SD for fold-change makes no sense because of two reasons:

1) SD is a property of the data - but your fold-change is an estimate

2) it has an interpretable meaning only for normally distributed variables - but fold-changes are not normally distributed

If you want a measure of precision for fold-changes then calculate the confidence intervals for these estimates.

I fully agree with the suggestion made by Jochen Wilhelm since SD should not be computed in this case since the data do not satisfy normal distribution .

Dear Wilhelm and Murthy,

Thanks for your answers. According to your suggestion, the bar diagram from fold change should not contain an error bar. is that right? Furthermore I would like to also like to know, if the diagram is from relative expression then I can use an error bar, is not it? thanks in advance

Why not? Sure is can contain an error bar. Reporting results should always be accompanied by some measure of precision or error! It's only that showing the SD as an indication for the errors of fold-changes is utter nonsense (as explained previousely). But there are other senesible choices, like confidence intervals.

You may also consider to show log fold-changes (or at least use a logarithmic scale on the ordinate) - this is a far better representation of the biological relevance and far less misguiding our intuitution about the effects (example: on the linear scale, a regulation to 10-fold looks huge, whereas a regulation to 0.1-fold looks much less strong - but the biological strength is similar, only to the opposite direction. On the log scale both (log) fold-changes would have the same numeric value but with opposite signs),

PS: I am not a fan of barplots in general, but that's a different topic.

Dear Wilhelm,

Thanks again. I also agree with you, log2(ddct) is much better option.

I have also found some articles regarding it.

Have a great day.

I am not sure if your last post contains a typo of you you really meant it the way you wrote it:

"log2(ddct)" is no sensible measure! ddct values can be positive and negative. Note that already practically logarithms of negative values are not defined (so this cannot make any sense).

In fact

     ddct == log2(fold-change)

(at least as approximation, given that the "true" amplification efficiency is not exactly 2.0)

Note that this equality assumes that dct values are calculated as ct[ref]-ct[goi]. If the dct values are calculated the other way around, the sign of the ddct value has to be changes, too.

Using log fold-changes means to use ddct values.

I wouldn't have minded if I was not aware of some (unfortunate) papers where the authors actually (and wrongly!) used the logarithms of dct or ddct values.

Dear Wilhelm,

Of course it was a typological error to write log(ddct) which has no sensible meaning.

I actually meant log2(fold change).

Thank you again for your concern and valuable comments and suggestions.

Dear Dr Wilhelm,

Currently I'm facing the similar issue where i did not include SD or the error bars in the bar charts of mean fold change ( 2 power to -∆∆Ct) of each gene for my gene expression study. But i was been advised to include the SD values in my study, unless i can find a good paper that none include the SD or the error bars in fold change in their study. And the paper i found almost all of them showed the SD or the error bars in their chart. Could you please advise me in this issue?

1) you should not use bar charts.

http://biostat.mc.vanderbilt.edu/wiki/pub/Main/TatsukiRcode/Poster3.pdf

2) SD is difficult to interpret (and prone to misinterpretations) when the distribution is skewed.

Article Linking Stochastic Dynamics to Population Distribution: An A...

http://www.uwe-mortensen.de/LinmpertStahelProblemsUsingNormalDistrib2011.PDF

3) SD refers to data, not to estimates. the mean LFC is an estimate, so I wonder how this goes with SD. Estimates have standard errors (SE). But again, showing SEs is meanigful only when the distribution is symmetric or in large-sample approximations (usually the sample sizes in biology are tiny rather than large). Instead, confidence intervals should be shown (thay are anyway to be preferred over showing SEs). Note: for assymetric distributions, confidence intervals are also assymetric.

Article Statistics notes: Standard deviations and standard errors

Dear Dr Wilhelm,

Thank you very much for your prompt reply. I plan to present the expression level using -∆∆Ct instead of 2 power to -∆∆Ct, so i can insert the error bars and show the expression in the down or up regulation trend for each gene.

Thank you very again.

I have 4 control samples and 4 treatment samples and each sample has 2 replicates. I've averaged out each ct value for the each sample and calculated the SD. Then when i got the value of dct(ct(GOI)- ct(housekeepinggene) i calculated the propagated error from the housekeeping gene ct and the gene of interest sqrt((s1^2)+(s2^2)) this is also the same SD for the ddct (dct GOI treatment-dct GOI control) correct? I now have for my control gene four ddct values and 4 SD how do i use this data to get confidence interval information to use for the error bars for the bar graph (standard how this data is presented for us). I averaged out the ddct then 2^-(average ddct) to get the fold change how do i handle the 4 SD values?

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Tafadzwa Chihanga , you should calculate the dCt values first. The SDs of ct values are not interesting. They depend on differences between the sample material / loading, and these differences should be controlled for by measureing a reference gene and taking the dCt values. So it's only the SD of the dCt values that is interesting, and it should be claculated from the dCt values (not propagated from the Ct values - at least not without considering the correlation [which can be quite high!])

The SD of the ddCt values can be calculated by error propagation of those of the dCt values (just as you described).

You should NOT use SDs or SEMs to express uncertainty about fold-changes (particularily not for such tiny sample sizes), since they have a log-normal distribution. Stay on the level of ddCt values or use confidence intervals.

Jochen Wilhelm okay got that so the CI is calculated from the average ddct value(4 controls) SD and then i use the CI or 2^-(CI) on the fold change(2^-(averageddct) in my bar chart?

Tafadzwa Chihanga seems correct. However, I recommend NOT showing fold-changes and NOT using barcharts. It would be so much better to show dCt or ddCt values insteadand to use scatterplots / dotplots.