Testing how much random is my sample

[fluidistic]

Hello guys,
I have a sample of about 400 natural numbers though I can get more numbers. To give you an idea the mean and the standard deviation are 29038031 and 1842882 respectively and I expect the numbers to follow a Gaussian distribution. I'd like to perform a test to tell me the probability that my sample is truly random. I just don't know which test to perform. I've read about diehard tests but I don't see how I could apply them.

So I'd like to hear some suggestions. Thanks!

Edit: 1st idea that I have: get more numbers. Then perform a Gaussian fit and calculate the residuals. Do the same for true random numbers following a Gaussian with the same mean and standard deviation and compare the residuals. I expect lower residuals with the true random numbers.

Edit2: Nevermind this idea would be useless. It would tell me how far from a Gaussian my distribution of numbers is, not how random they are...

 

[marcusl]

A simple and intuitive way is to plot your histogrammed data on Gaussian "probability paper," so named from the days when plots were made on actual graph paper. You'll see visually how close to a Gaussian you are.
http://en.wikipedia.org/wiki/Normal_probability_plot

Other simple, but quantitative, tests include:
a) Compute the skew and kurtosis and see how close to Gaussian they are (normal values are 0 and 3).
b) The mean, mode and median are all equal for a normal distribution.

More sophisticated tests abound. See, e.g.,
http://en.wikipedia.org/wiki/Normality_test

 

[fluidistic]

A simple and intuitive way is to plot your histogrammed data on Gaussian "probability paper," so named from the days when plots were made on actual graph paper. You'll see visually how close to a Gaussian you are.
http://en.wikipedia.org/wiki/Normal_probability_plot

Other simple, but quantitative, tests include:
a) Compute the skew and kurtosis and see how close to Gaussian they are (normal values are 0 and 3).
b) The mean, mode and median are all equal for a normal distribution.

More sophisticated tests abound. See, e.g.,
http://en.wikipedia.org/wiki/Normality_test

I've just used the software maxima which performed a Shapiro-Wilk test to check whether my data follows a Gaussian and I think there are high chances that it does: it returned a Kendall's W of over 0.99 with a p-value near 0.27.
The thing is that I am not sure that this is telling me anything about the randomness of my numbers which is what I'm looking for.

 

[WWGD]

I am not sure I understood; do you know that the original distribution is normal and then you want to know if the sample is random? Have you tried the runs test?

https://home.ubalt.edu/ntsbarsh/business-stat/opre504.htm#rrunstest

And a good thing is that the test is non-parametric.

 

[fluidistic]

I am not sure I understood; do you know that the original distribution is normal and then you want to know if the sample is random? Have you tried the runs test?

https://home.ubalt.edu/ntsbarsh/business-stat/opre504.htm#rrunstest

And a good thing is that the test is non-parametric.

Thanks a lot! That's exactly what I was looking for, I'll try tomorrow.
Meanwhile I tried a very similar test with R programming and the result was that there is a high probability that my data is random. (p-value was over 0.4 and the null hypothesis is that the data is random while the alternative hypothesis was non randomness in the data).

 

[WWGD]

Glad I could help.