# Can I use my high number of data points to check for normal distribution, if my sample size is too small?

I'm trying to find out if my signals are normally distributed. However, I only have a sample size of 10. This is not good enough to get a clear answer, so I was thinking of a different solution: what if I calculated the standard deviation for each signal (from different geological samples and at different positions in the spectra) and made a histogram where my bins are fractions of the standard deviation - and then made a sum of all these individual histograms?

My assumption is that all of my signals are normally distributed, and since they all come from the same type of measurement, I believe that this is feasible. (After all, if I had a large sample size and could see the normal distribution for each signal on its own already, I would still see it in the sum of these histograms.)

So that's what I did. I used 5928 signals in total, with 10 measurements for each signal. Here's the histogram (blue bars) along with a normal distribution of equal area (red line): As you can see, it seems to work, and the result does look similar to a normal distribution. What confuses me, though, is that it doesn't look more like a normal distribution - especially the tails seem to drop off really quickly, and there's this huge dip right in the center. (Also, it seems a little bit like it's leaning to the left.) As I said, I used almost 6000 data points for this, so if the idea was correct, I'd expect a much clearer normal distribution at this point.

So here are my questions:

• Do you see any problems with this idea in general?
• Do I introduce too much of an error by estimating the standard deviations from a sample size of 10? (The sample size can't be easily increased, unfortunately.)
• Is there maybe a better solution that I'm not aware of?

I have looked at the usual tests for normal distribution, but it seemed to me that they are designed to be used with measurements that all describe the same value (of which I only have 10).

(Please note that I'm a physicist with little experience in statistics. I'm perfectly aware that there are tons of pitfalls for beginners, but I usually can't tell if I'm walking right into them.)

## 1 Answer

Your question is difficult to understand, as you explain very little about your data. Obviously, you have about 6000 "signals" consisting of 10 measurements in all. If your assumption is correct, that all these measurements come from the same distribution, then yes, of course you can use all of them to check for normality. Your approach is as valid as your assumption of all these values stemming from one identical distribution is.

What you do not say is, why you are checking for normality. These appear to be real life measurements and those are never completely normal in a mathematical sense. The question is usually, whether they are "normal enough". Normality is a mathematical concept which includes values that are infinitely large or small and have perfect precision, none of that is true for real measurements.

So the next step in your thoughts should be: Why am I looking for normal distribution and what would I consider "normal enough". If this is about normal distribution as a requirement for statistical tests: at n=6000 this is rarely an issue. If you want to prove, that this is not normally distributed in order to check an underlying theory, than looking at histograms may be to crude an approach.

• I study compositions of different materials via spectroscopy. I measure at 10 positions for each material. Then I look at 8 different peaks in the spectra, these are my "signals". I do this for 86 different compositions. I hope this clears things up. – PoorYorick Oct 26 '16 at 16:18
• I do not see, how this adds up to 5928 nor do I have any knowledge of spectroscopy nor your compositions. Whether all peaks are identically distributed will depend on the compositions you analyse. Why do you care for the distribution of all those values thrown together? – Bernhard Oct 26 '16 at 16:51
• You do not need knowledge of spectroscopy to answer my question. The assumption that spectroscopists make is that these peaks have a normal distribution - and that's the hypothesis I'm working with. I care about the distribution of all those values thrown together, because to my understanding, it would show me whether this hypothesis is true. My question is completely grounded in statistics, not in physics. – PoorYorick Oct 27 '16 at 7:45
• Then my last sentence applies and you can do better than just looking at the histogram. Consider applying Lilliefors' test or Shapiro-Wilk test. – Bernhard Oct 27 '16 at 13:16
• As to the early dropping of the tails, there is a measure called kurtosis to quantify that. en.wikipedia.org/wiki/Kurtosis – Bernhard Oct 27 '16 at 13:23