# Seeking possible explanation for log2 normal distribution of sample standard deviations

It is my understanding that:

• The standard deviations of samples drawn from a normal distribution follow a chi-square distribution.

• A chi-square distribution with a large value of degrees of freedom approximates a normal distribution.

I have an experimental dataset of is protein expression of 2000 proteins from 25 biological samples, constituting 50K observations. This data takes the form of a matrix with 25 rows and 2000 columns.

I have simulated the dataset in R with the following code.

The population mean and sd represent those observed in the experimental data.

rNormPop <- rnorm(
50000, #50K observations
mean=28, #population mean
sd=2.5) #population sd

# generate an empty 25x2000 matrix
sampleMatrix <- matrix(
data = rep(0,50000),
nrow = 25,
ncol = 2000
)

#populate each row with a random sample of 2000
for(i in 1:25){
sampleMatrix[i,] <- sample(rNormPop,2000)
}

# calculate sd with respect to columns
sampleSD <- apply(
sampleMatrix,
2,
sd
)


Below are histograms of the simulated observations, sample means (with respect to matrix columns), and sample standard deviations (w.r.t. to matrix columns).

The experimental dataset resembles the simulated dataset in that the observations and sample means are normally distributed, however the sample standard deviations w.r.t. columns do not approximate a normal distribution. In the experimental dataset these sample standard deviations represent the observed variance in the expression of individual proteins across biological samples. Yet, the sample standard deviations appear to be normal following a log2 transformation.

Below are analogous plots of from the experimental dataset showing histograms of the experimental observations, sample means (with respect to matrix columns), sample standard deviations (w.r.t. to matrix columns), and finally the sample standard deviations following log2 transformation.

My question is whether the log2 normal distribution of the sample standard deviations is likely a coincidental, idiosyncratic feature of this dataset arising from some biological explanation, or is this expected behavior of any random variable based on some explanation that I am not accounting for?

• One what basis are you asserting normal distributions of observations or means? I don't see anything here that looks normal Jun 22 '18 at 3:57
• You seem to be asking for a biological explanation of a dataset that is incompletely described. In its present form the question comes down to "I observe this apparently lognormal distribution of standard deviations in my data. Why is that?" It's unclear how we are expected to be able to help.
– whuber
Jun 22 '18 at 14:07
• I have revised the text to attempt to better articulate the question. @Glen_b, I am confused by your comment. Do these distributions not appear Gaussian to you? Normal QQ plots show that these distributions closely follow a normal distribution. I opted not to include them as not to clutter the question with figures. Jun 22 '18 at 16:30
• Let me revise my comment; the first two histograms don't look inconsistent with normality (though I bet they aren't truly normal). The next several are pretty clearly skewed. However, that may not be consequential Jun 23 '18 at 3:56

## 1 Answer

The distribution of the sample standard deviation looks more like the sample of the variance. Dumb question, are you plotting the variance? The variance of a normal distribution follows a chi-square distribution. The bottom of the page on this site attached explains sample variance distributions.

• Thank you for the suggestion, I went back and doubled checked, and it is definitely the SD. The distribution of the variances is similar, but more dramatically skewed. At first I also thought that these values could have been following a Chi-square distribution (I also reviewed the link you shared just to make sure), but the sample sizes here are large (25), so with 24 degrees of freedom the chi-square itself nearly approximates a normal distribution. Jun 22 '18 at 3:07