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### AuthorTopic: Standard deviation & standard error  (Read 2584 times)

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#### Bri MT

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##### Standard deviation & standard error
« on: April 26, 2021, 12:35:55 pm »
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Heyo,

Going through feedback from student experiment write-ups it seems like there isn't much confidence about the difference between standard deviation and standard error. That's why I thought I'd put together this short guide on the both of them & how to interpret them

NOTE: you do not need to know all of this info for QCE science so do not try to memorise it but I'm hoping this improves your understanding of how standard deviation and standard error are used.  Jump to the end for a TLDR;

Lets look at the formula for sample standard deviation:

$\sigma = \sqrt{\frac{\sum_{i=1}^{n}{(x_{i} - \bar{x})^{2}}}{n-1}}$

The $\sum_{i=1}^{n}{(x_{i} - \bar{x})^{2}}$ part tells use that we're finding the difference between each value and the mean, squaring this value (they'll now all be positive + bigger deviations will have more impact) and adding all of those squared differences together. This is then divided by our number of samples minus one, and since we are finding standard deviation rather than variance we wrap the whole thing in a square-root. The effect of this is that we get a nifty measure of how far away our values are from the mean / how dispersed the values are. Unlike the variance, the standard deviation has the same units as we collected our data in.

Ok, so how is this different from the standard error?

To get the standard error, we get the standard deviation & then divide it by $\sqrt{n}$ i.e. $SE = \frac{\sigma}{\sqrt{n}}$. Note that this means that as you increase your number of samples, you decrease your standard error. If you increase the variability of your data, then you increase your standard error. Without going too much into stats, this helps us know how close the mean we found from our sample data is likely to be to the true value. For a normal distribution, if the means we are comparing are 2 x SE or more away from each other, there is less than a 5% probability that this is due to chance. This makes us confident that the means really are different.

TLDR;

The standard deviation tells us about the variability in our data, the standard error tells us about how confident we are that the mean we found is close to the true mean.

If you have overlapping standard deviation bars, not necessarily a big deal. if you have overlapping standard error bars, then you can't be very confident that your means are truly different.

If you have any questions or please feel free to comment below