If I were to collect ONE sample of size 20 from a normal population, how justified would I be in claiming that the sample is normally distributed? I'm getting a little confused since by the Central Limit Theorem, the sample size has to be *sufficiently large* and 20 is a relatively small number. On the other hand, if we have assumed the population to be normal than surely we are right to assume any sample from this population is also normal, despite its size.

The population in question consists of various times for a particle to travel from A to B. I have plotted the 20 samples as a histogram in R and they do not seem strongly normal. However, there are some negative values in the sample and once these are omitted, the plot more closely resembles a normal distribution. Am I right to omit the negative values if it makes the plot more normal, considering the population is normally distributed?

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#### Best Answer

Given your response to my comment: You can calculate the mean and variance of *any* quantitative variable, or, indeed, any set of numbers.

You can then calculate confidence intervals for the mean and variance with a bootstrap. It might be interesting to do this and then compare the result to the normal.

Should you exclude negative numbers? No, probably not.

Although the numbers are impossible, physically, I am guessing that they are very close to 0 and that the rest of the data is also very close to 0 in a positive direction. After all, unless the distance between the two points is very big indeed, the times will necessarily be close to 0. Thus, the times that are negative are important because they are due to measurement error.

Further, since the speed of light is known to quite impressive precision, it seems what you are really testing here is the precision of your equipment.

If not, please elucidate.

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