Why does the ecdf function return duplicate x values?

When calling the ecdf function (empirical cumulative distribution function), why is the first "x" value always duplicated? This means that the first 'F' output is always 0. For example:

 Trial>> y = 0:10;
 Trial>> [f,x] = ecdf( y );
 Trial>> disp(x')
      0     0     1     2     3     4     5     6     7     8     9    10
 Trial>> y = 5:15;
 Trial>> [f,x] = ecdf( y );
 Trial>> disp(x')
      5     5     6     7     8     9    10    11    12    13    14    15

In other words, the minimum value in the input array always appears twice, but the corresponding F value in the first instance is always 0. Since the ECDF is given by:

...is this correct?

Thanks!

Answer 1

[Tom Lane]

This is just for convenience in plotting. Compare these:

>> y = rand(10,1);
>> [f,x] = ecdf(y);
>> subplot(2,1,1); stairs(x,f)
>> subplot(2,1,2); stairs(x(2:end),f(2:end))

Your formula is correct. If you need only the values after each jump in the cdf, you can discard the first value from both x and f.

[Alex Knight]

Tom, many thanks - that makes a lot of sense!

Answer 2

[Huiru Li]

y = rand(10,1);

>> [f,x] = ecdf(y);

>> subplot(2,1,1); stairs(x,f)

>> subplot(2,1,2); stairs(x(2:end),f(2:end))