Why do I get non-zero standard deviation in data with no variability?

Hello,

I have been calculating means and standard deviations for columns of data, and I am getting some weird results.

I have a 1x4 cell array called Transmats that has a 6x4 matrix in each cell (type double) that contains 4 datapoints (columns) for 6 subjects (rows). Each cell in Transmats has data from a different group.

For testing, I have just used the same input data for each row in Transmats.

So for now, Transmats{1} is the same as Transmats{2}, Transmats{3}, and Transmats{4} and it looks like this:

44.4128701072509	0.302502520854340	0.302502520854340	54.9821248510404
44.4128701072509	0.302502520854340	0.302502520854340	54.9821248510404
44.4128701072509	0.302502520854340	0.302502520854340	54.9821248510404
44.4128701072509	0.302502520854340	0.302502520854340	54.9821248510404
44.4128701072509	0.302502520854340	0.302502520854340	54.9821248510404
44.4128701072509	0.302502520854340	0.302502520854340	54.9821248510404

I then calculate the group averages and standard deviations for each of my groups like so:

% make group averages and SDs
for g=1:4 %time*treatment
        for t =1:4 %transition type
        Avmats{g,t}=mean(Transmats{g}(:,t)); 
        Sdmats{g,t}=std(Transmats{g}(:,t)); 
        end
end

Because all my values for each subject are the same, I should get a standard deviation of 0. But this is not what happens. I get the following standard deviations in Sdmats:

7.78360568894479e-15	0	0	7.78360568894479e-15
7.78360568894479e-15	0	0	7.78360568894479e-15
7.78360568894479e-15	0	0	7.78360568894479e-15
7.78360568894479e-15	0	0	7.78360568894479e-15

Why?

The only thing I can think of is that matlab rounds the numbers differently between loop iterations for some reason (that's the only way to get different numbers in Transmats in my script, I think). Is there a way to stop this from happening?

Even weirder, if I run my script on a different PC, I get a different result. Then standard deviations for column 1 and 4 of Sdmats are 0, and columns 2 and 3 have very small numbers in them.

I'm very confused and I'd very much like my standard deviation to be 0 when I have no variability in my data.

Answer 1

[Steven Lord]

So for now, Transmats{1} is the same as Transmats{2}, Transmats{3}, and Transmats{4} and it looks like this:

They may be displayed the same but it's likely the stored values are very slightly different.

x = [1, 1+eps]

x = 1×2

1.0000 1.0000

areTheyTheSame = x(1) == x(2) % false

areTheyTheSame = logical
   0

difference = x(2)-x(1)

difference = 2.2204e-16

s = std(x)

s = 2.2204e-16

The two elements of x are displayed the same, but they do not contain the same value. Therefore the standard deviation of x is non-zero.

Answer 2

[the cyclist]

Your standard deviaions are zero -- to within the accuracy possible by calculation in floating point numbers.

The first reference at the bottom of that documentation page is a fairly accesssible introduction to the idea.

Answer 3

[Susan Leemburg]

Thank you for the replies, they are super helpful!

Luckily, these very small variations are not a problem for my data, but I'll definitely keep it in mind if I ever end up looking at very small values or very small differences.