# How to evaluate if a list obeys the Normal distribution?

I am processing a data when I encountered this problem. I use the way of plotting the "normal probability plot" to show the result

The list I have is:

t1={0.0330584, 0.0288738, 0.0303384, 0.0330584, 0.034523, 0.0311753,
0.029083, 0.0280369, 0.0332676, 0.0334768, 0.0338953, 0.0311753,
0.0305476, 0.0332676, 0.0269907, 0.0332676, 0.0278276, 0.0261538,
0.0286645, 0.0311753}


I want to use the following code to get the "Normal score" to shown if t1 is obeying normal distribution.

edist = EstimatedDistribution[t1, NormalDistribution[\[Mu],\[Sigma]]]

ListPlot[Transpose[{t1,
InverseCDF[edist, (Ordering[t1] + 1.0/6)/(
Length@t1 + 1/3)]}], Joined -> False, PlotRange -> All,
FrameLabel -> (Style[#, 22] & /@ {"Normal Score", "Data"})]


But the function is not giving me the correct result. So in Excel, the function RANK() can give the ordering of a certain value in a list. If there are two elements with same value, they will have same ordering after RANK() is applied. But in Mathematica, the function Ordering[] will give different value for same valued element. Is there another function can do this task?

What I used to do with Excel

When I use Excel to process it, I use two functions in Excel: RANK() and COUNT() the E column is obtained by

       (RANK(C1,$C$1:$C$20)+1/6)/(COUNT(C1,$C$1:$C$20)+1/3)


Then, I use =NORMSINV(E1) to get the final value of F column

After that, I can plot the C column and F column to shown the value of C column is obeying Normal distribution(whether if they are forming a line). • The problem is I don't know if there is functions in Mathematica can reproduce the results of the function RANK() and NORMSINV() in Excel. Dec 17 '15 at 1:40
• Also, is there better ways to shown if my data obeys Normal Distribution? Dec 17 '15 at 1:41

I think you are looking for a QuantilePlot

QuantilePlot[t1] There are others as well. Have a look at the Distribution Fits section Statistical Visualisation guide.

Hope this helps.

Use DistributionFitTest to test whether data is normally distributed. A small p-value suggests that it is unlikely that the data is normally distributed.

t1 = {0.0330584, 0.0288738, 0.0303384, 0.0330584, 0.034523, 0.0311753,
0.029083, 0.0280369, 0.0332676, 0.0334768, 0.0338953, 0.0311753, 0.0305476,
0.0332676, 0.0269907, 0.0332676, 0.0278276, 0.0261538, 0.0286645,
0.0311753};

dft = DistributionFitTest[t1, Automatic, "HypothesisTestData"];

dft["TestDataTable", All] edist = EstimatedDistribution[t1, NormalDistribution[μ, σ]];

{μ, σ} = List @@ edist;

Show[
Histogram[t1, Automatic, "PDF"],
SmoothHistogram[t1, PlotStyle -> Thick],
Plot[PDF[edist, x], {x, μ - 3 σ, μ + 3 σ},
PlotStyle -> Red]] To get a quick visual peek at the distribution of the data, we can do:

GraphicsRow[Table[
Show[
Histogram[Sort[(t1 - Mean@t1)/StandardDeviation@t1], k, "Probability"]
, Plot[PDF[NormalDistribution[0, 1], x], {x, -5, 5}]
]
, {k, 5, 15, 5]


resulting in It doesn't look all that normal, but the dataset is small, so of course it's hard to tell. Probably a real statistics test as mentioned in this answer to the question is more useful.