I have two samples X
($N$= 97) and X2
($N$=4782) drawn from the same population data. I like to test (using statistical visualizations such as normplot
and qqplot
and hypothesis tests such as jbtest
, chi2gof
and kstest
in matlab) if the data from each sample is normally distributed.
My first data is:
X = [8.13010235400000,13.6713071300000,14.0362434700000,18.4349488200000,26.5650511800000,30.9637565300000,34.3803447200000,40.6012946500000,45,49.3987053500000,58.6713071300000,59.0362434700000,59.0362434700000,59.0362434700000,61.9275130600000,61.9275130600000,63.4349488200000,63.4349488200000,63.4349488200000,63.4349488200000,63.4349488200000,64.4400348300000,71.5650511800000,71.5650511800000,71.5650511800000,71.5650511800000,75.9637565300000,75.9637565300000,75.9637565300000,75.9637565300000,75.9637565300000,75.9637565300000,75.9637565300000,75.9637565300000,75.9637565300000,75.9637565300000,77.4711922900000,77.4711922900000,77.4711922900000,77.4711922900000,77.4711922900000,77.4711922900000,77.4711922900000,77.4711922900000,77.4711922900000,78.6900675300000,90,90,90,90,90,90,90,90,90,90,90,90,90,90,90,93.1798301200000,97.1250163500000,97.7651660200000,102.528807700000,102.528807700000,102.528807700000,102.528807700000,102.528807700000,104.036243500000,104.036243500000,104.036243500000,104.036243500000,104.036243500000,104.036243500000,104.036243500000,105.255118700000,108.434948800000,108.434948800000,108.434948800000,108.434948800000,109.440034800000,116.565051200000,118.072486900000,120.963756500000,127.746805400000,130.601294600000,135,137.489552900000,139.398705400000,139.398705400000,149.036243500000,153.434948800000,159.227745300000,161.565051200000,179.999998800000,180];
The analyses using statistical visualizations in matlab show that the underlying distributions for both samples are normal. However, from the hypothesis tests, the null hypothesis for the first sample is not rejected using the same significance value (except for the chi-square test), but that for the second sample, X2 is completely rejected.
I am now confused as to how to prove my samples are normally distributed and as well come from the same population data. What can I do in this situation?
PS: sample X2
is too large for me to post, but if there is any suggestion on how I could show this, then I don’t mind.
EDIT: I have just collated another set of sample (N = 4700) from the same population data wherein the qqplots and cdf comparisons all look good (see new added image). Strangely, the hypothesis tests with jbtest and kstest in Matlab both rejects the null hypothesis. I am now beginning to believe that these hypothesis tests may not be trusted afterall, particularly for real case data.
PS: I couldn't try the Shapiro-Wilks test as Matlab do not have this.