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In scientific experiment, people often try to show an error bar based on the standard deviation around a mean (average) value.

My question is twofold:

  1. How to estimate the minimum but adequate sample sizes (space) to calculate the error bar in an experiment to make a meaningful and predictive plot? I guess it could depend on the accuracy of the equipment and the accuracy requirement of the given experiment.

  2. Does the minimum sample size depend on the type of the measured physical quantities. I mean whether the proper sample space will change as the quantity of concern, for example, velocity, temperature, thickness?

Thank you for any advice and reference recommended!

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    $\begingroup$ It's worth noting that if you don't think the error bar means something than you have admitted that you don't know anything about the validity of your data. Arguably you shouldn't be showing that kind of data at all. Also, obligatory comic link: phdcomics.com/comics.php?f=1816 $\endgroup$ May 3, 2016 at 17:40
  • $\begingroup$ Definitely look at Bayes's theorem. It's the only way to really understand this. $\endgroup$
    – DanielSank
    May 3, 2016 at 20:05

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Judging by your Q1, I think you are asking about error bars for data points in graphs, especially in relation to plotting a line or curve of best fit.

Error bars are not often calculated statistically for this purpose. To do so, a sample of N measurements must be made for each chosen value of the independent variable x and a mean and SD calculated. The standard error of the mean would then be SE = SD/sqrt(N). The error bar could then be placed at mean +/- 1 or 2 x SE, giving an approx. 70% or 95% confidence interval. The value of SE strictly applies only to this data point. Separate values should be calculated for each data point in the graph. However, if it is thought that the SE does not vary much with x, the calculation can be done for one value of x near the middle of the range and that value applied throughout.

However, very few experiments are made in such painstaking detail. Instead, the range of possible error is usually estimated from the accuracy of the least accurate component of the measurement. You would also repeat each measurement at least twice, if possible taking measurements in both directions (increase and decrease in the controlling variable). This would also give you more confidence in an appropriate size for the error bar, but even if you plot only the mean of these repeated readings, you would not usually bother calculating the SD for such a small sample.

An alternative is to plot all measurements instead of averaging them. If you are going to do a least-squares regression analysis on the data to fit a line or curve, then it is possible to include the size of the error bar for each data point (curve-fitting software usually includes this input), but it is far easier (especially if you are doing your own calculations) to include all repeated measurements (if you have them).

It is of course essential to have an error estimate for the output(s) of your experiment, eg the slope and intercept of the line of best fit.

Error bars are not particularly useful in graphs except as a pictorial summary of the repeated readings (to avoid the graph becoming too cluttered with multiple data points) or of the estimated uncertainty in accuracy of measurement devices. You can see the former type especially in particle physics, where each data point is a summary of millions of particle collisions. The error bar indicates to your reader your confidence in the accuracy of the data point. Whether this comes from the spread of repeated readings, or the accuracy obtainable from your measuring devices, doesn't matter very much; probably it will be a subjective combination of the two. What is important is that : (i) it should be a fair estimate - neither too big nor too small as to be misleading; and (ii) the same method of estimation should be used consistently throughout the graph (and also between any graphs which are being compared).

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  • $\begingroup$ "Error bars are not often based on standard deviation of the mean." I think this is overstated, this is probably subfield-dependent but I see plenty of such plots. I agree that it is more of a presentation choice though. $\endgroup$
    – Rococo
    May 3, 2016 at 17:51

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