

(5.1 mm +/- 0.1 mm) – (4.3 mm +/- 0.2 mm) =

= (5.1 mm – 4.3 mm)  +/-  (0.1 mm + 0.2 mm) =

= 0.8 mm +/- 0.3 mm

### Multiplication: The relative uncertainties are added (not multiplied!)

(5.1 mm +/- 0.1 mm) * (4.3 mm +/- 0.2 mm) =

= (5.1 mm +/- 1.96%) * (4.3 mm +/- 4.65%) =

= (5.1 mm * 4.3 mm)  +/-  (1.96% + 4.65%) =

= 21.9 mm^2  +/-  6.61%

### Division: The relative uncertainties are added (not divided!)

(5.1 mm +/- 0.1 mm) / (4.3 mm +/- 0.2 mm) =

= (5.1 mm +/- 1.96%) / (4.3 mm +/- 4.65%) =

= (5.1 mm / 4.3 mm)  +/-  (1.96% + 4.65%) =

= 1.2 +/-  6.61%

Here you can find an online calculator to perform these kinds of calculations.

## Why is uncertainty important in physics?

Uncertainty is a very important concept in science in general. Scientists use experiments to validate or refuse a hypothesis, and therefore a good understanding of uncertainty is crucial if we get results that might debunk a theory.

Some people think uncertainty means a lack of knowledge. However, uncertainty is rather a measure of how well something is known. If a theory has a certain uncertainty, it does not mean it is wrong. It just tells us within what bounds the theory can be assumed to be correct. As more research is made, the uncertainty can be reduced. However, some people interpret the uncertainty of a theory as a synonym for doubt. If one understands what uncertainty actually means, one would come to the realization that stating the uncertainty is simply the scientists wanting to be as honest as possible as to how confident they are of their results.

## How do you reduce uncertainty in physics?

Scientists are always trying to find ways to reduce the uncertainty in experiments, since a smaller uncertainty leads to more confident and more accurate results.

If you are taking experimental data, a way to reduce the uncertainty is to make several measurements and take the mean between them. Let us say for instance that you are studying a pendulum and want to calculate its period. Since you are the one stopping the clock, one can easily see that it is fairly difficult to measure the period precisely. Sometimes you might stop the clock a bit sooner or a bit later than you should. But if you take several measurements and take the mean, it is more likely that you will arrive to a more accurate estimate. Here we can also calculate the standard deviation. If it is very small, then you can conclude that you have taken very similar measurements. If on the other hand, the standard deviation is large, then the values are much spread apart.

Another source of uncertainty can be the device you are measuring with. If you use a very cheap ruler that seems to have inconsistencies, then the uncertainty will be higher than if you use a laser rangefinder. Also, if you are using a scale that is not correctly calibrated, you will get values that are not right. Therefore, you should be careful what you are measuring with, so that you know what level of confidence you have.

A last possible source of uncertainty are random errors. If you are taken very precise measurements but a train comes by, it can be that the vibrations affect the measurement. These environmental perturbations come into play when stating the uncertainty, since they can be so small that one can never be sure if the measurements were affected by them or not.

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