For example, in line #14, and many others, you create an array w/ ZERO element storage -> AX = new float[0];. So you can't put anything there!

Perhaps the confusion is derived that the index starts from zero, and goes up # of elements MINUS 1.
That is, if you had AX = new float[10]; then you'd have an index w/ range between 0 & 9.

If you're not sure how many elements you're gonna need, just create an array w/ 100, 1000 or even more element storage.

Even better, know how to use the powerful & dynamic ArrayList<> type!

AFAIK, indices themselves must be integer & positive.
And the content itself must be all of the same data type they were declared.

Just observed you're creating a local variable inside a for loop @ line #32 -> String[] pieces = split(lines[k], ',');

Then you just assume that split() of lines[k] generates indices between 1-6:

            AX[0] = (float(pieces[1]));
            AY[0] = (float(pieces[2]));
            AZ[0] = (float(pieces[3]));
            GX[0] = (float(pieces[4]));
            GY[0] = (float(pieces[5]));
            GZ[0] = (float(pieces[6]));

I believe you should use the iterator variable i for pieces[] instead of plain numbers.

Problem is I don't have that -> "imu-sensor-data-mconduct-new.csv" file, and dunno how to manipulate .csv format either. 
Since I can't run your program properly, most I can do is to just try to spot the obvious errors.

AFAIK = As Far As I Know

Since I can't run your code, I can only try to grasp its logic from the source text.

After:
for (int k = 0; k < lines.length; k++) {
    String[] pieces = split(lines[k], ',');

You create another loop inside that:
for (int i = 0; i < pieces.length; i++) {

Let's suppose that pieces.length = 7. It means pieces[] index has a range of 0 - 6.

Next you try to assign from AX[0] = ( float(pieces[1]) );
up to GZ[0] = ( float(pieces[6]) );

After that you make sums from sumAX += AX[i];
up to sumGZ += GZ[i];

There is a gross discrepancy there, b/c 1st you always get values from index 0, which in turn got its value from the very same pieces[1 to 6]!
They're all the same fixed values, nothing to do w/ the for iterator variable i!

Next, when i = 1:
sumAX += AX[i]; -> sumAX += AX[1];
But you have only assigned AX[0]! Where are AX[1 - 6]? And all of the others?

Maybe you don't really need that extra for i loop, just the for k one! 

Well, that's all I could get so far!

Right, progress has been made! Averages of values have been calculated, and ellipses have been plotted. The updated code is as follows:

Hello again Oskar!

Since I don't have access to your huge CSV file, I had to adapt your code to accept any improvised made-up one!
That is, any # of lines w/ any # of fields!!!
The 1st split line determines the # of fields for all of the lines of the whole CSV file.
And both the 1st line & each 1st field of each line are ignored for total sum & plotting.

But unfortunately, I can't figure it out the logic behind your graph-plot algorithm. 
You have to be more visually specific about what you expect from your data display!

Anyways, check it out your modded code:

/**
 * CSV (v1.1)
 * by Oskar.R.Winter (2012/Nov)
 * tweaked by GoToLoop
 * 
 * http://forum.processing.org/topic/error-reading-arrayindexoutofboundsexception-0
 */

private static final int    sc = 80, sz = 4;   // scalar & size
private static final color  bg = 0200, fg = 0; // back&fore-ground colors

private static final String file = "imu-sensor-data-mconduct-new.csv";

private int linesNum, fieldsNum;

private String[]  lines;  // stores all lines of CSV file
private float[][] csv;    // stores all fields from the lines of CSV file

final void setup() {

    size(1100, 750);
    smooth();
    noLoop();
    colorMode(RGB);

    background(bg);
    fill(fg);
    strokeWeight(sz);

    LoadCSV_File();
    CreateCSV_Array();

    lines = null;   // marks lines[] to be dealt with by Java's Garbage Collector

    PlotCSV_Array();

    println("\n1st csv:");
    println( csv[1] );
    println("\nLast csv:");
    println( csv[linesNum - 1] );
}

private final void LoadCSV_File() {

    lines = loadStrings(file);   // loads CSV file split up by lines
    linesNum  = lines.length;

    fieldsNum = split(lines[0], ',').length;  // its 1st line determines how many CSV's fields
}

private final void CreateCSV_Array() {

    csv = new float[linesNum][fieldsNum];

    for (short row=1; row<linesNum; ++row)
        csv[row] = float( split(lines[row], ',') ); // each row/line is split by columns/fields
}

private final void PlotCSV_Array() {

    // temporary array to store all fields from a line# of csv[][]:
    float[] fields;

    // temp. array to store total individual sum of each separate field element from fields[]:
    final float[] sum = new float[fieldsNum];

    translate(0, height>>1);

    for (short row=1; row<linesNum; ++row) {
        fields = csv[row];  // gets all fields from a line#

        for (short col=1; col<fieldsNum; ++col) {
            sum[col] += fields[col];  //  accumulates the sum of each field from all lines
            point( 150*col, sc*sum[col] / linesNum );
        }
    }

    println("Sum:");
    println(sum);
}

Problem solved! Confusion with for loops causing the main problem, here is the code I have to date: