Intro: Making a Meaningful Measurement

Why uncertainties really matter and how to deal with them.

PS2 - Lab 2

Measurement Error

Intro to: Your vet says your cat, Hoagie, needs to lose weight. Your scale only weighs to the nearest pound. You weigh Hoagie in at 18 pounds, and start him on a diet. After a week you weigh Hoagie again, and the scale reads 17 pounds.

Is the diet working?

__________________________________________________________________________________

Which of the following could be true?

(a) Hoagie lost weight(b) Hoagie gained weight(c) Hoagie’s weight didn’t change(d) We could improve our assessment by taking Hoagie off the scale and immediately

put him back on it to measure his weight again.(e) We could improve our assessment by repeatedly measuring Hoagie’s weight before

and after the diet(f) Because the scale only reads to the nearest pound, Hoagie could have lost just a

small amount of weight, taking him from just over to just under 17.5 pounds(g) None of these are necessary, Hoagie for sure lost 1 pound.(h) We can’t say whether Hoagie lost weight or not.

● Meaningful measurements MUST come with uncertainty

Measurement Error● Meaningful measurements MUST

come with uncertainty

● There are several types of uncertainty - today’s focus is on uncertainty that comes from the resolution of your measuring device

Intro to:When making a measurement, we always need to ascribe uncertainties to make our measurements meaningful and quantitative. You’ll commonly make either an analog or digital measurement, and we’ll follow the rules below:

Analogue:

A ruler has notches down to 0.1 cm, and since you cannot resolve the measurement less than the notches, your measurement would have an uncertainty of 0.1 cm.

Digital:

With digital outputs there is minimal experimentalist uncertainty, but the machine must round the decimal point not shown. The uncertainty of the measurement is half of the last decimal place of the output.

Measured value, x

Length of this stick represents the uncertainty in the measured value, 𝛿x

● When an uncertainty is estimated for a measured value x, it is designated ± 𝛿x (‘plus or minus delta x’)

● This is reported graphically on a data point with ‘error bars’ as shown to the right. The length of the bar on one side of the dot corresponds to the uncertainty, delta x

Intro to: Uncertainty should convey to others how confident we are with our measurements. But simple number like ± 1 cm might not always do the trick.

● Example: A suspicious ice cream truck sells ice cream by the scoop, and their uncertainty on one scoop is ± 1 scoop. Two of your friends are looking to buy ice cream: Ben is looking is looking to get a 3 gallon (96 scoops) bucket, but Jerry just wants a cone with a scoop. Even though the uncertainty in how much ice cream each person is getting is the same, who are you more “certain” will be happy? Ben or Jerry?

To say whether an uncertainty is large or a measurement is reliable, you need to compare the measured value with the uncertainty- relative uncertainty gives us that measure nicely.

Absolute Uncertainty: 𝛿x has the same units of x and is called the absolute uncertainty, or simply the uncertainty of x.*

Relative Uncertainty: The ratio of the absolute uncertainty to the measured value (𝛿x/x) is called the relative uncertainty, and is usually converted to a percent: 𝛿x/x ⨉ 100%.

If my absolute error is 1 cm but my measurement is also 1 cm, my relative error is very high, and thus my confidence in my measurement is quite low. However if my error is 1 cm, but my measurement is 10 m then my relative uncertainty is low, my measurement is very good, and I am confident in my measured value - relative error very easily gives us this information.

* We only report one significant figure of absolute uncertainty, and round our measured value to the significance of the uncertainty. It doesn't make sense to report a value to higher precision than our uncertainty (e.g 10.6 ± 1 cm is improper).

● Meaningful measurements MUST come with uncertainty

● There are several types of uncertainty - today’s focus is on uncertainty that comes from the resolution of your measuring device

● When an uncertainty is estimated for a measured value x, it is designated ± 𝛿x (‘plus or minus delta x’). Relative uncertainty is denoted ± 𝛿x/x.

Measurement Error

Outline:1. Practice assigning uncertainties to measurements2. Learn how to compare measurements quantitatively

with the significance test3. Learn how to propagate uncertainties from a

measured quantity to a quantity of interest4. Use Python to easily propagate uncertainties5. Big Question for the day: Why does ice float?

Uncertainties

● Can you identify the resolution uncertainty in your measurement?

Your kit has three measuring devices - a tape measure, a scale, and a beaker. Each person in the group should measure and record the following. Make sure you are reporting uncertainty!

1. the length and width of the blood pressure box

2. the mass of the superball

3. the volume of the superball

Once you are all done compare your values to others in the group. Do your values agree?

Activity 1:

Comparing Values

● We can never 100% certain of measurements, so we can also never be 100% sure two measurements are different. Instead we determine the likelihood the measurements are different.

Intro to:It’s important to have a quantitative metric that we use to decide if two measurements are the same or not. We’ll use the significance test, s.t., for comparing two measured values with uncertainties, A ± 𝛿A and B ± 𝛿B:

Write and test a function in Python that computes the s.t. for any two values and their uncertainties. Your code should look something like the following. (Hint: If you code up line 8 correctly and you use the test values shown below, you should get the value printed out at the bottom. You’ll also need to import the numpy library.)

You get a better scale to weigh Hoagie and his sister, Noodles. You weigh them both over the course of a month:

Hoagie Noodles .

September: 18.5 ± 0.2 lbs 14.0 ± 0.2 lbsOctober: 18.3 ± 0.2 lbs 12.0 ± 0.2 lbs

Activity 1

(1) Calculate the s.t. for Hoagie’s weight change and Noodles’ weight change. (Use your function!)

(2) Which cat’s weight change was more significant?

(3) What indicates a significant difference between two measurements - a low value of st, or a high value of st?

A Signifi-CAT Change? - Hoagie Revisited

1. Someone has measured the speeds of five pairs of snails participating in the 2020 Tour de Garden. Rank the pairs according to the significance test of each pair. (In other words, rank from “most same” to “most different.”

2. Are your measured values from activity 1 consistent with those of your group members (provide quantitative support to your answer)?

Am I Getting It? #1 Measured value, x

Length of this stick represents the uncertainty in the measured value, 𝛿x

Error Propagation

● When you can’t measure the thing you’re actually interested in - you can follow a process called “error propagation.”

Intro to:

xkcd

(This is not how you propagate error.)

Error Propagation

Intro and Am I Getting It? #2:

Example: Say you want to measure the area of something square, such as the face on one of the dice. You don’t have an area-ometer, but you do have a tape measure. How are you going to get the error in the area if all you can measure are lengths? (this is rhetorical)

Think about how the two quantities are related - we find the area by squaring the side length, A = x2. If we plot x vs A, our plot looks like this:

● When you can’t measure the thing you’re actually interested in - you can follow a process called “error propagation.”

Question: Which of the following will the

uncertainty of the area, 𝛿A, depend on?

(a) The measured side length, x0

(b) Side length uncertainty, 𝛿x(c) The derivative (slope) of A(x) at x0(d) The integral (area) of A(x) up to x0

Question: Look at the graph to the right – which has larger uncertainty in the area, a square whose side length is 1 ± 0.1cm or a square whose side length is 3.5 ± 0.1cm?

(a) The larger square(b) The smaller square(c) Their areas have the same error

Uncertainty in your measured value

Uncertainty in the

quantity you wan

t

Slope of the function that

relates what you want to what

you measured (evaluated at the

value you measured.)

PS - If you have a function of multiple measurements, f(x,y, ..), each with uncertainty, the formula is just an extension of that above:

The only Error Propagation Formula(s) you ever need!

Calculate the perimeter of one face of a die, the total surface area of a die, and the volume of the die. Use the chart below to help you calculate the uncertainty in each quantity. Once you have done so, input the resulting values for each quantity on Canvas. Assume that we measured a side length of x = 1.0 ± 0.1 cm.

Quantity Relationship between what you measured and what you

want

Derivative Expression for the propagated absolute uncertainty in the Quantity

in terms of x and delta x

Expression for the propagated relative uncertainty in the

quantity in terms of x and delta x

Perimeter (P) P(x) = 4xP(1.0) = 4 cm

dP/dx = 4 𝛿P = 4𝛿x𝛿P# = 4(0.1) cm = 0.4 cm

𝛿P/P = 4𝛿x/(4x) = 𝛿x/x𝛿P/P# = 0.1 cm/1 cm = 0.1 = 10%

TotalSurface area

(A)

A(x) = 6x^2 dA/dx = 𝛿A =

Volume (V) V(x) = x^3 dV/dx = 𝛿V =

Am I Getting It? #3

#3b Using python

● Use the uncertainty python package to do these calculations for you!

This is kinda tedious. Luckily python can do the tedious tasks for you. Now that you know the reasoning behind error propagation, use the notebook below to verify that your calculations from the previous slide are correct.

TRY IT HERE

Am I Getting It?

Why does ice float?● The rest of the lab will be this activity

● You get to design and carry out the activity however you see fit!

● You will hand in the colab notebook. Make notes in the notebook like you would a lab book.

Lab Activity: Is ice less dense than water?

● Design an experiment to calculate and report the density of water and the density of ice with uncertainty. Write your plan for the experiment in your Colab notebook.

● Try to make your measurements as precise as possible! (Please talk to a TF if your densities have high relative uncertainties)

● Each group member should carry out the experiment.● Choose one group member’s data, and use it to

compare the densities of ice and water, quantitatively.

BONUS: Can you measure variations in the density of ice?

● Ice has different densities at different temperatures. So, you can figure out if your ice cubes are a different temperature than your teammates’ ice cubes by comparing their densities. Comparing you teammates’ data with yours, can you conclude you have different temperature ice cubes?

● Discuss your confidence in this conclusion.

Learning objectives

Reflect. Can you now …

● Assign absolute and relative error to your measurement based on the resolution of your device?

● Use uncertainty to compare two measured values?

● Propagate error from the quantity you measure to the quantity you want?

If you have remaining questions,

please ask your TF!

Error Propagation

Bonus! (Reference Page) The most-used error propagation is for addition/subtraction and multiplication/division.

If we have measured values, x, y, and z, each with an associated uncertainty:

For addition or subtraction:

For multiplication or division: