Physics Lab 4

Introduction to Uncertainty

What is experimental uncertainty?

In experimental work, we generally assume there is a “true” value for the quantity we measure if

we have a perfect experimental design, and perfect instruments in a perfect laboratory. However,

thanks to various imperfections in real life, real measurements always turn out to be different from

the “true” value to a certain degree. We say that the measurement results carry uncertainty.

Is experimental uncertainty the same as human error?

Experimental uncertainty is NOT human error.

Experimental uncertainty is caused by inevitable factors, for example limited instrument precision,

experimental design limited by budget or technology, etc. We can tweak our experimental method to

minimize uncertainty, but there is no way to completely eliminate it. Therefore, it is not a bad thing to

report experiment results with uncertainties; in fact, it is ethically right to report experimental results

with uncertainties.

On the other hand, human error is a gross mistake in experiment design or operation, it is

avoidable, and should be eliminated.

We hope this explanation can help you distinguish the difference between the two terms and use

them correctly from now on. Throughout the physics labs this distinction will be continuously

assessed.

How to do uncertainty analysis

Since we do not know the “true” value, (otherwise, we do not need to measure it in the first place)

what can we do to find out how reliable our measurement is? We need to resort to uncertainty

analysis. There are many different ways for uncertainty estimation, we will learn one method in this

class.

Mean (average), standard deviation, standard error of the mean and confidence level

When we repeat the same measurement multiple times, we usually will not obtain the same

reading every single time. In the first experiment, we measure the diameter of a metal rod multiple

times, let us use it as an example. The table below summarizes a total number of counts 𝑁 = 10

reading results.

Reading count 𝑛 1 2 3 4 5 6 7 8 9 10

Diameter (mm) 6.012 6.016 6.008 6.021 6.023 6.009 6.010 6.018 6.015 6.013

Let us use 𝑥𝑖 to represent the readings, so 𝑥1 = 6.012mm, 𝑥2 = 6.016mm … 𝑥10 = 6.013mm.

Studies show that when there are a large number of readings, however random the variation may

seem, we can expect that roughly half of the readings are greater than the “true” value, and the other

half are smaller; therefore, we can use the mean �̅� of all readings to represent the “true” value. In this

case.

�̅� = 𝑥1 + 𝑥2 + 𝑥3 … + 𝑥𝑁

𝑁 = 6.015mm

In Microsoft Excel, you can call the function =AVERAGE() to facilitate the calculation of the mean

�̅�.

Each reading 𝑥𝑖 deviates from the mean �̅� by 𝑥𝑖 − �̅�. We define the standard deviation 𝝈 (Greek

letter sigma) to represent how much variation exists. The formula to calculate 𝜎 is given below

𝜎 = √ (𝑥1 − �̅�)

2 + (𝑥2 − �̅�) 2 … + (𝑥𝑁 − �̅�)

2

𝑁 − 1 = √

Σ(𝑥𝑖 − �̅�) 2

𝑁 − 1 .

In Microsoft Excel, you can call the function =STDEV.S() for this calculation. Please note there are

different standard deviation functions, STDEV.S() is the one for this purpose.

For the sample data, the standard deviation turns out to be 𝜎 = 0.005mm. Small standard

deviation represents low variation, and high consistency measurements, which is an indicator of good

quality measurements. On the other hand, large standard deviation means measurements are

scattered everywhere, which represents poor quality measurements.

Next, let us introduce another concept, 𝝈�̅�, the standard error of the mean, which can be

calculated using the formula

𝜎�̅� = 𝜎

√𝑁 .

Microsoft Excel does not have a dedicated function to calculate 𝜎�̅�, but if we can calculate 𝜎 using

STDEV.S() and know the number of readings 𝑁, we can easily determine the value of 𝜎�̅�. For the

sample data,

𝜎�̅� = 𝜎

√𝑁 =

0.005

√10 = 0.002mm.

Standard error is directly related to the confidence level. If we multiply 1.96 to 𝜎�̅�, 1.96 × 𝜎�̅� =

0.003mm, it means we have 95% confidence the “true” value we want to determine lies within

±0.003mm range of the mean �̅�, i.e. between 6.015 − 0.003 = 6.012mm and 6.015 + 0.003 =

6.018mm. In other words, there is only 1 out of 20 chance ( 1

20 = 5%) the true value is outside the

range. If we can accept a loose confidence level, say 90% (equivalent to 1 out 10 chance), we just need

to multiply 1.645 to 𝜎�̅�. You may have a question about the particular values 1.96 and 1.645. To learn

more about where they come from, you need to take the course of statistics.

Sometimes, the values of standard deviation 𝜎 and standard error 𝜎�̅� themselves do not paint a

complete picture of the measurement quality; we also need to compare them to the mean �̅�, in the

forms of relative standard deviation and relative standard error. Let us use the same diameter

measurements as an example. We see that

relative standard deviation = 𝜎

�̅� × 100% =

0.0051

6.015 × 100% = 0.09%

and

relative standard error = 𝜎�̅� �̅�

× 100% = 0.002

6.015 × 100% = 0.03%

Both numbers are very small, so these 10 measurements are highly consistent.

How to report uncertainty

When we need to report uncertainty analysis results, we can follow this convention.

1. Keep 1 significant figure for both the standard deviation 𝜎 and the standard error of the mean 𝜎�̅�. In the previous calculation, your calculator probably reports 𝜎 = 0.005061mm and 𝜎�̅� = 0.0016003mm, we record them as 𝜎 = 0.005mm and 𝜎�̅� = 0.002mm.

2. Match the significant figure of the mean �̅� to the same decimal place as the standard error of the mean 𝜎�̅�. In this case, since the standard error 𝜎�̅� = 0.002mm has its significant figure at the thousandth place, even though your calculator says the mean �̅� = 6.0145mm, we will round it up to the thousandth place, thus �̅� = 6.015mm.

Then, we can report the final result 𝑥 = �̅� ± 𝜎�̅� = 6.015 ± 0.002mm in a neat form.

Percent difference and percent error

In experimental work, we often need to compare two measurement values, none of which is more

authoritative than the other; percent difference is the tool. We use percent error when we need to

compare our result to an established reference value.

If you have two measurement values, Value1 and Value2, to compare, here is the percent

difference formula

% difference = |Value1 − Value2|

( Value1 + Value2

2 )

× 100%

Basically, you divide the difference of the values |Value1 − Value2| by their average Value1+Value2

2 .

If you have an experimental result, 𝑅𝑒𝑠𝑢𝑙𝑡, and you want to compare it to an established

reference value 𝑅𝑒𝑓𝑒𝑟𝑒𝑛𝑐𝑒, here is the percent error formula

% error = 𝑅𝑒𝑠𝑢𝑙𝑡 − 𝑅𝑒𝑓𝑒𝑟𝑒𝑛𝑐𝑒

𝑅𝑒𝑓𝑒𝑟𝑒𝑛𝑐𝑒 × 100%

Summary Quantity Symbol Notes

Number of samples 𝑁

Mean �̅� Excel function =AVERAGE()

Standard deviation 𝜎 Excel function =STDEV.S()

Standard error of the mean 𝜎�̅� 𝜎�̅� =

𝜎

√𝑁

95% confidence level 1.96 × 𝜎�̅�

90% confidence level 1.645 × 𝜎�̅�