Normal distribution array function [on hold]

As the conditions of the if statements only depending on some fixed numbers and the arraysize input parameter these values can be precalculated.

int oneThirdArraySize = (int)((1.00 / 3.00) * arraySize);

and then used in the loop like

for (var i = 0; i < arraySize; i++)
{

    if (i <= oneThirdArraySize)
    {
        startNumber = startNumber + 1;
        arr[i] = startNumber;
    }
    // and so on
}

By precalculation of the right side values of the if conditions outside of the loop you can speed this up because right now you do these calculation up to two times for every iteration.

But we can do better because you are using some magic numbers here which we can hide behind some meaningful const variables.

private static const double oneThird = 1d / 3d;
private static const double twoThird = 2d / 3d;  

In this way the calculations need to be changed to

int oneThirdArraySize = (int)(oneThird * arraySize);

I don't understand the logic or how it generates a Gaussian distribution.

Assuming the logic itself is correct, you could simplify the code.

Key points:

• assignment statement identical in all if branches, so pull it out of the conditional logic (DRY principle)
• No reason to do floating-point arithmetic - array indexes have to be integers so it's not clear what you're accomplishing
• use if/else to avoid repeating the inverse of the first if branch (DRY principle again)
• since there are only 3 mutually exclusive possibilities of the array index (first third, second third, final third) you don't even need to specify the final condition explicitly.

Improved version:

private static IComparable[] NonUniformDistributionsGaussian(int startNumber, int arraySize)
{
    IComparable [] arr = new IComparable[arraySize];
    for (var i = 0; i < arraySize; i++)
    {

        if (i <= arraySize / 3)
        {
            startNumber = startNumber + 1;
        }
        else if (i <= 2*arraySize/3)
        {
             startNumber = startNumber  + 2; 
        }
        else
        {
             startNumber = startNumber  - 2; 
        }
        arr[i] = startNumber;
    }
    return arr;
}