How can I calculate uncertainty in my data?

[Freeze]

I just did a physics lab and now I have to fill up this uncertainity data sheet. I finished the sheet but I am having doubts about some of my answers. So I am just going to state how I solved the problems, if any of you can just tell me if I did them right or wrong (if wrong, what did I do wrong?) it would be much appreciated. ( I know..this stuff can be a pain to read over forum)

3 of us measured 3 aspect of one same object and we received the following data

D1 = 51.56 mm , 51.68mm and 51.60mm +- 0.01 mm (<==uncertainity)

Average D1 = 51.61mm = .5161cm
$$$
(not sure about the formula) Uncertainity of Average D1 = ((max-Average) + (average-min) )/2 = +- .06mm = +- .0006cm (it asks us to put the uncertainty as cm).

$$$
% relative error of the average uncertainty (again not sure about the formula) - (uncertainity of Average D1/ Average D1) * 100 = .12%


I am not going to go through D2 and T (the other two objects) but let me just give their average and uncertainty of their average


Average D2 = .2041cm +-.0001 cm
Relative Uncertainty = .0005

Average T = .0312 +- .0007cm
Relative Uncertainty = .0224

Now it Asks for

Average D2^2, it's uncertainty and %relative error

Since AverageD2^2 = Avg D2*Avg D2

$$$ (not sure if the method of getting the relative error for D2^2 is correct) To

So - .0005+.0005 = .001 = .1 % (relative error %)

$$$ (not sure if the method of getting the absolute uncertainty here is correct) And to get the absolute uncertainty I have to multiply relative uncertainty by the quantity, so = (Avg D2 * Avg D2) * (relative uncertainty) = (.2041 cm * .2041 cm ) * (.001) = .00004 cm^2 .

Now it asks for

D1^2 - D2 ^ 2 's relative error % and uncertainty.

To get that I have to add the absolute uncertainty of D1^2 to absolute uncertainty of D2^2. Average D1 ^ 2's absolute uncertainty was .0006 cm.

So Absolute uncertainty for D1^2 - D2 ^ 2 = .001+.001 = +-.002.



Now it asks for the volume of the object for which we have to use the following formula = pie ( (D1 ^2 - D2 ^ 2)t )/ 4 (i am sure about this formula as it was given in the sheet) and it asks us to find it's uncertainty.

This is where it got a little more confusing for me. However, I realized I will only have to find the uncertainty of the following part - (D1 ^2 - D2 ^ 2)t.

I need to know the relative uncertainty of D1 ^ 2 - D2 ^ 2 , and since I already know the absolute uncertainty for that which is .002, I will just have to divide that by the product of D1 ^2 - D2 ^ 2 which is .225, so the relative uncertainty is = .002/.225 = .008

Now i have to multiply the product of (D1^2 - D2^2) (t) with (.008+relative uncertainty of T, which is .0224) and that will give me the the volume.

Wow..ok I am just going to stop there. If all these are right, then I am probably never going to have problem with uncertainties again.

 

[Freeze]

Ok made a silly mistake there, in all those figures given in centimeters, the decimal goes one time to the right. I accidentally divided by 100 instead of 10.

 

[M98Ranger]

Calculating uncertainty in data is an important aspect of any scientific experiment or measurement. It allows us to understand the limitations of our data and the reliability of our results. To calculate uncertainty in your data, you need to consider the sources of error and variability in your measurements.

In your example, you have measured three aspects of one object and received three different values for each measurement. To calculate the average, you add all three values and divide by the number of measurements (in this case, 3). This gives you an average value of 51.61 mm. To calculate the uncertainty of the average, you need to consider the range of values you obtained (51.56 mm to 51.68 mm) and divide that by 2. This gives you an uncertainty of 0.06 mm.

To express the uncertainty in cm, you need to convert the value to cm by dividing it by 10. So, the uncertainty of the average in cm is 0.006 cm.

To calculate the relative error, you need to divide the uncertainty of the average by the average value and multiply by 100. In this case, the relative error is 0.12%.

For D2 and T, you have correctly calculated the average and the uncertainty of the average. To calculate the uncertainty of D2^2, you need to multiply the uncertainty of the average by 2 (since you are squaring the value). This gives you an uncertainty of 0.0002 cm^2. To calculate the relative error, you divide the uncertainty by the value and multiply by 100. In this case, the relative error is 0.1%.

To calculate the uncertainty of D1^2 - D2^2, you need to add the uncertainties of the individual values (0.0006 cm and 0.0002 cm). This gives you an uncertainty of 0.0008 cm. To calculate the relative error, you divide the uncertainty by the value and multiply by 100. In this case, the relative error is 0.36%.

For the volume calculation, you have correctly identified that you only need to consider the uncertainty of (D1^2 - D2^2)t. To calculate the relative uncertainty, you need to divide the uncertainty by the value and add the relative uncertainty of t (0.0224). This gives you a relative uncertainty of 0.03%. Multiplying this by the value of (D1