Before explanation, please, evaluate some information in attachment, and, please read a text below.

http://bobhall.tamu.edu/FiniteMath/Module8/Introduction.html

Your question is confusing.

What do you mean with "asymmetrical error bars" and with "normal error bars"?

The SD is the square root of the variance. So far this is simply a (single!) value being related to the variability of the values. Up to this point it has no useful meaning.

Given the distribution of the values is normal, the standard deviation gains some more meaning, namely that one expects about 63% of the values in the interval from (mean-SD) to (mean+SD), about 95% of the values in the interval (mean-2*SD) to (mean+2*SD), and about 99% of the values in the interval (mean-3*SD) to (mean+3*SD).

IF the distribution of the values is normal, THEN I can show the interval from, for instance, mean-2*SD to mean+2*SD to indicate the range in which I expect about 95% of the values to be (this certainly refers to "future", yet unobserved values). This interval is symmetric around the mean, that is, its upper limit is the same distance away from the mean as its lower limit. The interval MUST be symmetric around the mean because the normal distribution is symmetric around the mean.

If the distribution is not symmetric, then a summary statistic like the SD has no meaningful (practical) interpretation. If makes no sense to show an interval based on the SD, because is is not clear what this interval means (for example it is not clear what fraction of values are expected to be in such an interval). So this interval has no meaning anyway. In such cases one can indicate the interval containing 95% of the (expected) values as the interval ranging from the 2.5%-percentile to the 97.5%-percentile of the data. The central tendency of the values is then best indicated by the 50%-percentile (=median). Such an interval may then be asymmetric, the distance from the median to the upper limit may be different to the distance from the median to the lower limit. Exactly this gives the visual inormation about the skewedness of the distribution.

There are no "normal error bars". Error bars can have very different meanings. They can indicate inter-percentile ranges or SDs, but also confidence intervals, likelihood intervals, prediction intervals, standard errors, and I don't know what else. It is therefore important to always clearly state what the error bars mean (how they are obtained).

A final remark, regarding "error bar look very large":

If THIS is your concern, that error bars look too large, then why don't you just omit them? Using error bars just to make a diagram "look nice" is completely off-topic. Such error bars are useless. So better omit them completely. I know many people who use standard errors instead of confidence intervals to indicate the precisions of estimates ONLY because the error bars based on standard errors are smaller than those shwoing the confidence intervals. What a kindergarten...

Josh, out of curiosity, can you explain where these asymmetric error bars come from?  Like are they for means, or medians, or geometric means?  And they're supposed to represent the standard deviation?  Or some other measure of dispersion?

Is there a specific software package you are using to generate the values?