# Compute Standard Deviation to Estimate Uncertainty

• tony873004
In summary: I'm kinda surprised he liked it, cause it's not technically correct to say that the uncertainty on the distance to the center (presumably calculated to be the mean of your data points) is equal to the spread. Actually, the uncertainty on the mean of a set of data points is:\sigma_{\bar{x}}^2=\frac{\sigma^2}{N}where N is the size of the sample. This would mean that your uncertainty is a factor of two smaller than the standard deviation of the sample:\sigma_{\bar{x}}=\frac{\sigma}{\sqrt{4}}
tony873004
Gold Member
$$\textit{To get an estimate of your uncertainty, compute the standard deviation. My 4 distances are 150, 120, 100 and 100 parsecs (pc)} The average of my 4 distances is $\bar {x}=\frac{\sum\limits_{i=1}^n {x_i } }{n}$ $\bar {x}=\frac{150pc+120pc+100pc+100pc}{4}=117.5pc$ The average =  120pc \textit{The standard deviation is } $\sigma =\sqrt {\frac{\sum\limits_{i=1}^n {\left( {x_i -\bar {x}} \right)^2} }{n-1}}$ \[ \sigma =\sqrt {\frac{\left( {150-120} \right)^2+\left( {120-120} \right)^2+\left( {100-120} \right)^2+\left( {100-120} \right)^2}{n-1}} =23.805$$

I've computed it but what does it mean? How do I estimate my uncertainty from this number? The book doesn't explain this.

The standard deviation IS a measure of your uncertainty.

Standard deviation is sort of the average (or expected) away from the average. If you input a lot of data and after a while you get a stable average and standard deviation, any other point inputted will on average be the standard deviation away from the average.

In other words the standard deviation is how far the points deviate away from the average standardly

tony873004 said:
I've computed it but what does it mean? How do I estimate my uncertainty from this number? The book doesn't explain this.

Generally, when people report their errors, they report $$\sigma$$, but occasionally you'll see them report $$2\sigma$$ or $$3\sigma$$ (or some other statistics, but $$\sigma$$ is usually standard). These choices for error quoting are fairly arbitrary, it's just a matter of what you're trying to communicate (unfortunately, it can also be used for deception). If the errors are distributed normally, then a 1$$\sigma$$ error bar means that there's about a ~68% chance that the true value lies in that interval, while 2$$\sigma$$ and 3$$\sigma$$ represent about a ~95% and ~99.7% chance that the true value is in the interval. You might quote the latter if you wanted to be conservative with your results. Of course, there are errors on your estimate of the error as well, but let's not get into that.

I don't know what the case is for you, but definitely consider the above before quoting an error.

Thanks for all your replies to a topic not covered by my book!

SpaceTiger, I actually included what you said in my final answer. I got an uncertainty of about 24 parsecs, And there's a 1 pc difference between the closest star and the furthest star in the Pleadies star cluster. I don't know if I should add the intrinsic difference to the 24 or subtract it. I'm guessing subtract. I just concluded it in a sentence to the effect "In the face of a 24 pc uncertainty, the 1 parsec difference in actual star differences is negligable." The teacher liked my answer.

tony873004 said:
SpaceTiger, I actually included what you said in my final answer. I got an uncertainty of about 24 parsecs, And there's a 1 pc difference between the closest star and the furthest star in the Pleadies star cluster. I don't know if I should add the intrinsic difference to the 24 or subtract it. I'm guessing subtract. I just concluded it in a sentence to the effect "In the face of a 24 pc uncertainty, the 1 parsec difference in actual star differences is negligable." The teacher liked my answer.

I'm kinda surprised he liked it, cause it's not technically correct to say that the uncertainty on the distance to the center (presumably calculated to be the mean of your data points) is equal to the spread. Actually, the uncertainty on the mean of a set of data points is:

$$\sigma_{\bar{x}}^2=\frac{\sigma^2}{N}$$

where N is the size of the sample. This would mean that your uncertainty is a factor of two smaller than the standard deviation of the sample:

$$\sigma_{\bar{x}}=\frac{\sigma}{\sqrt{4}}$$

This is simply because more measurements will give you more independent data points with which you can estimate the distance, making your final estimate more accurate. The "error" that is given by $$\sigma$$ alone would be the error associated with a single measurement.

Anyway, I'm glad this didn't turn out to be a problem in the end. My apologies for not being clearer about it.

hmmm... Well 1 pc is still insignificant in the face of an uncertainty of 12pc.

Being that we never covered this in class and the book doesn't cover it either, I don't think the teacher had any choice other than to like my conclusion which is still pretty much correct even though I used 24 instead of 12.

Google failed me on this one, so thanks for the extended explanation and for the formula.

## What is the purpose of computing standard deviation to estimate uncertainty?

The purpose of computing standard deviation to estimate uncertainty is to quantify the amount of variation or spread in a set of data. It provides a measure of how much the individual data points deviate from the mean, which can help determine the reliability and accuracy of the data.

## What is the formula for computing standard deviation to estimate uncertainty?

The formula for computing standard deviation to estimate uncertainty is √(Σ(x - μ)² / N) , where x is each individual data point, μ is the mean of the data, and N is the total number of data points.

## What does a high standard deviation indicate?

A high standard deviation indicates that the data points are spread out over a larger range, signifying a higher degree of uncertainty in the data. This could be due to a variety of factors, such as measurement error, variability in the population being studied, or a small sample size.

## How is standard deviation related to uncertainty?

Standard deviation is directly related to uncertainty, as it measures the amount of variation or spread in a set of data. A larger standard deviation indicates a higher degree of uncertainty, while a smaller standard deviation suggests lower uncertainty and a more precise measurement.

## How can computing standard deviation to estimate uncertainty be useful in scientific research?

Computing standard deviation to estimate uncertainty is useful in scientific research as it allows for the assessment of the reliability and accuracy of data. It can also help identify outliers or unusual data points that may need further investigation. Additionally, standard deviation can be used to compare the variability of different sets of data, which can provide insights into potential patterns or relationships.

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