# Averaging Errors: Calculating Error for 2 Values

• tom717
In summary, averaging the two values results in an error that is proportional to the variance of the errors.
tom717
This isn't a set problem itself, rather it is just a small part of my chemisty coursework.

I have two values with known errors

a) -0.000379272 ± 1.75895E-05
b) -0.000576206 ± 4.13448E-05

What happens to the errors when i average the two values?

Good question and one which too few people ask. The first question to ask is what kind of errors are they. Are they Gaussian distributed or evenly distributed? If Gaussian, how many standard deviations does the given error value represent?

You might want to start by just adding and subtracting the error values to the data and averaging and see what happens. Using a random number generator or a Gaussian random number generator might give you a better picture what you might encounter with real data. Maybe after doing it a few times you can come up with a rule for what happens.

Thanks for your reply. I was hoping it would be something straight-forward. It's not hugely important, so I won't put much more effort into now.

I know you were, but lots of times you learn a lot more by investigating little things like these and you're able to use your findings your whole career. Since very few other people investigate these things, it puts you slightly above the rest.

tom717 said:
This isn't a set problem itself, rather it is just a small part of my chemisty coursework.

I have two values with known errors

a) -0.000379272 ± 1.75895E-05
b) -0.000576206 ± 4.13448E-05

What happens to the errors when i average the two values?

Before you even start to do something like averaging these values, you should ask whether it makes sense to do so. In this case it does not. The two values, along with their errors, are incommensurate. It would be better in this case to throw one out than to average them. Which one? Better make another measurement.

If you insist in proceeding, you need to ask whether the errors are statistically independent and how they are distributed. You also should be careful how you do the averaging. Value (a) has a much smaller error than value (b), so the "average" should be closer to (a) than (b). What you want is a weighted average, with the value with the smaller error receiving the greater weight. One widely used weight is the inverse of the variance.

If you compute the average as the variance-weighted mean and if the errors are uncorrelated from each other, the error in the weighted mean is given by

$$\frac 1 {\sigma^2} = \sum_i \frac 1 {{\sigma_i}^2}$$

## 1. What is the purpose of calculating error for 2 values?

Calculating error for 2 values allows us to determine the accuracy of our measurements or calculations. It helps us understand the degree to which our results may deviate from the true value.

## 2. How is error for 2 values typically calculated?

The most common method for calculating error is by using the formula: (|experimental value - true value| / true value) * 100%. This gives us the percentage error between the two values.

## 3. Can error for 2 values ever be completely eliminated?

No, error is an inherent part of any measurement or calculation. However, by carefully controlling variables and using precise instruments, we can minimize the amount of error present.

## 4. What is the difference between systematic and random error?

Systematic error is consistent and predictable, often caused by a flaw in the experimental setup or measurement technique. Random error is unpredictable and can be caused by factors such as human error or equipment limitations.

## 5. How can we use error for 2 values to improve our results?

By calculating error for 2 values, we can identify areas where our measurements or calculations may be inaccurate and make adjustments to improve the reliability of our results. Additionally, comparing the error of multiple trials can help us determine the precision of our data.

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