Uncertainty error calculations

In summary, after measurements, the following values and their uncertainties were recorded: x = 8.3 ± 0.1 and y = 2.72 ± 0.07. From these values, we can find the following values of Z: a) Z = x + y = 11.02 ± 0.17, b) Z = x - y = 5.4 ± 0.2, c) Z = xy = 22.576 ± 0.007, d) Z = x/y = 3.1 ± 1.4, e) Z = y/[(x)^0.5] = 0.94 ± 0.22, f) Z
  • #1
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After measurements, the following values and their uncertainties were recorded:

x = 8.3 ± 0.1 and y = 2.72 ± 0.07

Find the following values of Z

a) Z = x + y

b) Z = x - y

c) Z = xy

d) Z = x/y

e) Z = y/[(x)^0.5] (this is y divided by the square root of x)

f) Z = exp(y)

g) Z = ln(x + y)

h) tan(x/y)

My attempts: a) 11.02 ± 0.17 ; b) 5.4 ± 0.2 ; c) 22.576 ± 0.007 ; d) 3.1 ± 1.4 ; e) 0.94 ± 0.22 ; f) exp(2.72 ± 0.07) ; g) ln(11.02 ± 0.17) ; h) tan(3.1 ± 1.4)

Any help would be appreciative, I'm a bit rusty, having not done any physics in two years, and apparently, my attempts are all incorrect!
 
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  • #3
Thank you, I'll have a read of that, I have a recommended textbook entitled measurements and their uncertainties, but it doesn't have any of this fundamental stuff in, I'm finding it really annoying! :(
 
  • #4
No more help guys?
 
  • #5


As a scientist, it is important to understand and accurately calculate uncertainty error in order to properly analyze and interpret experimental results. In this scenario, the values of x and y have been measured with their respective uncertainties. It is important to note that the uncertainty values represent the potential error in the measurement and should be taken into consideration when calculating any derived quantities.

a) Z = x + y = (8.3 ± 0.1) + (2.72 ± 0.07) = 11.02 ± 0.17

b) Z = x - y = (8.3 ± 0.1) - (2.72 ± 0.07) = 5.58 ± 0.17

c) Z = xy = (8.3 ± 0.1) * (2.72 ± 0.07) = 22.576 ± 0.73

d) Z = x/y = (8.3 ± 0.1) / (2.72 ± 0.07) = 3.05 ± 0.13

e) Z = y/[(x)^0.5] = (2.72 ± 0.07) / [(8.3 ± 0.1)^0.5] = 0.95 ± 0.02

f) Z = exp(y) = exp(2.72 ± 0.07) = 15.2 ± 0.6

g) Z = ln(x + y) = ln(8.3 ± 0.1 + 2.72 ± 0.07) = 2.4 ± 0.1

h) Z = tan(x/y) = tan(8.3 ± 0.1 / 2.72 ± 0.07) = 3.1 ± 0.2

It is important to keep in mind that the uncertainty in the final result (Z) is a combination of the individual uncertainties in x and y. Therefore, when performing any calculations, it is important to properly propagate the uncertainties in order to obtain an accurate and meaningful result. Additionally, it is always a good practice to report the final result with the appropriate number of significant figures, taking into account the uncertainty.
 

1. What is the definition of uncertainty in scientific measurements?

Uncertainty refers to the range of values within which the true value of a measurement is likely to fall. It is a measure of the potential error or variation in a measurement due to limitations in the measurement process or equipment.

2. How is uncertainty calculated?

Uncertainty is calculated by considering the sources of error in a measurement and estimating their potential impact on the final result. This can be done using statistical methods, such as standard deviation or confidence intervals, or through a detailed analysis of the measurement process.

3. What is the difference between systematic and random uncertainty?

Systematic uncertainty, also known as bias, is a consistent error that affects all measurements in the same way. It can be minimized by improving the measurement process. Random uncertainty, on the other hand, refers to the natural variation in measurements and cannot be eliminated, but can be reduced through repeated measurements and statistical analysis.

4. How do you express uncertainty in a measurement?

Uncertainty is typically expressed as a range of values, often with a confidence level, such as ± 0.5 cm with a 95% confidence level. It can also be expressed as a percentage or as a fraction of the measured value.

5. Why is it important to consider uncertainty in scientific measurements?

Uncertainty is important to consider in scientific measurements because it provides a measure of the reliability and accuracy of the results. It allows for proper interpretation of the data and helps to determine if the results are within an acceptable range of error. It also allows for comparison of results between different experiments and helps to identify areas for improvement in the measurement process.

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